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Combinatorics - Science topic

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size (enumerative combinatorics), deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization), and studying combinatorial structures arising in an algebraic context, or applying algebraic techniques to combinatorial problems (algebraic combinatorics).
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Dear Researchers,
Do researchers/universities value students/researchers having published sequences to the OEIS?
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Dear Marco Ripà ,
I have done both. I cited my work in the sequences and the sequences in my work.
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Let G be cubic plane graph such that every face boundary of G is of length divisible by four. By an indirect proof, it follows that the number of vertices in G must be divisible by eight.
Can you prove this by an elementary method? (which should be possible I think, though I don't have such a proof).
Disclaimer: Cross-posted in stackexchange (https://cstheory.stackexchange.com/q/50959/47855).
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This is NOT an answer to the question; rather a comment on the answer by Peter Breuer .
Yes, a cubic graph is graph such that every vertex has degree three. And, by "face boundary" I mean the edges that form the boundary of a face, and "length" means the number of edges in that boundary.
Thank you for your effort. But, the argument "he number of edges must be some multiple of 4 times the number of faces" seems wrong to me. For instance, a graph can have exactly 3 faces with boundary lengths 4,8 and 8 respectively, yet 4+8+8 is not a multiple of F=3.
Remark: The above one is not an actual example though; a graph with 10 faces and face boundaries 8,8,4,4,...,4 is a proper counterexample to the argument.
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Is the reciprocal of the inverse tangent $\frac{1}{\arctan x}$ a (logarithmically) completely monotonic function on the right-half line?
If $\frac{1}{\arctan x}$ is a (logarithmically) completely monotonic function on $(0,\infty)$, can one give an explicit expression of the measure $\mu(t)$ in the integral representation in the Bernstein--Widder theorem for $f(x)=\frac{1}{\arctan x}$?
These questions have been stated in details at the website https://math.stackexchange.com/questions/4247090
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It seems that a correct proof for this question has been announced at arxiv.org/abs/2112.09960v1.
Qi’s conjecture on logarithmically complete monotonicity of the reciprocal of the inverse tangent function
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Dear Colleagues
I need an inequality for the ratio of two Bernoulli numbers, see attached picture. Could you please help me to find it? Thank you very much.
Best regards
Feng Qi (F. Qi)
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Dear Collegues,
Could you please recommend conferences in 2019-2021 having Finite Group theory and Combinatorics in their topics?
I will be very thankful for any references!
Thanks in advance!
Best wishes,
Natalia
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In addition to the excellent suggestions of Mohamed Amine Bahayou I would recommend this conference list: faculty.math.illinois.edu/~west/meetlist.html
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Do you know the web sites of the journals Ars Combinatoria and Utilitas Mathematica? Are these two journals ceased? One of my manuscripts was accepted a long time ago for publishing in the first journal, but now I cannot contact any editor, I cannot find its website, and I cannot get any message of these two journals. So I wonder if these two journals have been ceased.
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Dear Dr.,
The information above is provided for possible search. That is, the corresponding transitions to other sites needed, that are possible from the links provided by me.
Unfortunately, I don't have anything more yet.
Sincerely,
Sergey Klykov
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I need some suggestions what are the growing topics in algebraic combinatorics and graph theory for research? Thank you in advance to everyone who will answer.
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(enhanced) power graphs of groups
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I have written an analysis article but cannot seem to find an appropriate journal.
Could you recommend average level journals as well as low level journals in analysis?
Thank you for participating in this discussion.
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The journal “American Journal of Mathematical Analysis” http://www.sciepub.com/journal/ajmais published by Science and Education Publishing (SciEP) a publisher included in the Beall’s list of potentially predatory publishers: This is a red flag. There are more, for example: https://beallslist.net/
-Misleading contact address suggesting US origin while they are not
Better avoid.
Roudy El Haddad indeed the journal “Moroccan Journal of Pure and Applied Mathematics” 2605-6364 https://sciendo.com/journal/MJPAA is having a pretty solid publisher behind it and received a Scopus indexing https://www.scopus.com/sourceid/21101026926, so most likely a safe choice.
The journal “International Journal of Nonlinear Analysis and Applications” https://ijnaa.semnan.ac.ir/ seems a pretty safe choice as well. Scopus indexed https://www.scopus.com/sourceid/21100873480 and indexed in ESCI.
Best regards.
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I have confirmed that the Hessenberg determinant whose elements are the Bernoulli numbers $B_{2r}$ is negative. See the picture uploaded here. My question is: What is the accurate value of the Hessenberg determinant in the equation (10) in the picture? Can one find a simple formula for the Hessenberg determinant in the equation (10) in the picture? Perhaps it is easy for you, but right now it is difficult for me.
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In the 19 hundreds, many though that an explicit formula for the partition function was never going to be found. In 2011, finally, an explicit formula for the partition function was discovered.
For that reason, I am fascinated by how close do mathematicians think we are currently to discovering an explicit formula for prime numbers.
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Dear one of the way is using "Using the Stirling approximation"
etc
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How many more years do you predict it will take before the Riemann Hypothesis is solved?
Do you think we are close or does it seem that we are still very far?
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I agree with you. Although year after year we are getting closer. A few year ago, it was proven that at least 40% of zeros have to be on the critical line. So a lot of progress has been done. Many conjecture have also been presented, which if proven, they would imply the Riemann Hypothesis. So have gotten some important result, however, we have not quite solved it yet.
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I find that multiple sums are a very interesting concept. However, an even more interesting concept for me is the sum of such sums (the sum of multiple sums). The sum of multiple sums can be turned into a simple product by the formula in the first attached image.
What mathematical applications could this formula have? Where could it be useful?
I have found 2 interesting applications:
1- sum of multiple zeta values (see image 2).
2- sum of multiple power sums (see image 3).
Could you suggest any applications for these 2 particular cases? Could you suggest additional particular cases that would be of interest to mathematicians or physicists?
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Why are you asking?
Please respond when you read this.
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I have written two articles about a generalization of Multiple zeta values and Multiple zeta star values. I also presented applications for this generalization including partition identities, polynomial identities, a generalization of the Faulhaber formula, as well as MZV identities. If you are intrested check them out on my profile and give me your opinion.
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If anyone read my articles and has recommendations or comment, please inform me of these comments. Any comments or suggestions would be very helpful.
Also if anyone knows someone that would be interested in my articles please do recommend it to them.
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I am considering to distribute N-kinds of different parts among M-different countries and I wan to know the "most probable" pattern of distribution. My question is in fact ambiguous, because I am not very sure how I can distinguish types or patterns.
Let me give an example. If I were to distribute 3 kinds of parts to 3 countries, the set of all distribution is given by a set
{aaa, aab, aac, aba, abb, abc aca, acb, acc, baa, bab, bac, bba, bbb, bbc, bca, bcb, bcc, caa, cab, cac, cba, cbb, cbc, cca, ccb, ccc}.
The number of elements is of course 33 = 27. I may distinguish three types of patterns:
(1) One country receives all parts:
aaa, bbb, ccc 3 cases
(2) One country receives 2 parts and another country receives 1 part:
aab, aac, aba, abb, aca, acc, baa, bab, bba, bbc, bcb, caa, cac, cbb, cbc, cca, ccb 17 cases
(3) Each county rceives one part respectively:
abc, acb, bac, bca, cab, cba 6 cases
These types may correspond to a partition of integer 3 with the condition that (a) number of summands must not exceed 3 (in general M). In fact, 3 have three partitions:
3, 2+1, 1+1+1
In the above case of 3×3, the number of types was the number of partitions of 3 (which is often noted p(n)). But I have to consider the case when M is smaller than N.
If I am right, the number of "different types" of distributions is the number of partitions of N with the number of summands less than M+1. Let us denote it as
p*(N, M) = p( N | the number of summands must not exceed M. )
N.B. * is added in order to avoid confusion with p(N, M), wwhich is the number of partitions with summands smaller than M+1.
Now, my question is the following:
Which type (a partition among p*(N, M)) has the greatest number of distributions?
Are there any results already known? If so, would you kindly teach me a paper or a book that explains the results and how to approach to the question?
A typical case that I want to know is N = 100, M = 10. In this simple case, is it most probable that each country receives 10 parts? But, I am also interested to cases when M and N are small, for example when M and N is less than 10.
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Thank, Luis Daniel Torres Gonzalez , you for your contribution. My question does not ask the probability distribution. It asks what is the most probable "pattern" when we distribute N-items among M-boxes. I have illustrated the meaning of "pattern" by examples, but it seems it was not sufficient. Please read Romeo Meštrović 's comments above posted in March, 2019.
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I am a researcher in combinatorics and Semigroup in Ahmadu Bello University, Zaria seeking to know the binding line between both.
Thank you.
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Thank you for your answers and recommendations.
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The following formally published papers are related to this question:
[1] Feng Qi, Diagonal recurrence relations for the Stirling numbers of the first kind, Contributions to Discrete Mathematics 11 (2016), no. 1, 22--30; available online at https://doi.org/10.11575/cdm.v11i1.62389
[2] Feng Qi and Jiao-Lian Zhao, Some properties of the Bernoulli numbers of the second kind and their generating function, Bulletin of the Korean Mathematical Society 55 (2018), no. 6, 1909--1920; available online at https://doi.org/10.4134/bkms.b180039
[3] Feng Qi and Bai-Ni Guo, A diagonal recurrence relation for the Stirling numbers of the first kind, Applicable Analysis and Discrete Mathematics 12 (2018), no. 1, 153--165; available online at https://doi.org/10.2298/AADM170405004Q
[4] Feng Qi and Bai-Ni Guo, Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials, Mediterranean Journal of Mathematics 14 (2017), no. 3, Article 140, 14 pages; available online at https://doi.org/10.1007/s00009-017-0939-1
[5] Feng Qi, Da-Wei Niu, Dongkyu Lim, and Yong-Hong Yao, Special values of the Bell polynomials of the second kind for some sequences and functions, Journal of Mathematical Analysis and Applications 491 (2020), no. 2, Paper No. 124382, 31 pages; available online at https://doi.org/10.1016/j.jmaa.2020.124382
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Where to find the answer?
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The following formally published papers are related to this question:
[1] Feng Qi, Diagonal recurrence relations for the Stirling numbers of the first kind, Contributions to Discrete Mathematics 11 (2016), no. 1, 22--30; available online at https://doi.org/10.11575/cdm.v11i1.62389
[2] Feng Qi and Jiao-Lian Zhao, Some properties of the Bernoulli numbers of the second kind and their generating function, Bulletin of the Korean Mathematical Society 55 (2018), no. 6, 1909--1920; available online at https://doi.org/10.4134/bkms.b180039
[3] Feng Qi and Bai-Ni Guo, A diagonal recurrence relation for the Stirling numbers of the first kind, Applicable Analysis and Discrete Mathematics 12 (2018), no. 1, 153--165; available online at https://doi.org/10.2298/AADM170405004Q
[4] Feng Qi and Bai-Ni Guo, Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials, Mediterranean Journal of Mathematics 14 (2017), no. 3, Article 140, 14 pages; available online at https://doi.org/10.1007/s00009-017-0939-1
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Bell polynomials of the second kind Bn,k(x1,x2,...,xn-k+1) are also called the partial Bell polynomials, where n and k are positive integers. It is known that Bn,k(1,1,...,1) equals Stirling numbers of the second kind S(n,k).
What are the values of the special Bell polynomials of the second kind Bn,k(0,1,0,1,0,1,0,...) and Bn,k(1,0,1,0,1,0,...)? Where can I find answers to Bn,k(0,1,0,1,0,1,0,...) and Bn,k(1,0,1,0,1,0,...)? Do they exist somewhere?
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The following formally published papers are related to this question:
[1] Feng Qi and Bai-Ni Guo, Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials, Mediterranean Journal of Mathematics 14 (2017), no. 3, Article 140, 14 pages; available online at https://doi.org/10.1007/s00009-017-0939-1
[2] Feng Qi, Da-Wei Niu, Dongkyu Lim, and Yong-Hong Yao, Special values of the Bell polynomials of the second kind for some sequences and functions, Journal of Mathematical Analysis and Applications 491 (2020), no. 2, Paper No. 124382, 31 pages; available online at https://doi.org/10.1016/j.jmaa.2020.124382
[3] Feng Qi, Da-Wei Niu, Dongkyu Lim, and Bai-Ni Guo, Closed formulas and identities for the Bell polynomials and falling factorials, Contributions to Discrete Mathematics 15 (2020), no. 1, 163--174; available online at https://doi.org/10.11575/cdm.v15i1.68111
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Dear Researchers,
Kindly let me know about 5 top most research problems in number theory which are concerning to
Double series and / or combinatorics
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How to prove or where to find a proof of the lower Hessenberg determinant showed by two pictures uploaded here?
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Dear All Colleagues
The final version has been accepted for publication in the Mathematica Slovaca. For details, please click at the website:
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It is well known that for every nonnegative n, there exist Steiner triple systems STS(6n+1) and STS(6n+3). I see similarities between this statement and the one that I included in my question, but perhaps you can shed light on this issue. Thanks.
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According to a theorem of Doyen and Wilson, the answer to this question is "yes".
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From a combinatorics perspective, what if what you are requiring of the optimization algorithm is too scattered?
In other words, from a combinatorics perspective, what if what you are requiring in terms of the optimization problem specification, also referring to the constraints, including equality constraints, implies too scattered and sparse coefficient (variable cell) ranges.
Is this possible? The algorithm may then struggle to locate the feasible regions or ranges.
I am particularly referring to a nonconvex nonlinear optimization problem.
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Do you have bounded, e.g. binary, variables or unbounded ones. I the case of unbounded variables, the problem may not have an algorithm.
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I have three tasks for 3 participants - task 1 (EO), task 2 (BCST), task 3 (RMET). I use six measures - M1 (CD), M2 (LLE), M3 (HFD), M4 (MSE), M5 (MFDFA), M6 (Kc). I have 20 EEG channels. I use Wilcoxon signed rank to compare each channel pair between two states. I get p-values for 3 H: tests -1H0: EO=BCST, 2HO: BCST=RMET, 3H0: EO=BCST. The other three are reciprocal adding to 1. So comparing p-values from the tests I get 12 possible outcomes - 1: EO=BCST>RMET, 2: EO=BCST<RMET, 3: BCST=RMET>EO, 4: BCST=RMET<EO, 5: EO=BCST>RMET, 6: EO=BCST<RMET, 7: EO>BCST>RMET, 8: BCST>RMET>EO, 9: RMET>EO>BCST, 10: EO>RMET>BCST, 11: RMET>BCST>EO 12: BCST>EO>RMET.
Question: Given a confidence level of 0.05 is the probability of getting all 3 participants to have the same order based on p-values below the 0.05 CL threshold thus excluding any two tasks being equal based on the following: 0.05 x 0.05 x 0.05 x 6 x 6 = 0.0045?
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I have N lattice points which are arranged linearly and equally spaced. I want to make connections(say with some wire or thread) with each lattice site with another. The first one has N-1 possibilities and the second one has N-3 as each one cannot make connections itself and the first one has already formed one. So the total possibility is (N-1)(N-3)(N-5)(N-7)..... and so on. Now I want to impose two conditions.
Case(i): If I impose a condition that each site cannot be connected with the nearest neighbor, how many ways I can make the connections. How do complete this counting problem with this condition?
Case(ii): Apart from the above condition(immediate neighbors should not be connected), If I impose a further condition that each site can be connected with other or it can also be left unconnected. How do I count the number of ways doing this?
I know both case(i) and case(ii) will have different answers. I really don't know where to start this problem at all
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Is your lattice a line with N vertices?
And you want to connect them by forwarding paths? To make your question is clear. Assume that your lattice includes 4 vertices.
Sketch some graph, then attach it as a pdf file. Then we may help.
Best regards
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I am working on the  construction of Barnette graphs for given diameter. I would like to know the reason why many cubic 3 connected planar , (not  a bipartite)  are  both non-Hamiltonian and Hamiltonian graphs. I found a unique property of those Hamiltonian graphs. I need the latest results related to my question.
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If any one property of Barnatte graph is dropped it is non hamiltonion
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for k=1, 2, 3 we get 7, 13, 19 as primes.
for k=5,  6, 7 we get  31, 37,  43,  as primes.
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Such arithmetic progression does not exist. Namely, suppose that k_n=a+(n-1)d is a desired arithmetic progression, where a is a fixed integer, d is a fixed positive integer, such that 6k_n+1 is a prime number for all n=1,2,3,… . This means that 6k_n+1=6(a+(n-1)d)+1=(6a+1)+(n-1)(6d) would be an arithmetic progression containing only prime numbers. However, this is obviously impossible by the well known Prime Number Theorem for Arithmetic Progressions.
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Do you think that the iThenticate/CrossCheck/Similarity Index would cause heavy and serious confusion in mathematics? Even destroy, ruin, damage Mathematics? Our mathematics and mathematicians should follow and inherite symbols, phrases, terminology, notions, notations in previous papers, but now we have to change these to avoid, to escape, to hide, to decrease the iThenticate/CrossCheck/Similarity Index! It’s very ridiculous for mathematics and mathematicians! Mathematics is disappearing! being damaged!
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Yes! Even standard mathematical symbols and notations are captured in similarity index. The habit of using unconventional symbols and notations just to reduce similarity index is destroying the beauty and taste of mathematics.
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Hi.
Please change the picture of header site. That is "2th International Conference on Combinatorics, Cryptography and Computation".
Thank you.
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:) They should change to 3rd (and it's 2nd not 2th).
Cheers.
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Many text books claim that particles that obey Boltzmann statistics have to be indistinguishable in order to ensure an extensive expression for entropy. However a first principle derivation using combinatorics gives the Boltzmann only for distinguishable and the Bose Einstein distribution for indistinguishable particles (see Beiser, Atkins or my own text on Research Gate). Is there any direct evidence that indistinguishable particles can obey Boltzmann statistics?
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Is it a variant of Vandermonde convolution formula for falling factorials? What is the answer? See the picture.
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See the picture uploaded here.
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State dependent additivity and state independent additivity? ;
akin to more to cauchy additivity versus local  kolmorgov additivity/normalization of subjective credence/utility,  in a simplex representation of subjective probability or utility ranked by objective probability  distinction? Ie in the 2 or more unit simplex (at least three atomic outcomes on each unit probability vector, finitely additive space) where every events is ranked globally within vectors and between distinct vectors by < > and especially '='
i [resume that one is mere representability and the other unique-ness
the distinction between the trival
(1)x+y+z x,y,z mutually exclusive and exhaustive F(x)+F(y)+F(z)=1
(2)or F(x u y) = F(x)+F(y) xu y ie F(A V B)=F(A)+F(B)=F(A)+ F(B) A, B disjoint
disjoint on samevector)
(3)F(A)+F(AC)=1 disjoint and mutually excluisve on the same unit vector
and more like this or the properties below something more these
to these (3a, 3b, 3C) which are uniqueness properties
forall x,y events in the simplex
(3.A) F(x+y)=F(x)+F(y) cauchy addivity(same vector or probability state, or not)
This needs no explaining
aritrarily in the simplex of interest (ie whether they are on the same vector or not)
or(B)  x+y =z+m=F(z)+F(m) (any arbitary two or more events with teh same objective sum must have the same credence sum, same vector or not) disjoint or not (almost jensens equality)
or (C)F(1-x-y)+F(x)+F(y)=1 *(any arbitrary three events in the simplex, same vector or not, must to one in credence if they sum to one in objective chance)
(D) F(1-x)+F(x)=1 any arbitary two events whose sum is one,in chance must sum to 1 in credence same probability space,/state/vector or not
global symmetry (distinct from complement additivity) it applies to non disjoint events on disitnct vectors to the equalities in the rank. 'rank equalities, plus complement addivitity' gives rise to this in a two outcome system, a
. It seems to be entailed by global modal cross world rank, so long as there at least three outcome, without use of mixtures, unions or tradeoffs. Iff ones domain is the entire simplex
that is adding up function values of sums of evenst on distinct vectors to the value of some other event on some non commutting (arguably) probability vector
 F(x+y)=F(x)+F(y)
In the context of certain probabilistic and/or utility unique-ness theorems,where one takes one objective probability function  and tries to show that any other probability function, given ones' constraints, must be the same function.
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and F(x+y)=F(x)+F(y)
In the context of certain probabilistic and/or utility unique-ness theorems,where one takes one objective probability function  and tries to show that any other probability function, given ones' constraints, must be the same function.
what is meant by state dependent additivity does that mean that instead of F(x U y)=F(x)+F(y) x,y disjoint (lie on the same vector) that (same finite probability triple) ; or instead of F(x)+F(y)+F(z)=1 iff x,y,z are elements of the very same vector (same triple)
in the simplex one literally instead has that F(x+y)=F(x)+F(y) over the entire domain of the function. ie adding up (arguably non commuting) elements of distinct vectors,   or F(x)+F(y)+F(1-x-y)=1 arbitarily over the simplex; or domain of interest, where the only restriction is that one can only add up elements as many times as they are present in the domain. Whilst with cauchy additivity one if one domain is merely a single vector. <1,3, 1,6, 1,2, unit event=1 > one so long 1/6 is in the domain (supposing that entire domain, probability vector space, just is that vector dom(F)={1,3 1,6, 1,2 1),  where if F(1)=1 one can arbitrarily add up 1=F(1)=F(1/6+1/6+1/6 ) six times= 6F(1/6), so F(1/6)=1/6.
I presume however, that if ones domain is the entire simplex there would not be any relevant difference, between outright Cauchy additivity and state independent additivity; and thus to presume would be outright presumptuous.  Or this a name for cross world global rank which entails its long as there at least three atomic events on each simplex (so long as the simplex is well constructed, and the rank is global and modal), even if finite local standard additivity is presumed). As one can transfer values of equiprobable events onto other vectors where they are disjoint?
if by state independent addivity  this means one can arbitrary add up the function values of F(1/6) for objective probability six times, to F(1)=1 the function value lets say at chance=1 to attain that F(1/6)=6 so long as those events are ranked equal and are present at least six times somewhere or other, even if in distinct state, or vectors (in the same system).
or does this apply to local additivity, where one has a global modally transitive rank over the simplex where n>=2 (ie the number of elements in each triple is at least three) because a cross world rank, with equalities, will entail this in any case, if justified. So if one can derive that cross world additivity must hold given finite additivity and global modal rank including equalities cross world/vector, on pain of either either local addivitity failing (probabilism)  or ones justified global and local  total rank  is violated, justified for whatever reason is violated)  including equalities must hold (for whatever reason) does this count as presumptious.
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Points are two-combinations of n phase names. Lines are three-combinations of phase names. Point and line are incident upon one another if the two phase names comprising the point are part of the name of the line (e.g., point AB and line ABE are incident upon one another). In attached figures, lines of perspective are red, perspective triangles are green, and points of perspective of pairs of green triangles are connected with orange lines. Brute force counting of adjacencies is doable for Desargues configuration (n=5) , but impractical as system number of phases increases.
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It's been a while, but I've finally achieved an answer to my question. In the Euclidean plane, given points as two-combinations of n phase names and lines are three-combinations of the phase names there must be 1 point adjacent to 2n-4, 2 points adjacent to 2n-5, 3 points adjacent to 2n-6, ..., n-1 points adjacent to n-2.
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A-B-C ABOUT IDENTIFICATION OF GRAPHS
John-Tagore Tevet
Let us try to open the essence of graphs, from that's so far tried to circumvent.
1. What is a graph
Graph is an association of elements with relationships between these that has a certain structure.
Graphs are represented for different purposes. On the early rock paintings have been found the constellations show schemes. Graphs was used also for explain the theological tenets.
Example 1. Graph (structural formula) of isobutane C4H10:
Graphs began to investigate after then when Leonhard Euler in 1736 has solved the problem of routing on the seven bridges between four banks of Königsberg [1].
Example 2. Königsberg’s bridges and corresponding graph:
Also in present time used the graphs mainly for solving the problems of routing and flowing. Already in 1976 considered that such one-sided approach is a hindering factor for studying of graphs [2]. To the essence of graph, to its structure and symmetry properties has the interest practically non-existent. The last explorer was evidently Boris Weisfeiler in 1976 [9].
Definition a graph as an object consisting in node set V and edge set E, G=(V, E), is a half-truth that beget confusions. Essential is to explain the properties of inner organizing (inner building) or structure, i.e. identification of graphs.
Graph is presentable: 1) as a list L of adjacencies; 2) in the form of adjacency matrix E; 3) graphically G, where the elements to “nodes” and relations to “edges” called.
Example 3. List of adjacencies L, corresponding adjacency matrix E and for both corresponded graphs GA and GB:
Explanations:
The outward look and location of the enumerated elements in graph not have something meaning. But on the emotional level it rather engenders some confusion.
One graph can be differs from the other on its looking or its inner organizing (inner-building) or structure S what in ordinarily visually not be opened. Maybe just due to this is to the present days the existence of structure ignored.
We can here make sure that graphs GA and GB have the same structure and these are isomorphic GA @ GB. Ordinarily differentiate in the objects just the “outward” differences and refuse to see some common structure.
 
Propositions 1. Structure axioms:
P1.1.    Structure S is presentable as a graph G and each graph G has its certain structure S.
P1.2.    Isomorphic graphs have the same structure – structure is the complete invariant of isomorphic graphs.
 
Identification of graph is based on identification the binary relations between elements [3 - 8]. Binary relation can a “distance relation”, “circle relation”, “clique relation” etc. and is measurable. Binary relation characterized by corresponding binary sign.
2. Identification of the graph
For identification of the graphs uses two each others complementary ways:
Multiplicative identification (products of adjacency matrixes);
Heuristic identification.
 
Propositions 2. Multiplicative identification: multiplication the adjacency matrixes:
P2.1.       To multiplying the adjacency matrix with itself E´E´E´…=En and fixing in case of each degree n the number p of different multiplicative binary signs enij that as rule enlarges. Forming the sequence vectors ui of different multiplicative binary signs.
P2.2.       In each case if p enlarges (change) must transpose the rows and columns of En correspondingly to the obtained frequency vectors ui.
P2.3.       Stop the multiply if p more no enlarges and to present the current En and the following En+1.
Explanation: Multiplicative signs differentiate the binary signs but no characterize these.
Example 4. Adjacency matrix E and its transposed products E2, E3 of graphs on example 3:
1  2  3  4  5  6| i
                0  1  0  1  0  1| 1
                1  0  1  0  1  1| 2
          E    0  1  0  1  0  1| 3
                1  0  1  0  1  0| 4
                0  1  0  1  0  1| 5
                1  1  1  0  1  0| 6
                                         ui
2  6| 1  3  5| 4     |    i    0 1 3 4   k
4  3| 1  1  1| 3     |    2    0 3 2 1   1
3  4| 1  1  1| 3     |    6    0 3 2 1   1
E2   1  1| 3  3  3| 0     |    1    1 2 3 0   2
1  1| 3  3  3| 0     |    3    1 2 3 0   2
1  1| 3  3  3| 0     |    5    1 2 3 0   2
3  3| 0  0  0| 3     |    4    3 0 3 0   3
ui
 2  6| 1  3  5| 4|   i    0 2 3 6 7 9 10  k
 6  7|10 10 10| 3|   2    0 0 1 1 1 0 3   1
 7  6|10 10 10| 3|   6    0 0 1 1 1 0 3   1
E3   10 10| 2  2  2| 9|   1    0 3 0 0 0 1 2   2
10 10| 2  2  2| 9|   3    0 3 0 0 0 1 2   2
10 10| 2  2  2| 9|   5    0 3 0 0 0 1 2   2
 3  3| 9  9  9| 0|   4    1 0 2 0 0 3 0   3
Explanations:
a)      The set of similar relations (and elements) recognize their position W in the structure. Position W is in group theory known as transitivity domain of automorphisnsms, equivalence class or orbit.
Multiplicative binary signs enij recognize here five positions of binary relations WR and on they base three positions of elements WV.
 
Propositions 3. Position axioms:
P3.1.       If structural elements (graph nodes) vi , vj , … have in graph G the same position WVk then corresponding sub-graphs (Gi=G\vi) @ (Gj=G\vj) @....   are isomorphic.
P3.2.       If relations (edges) eij, ei*j*, … have in graph G the same binary(+)position WR+n then corresponding greatest subgraphs (Gij=G\eij) @ (Gi*j*=G\ei*j*) @....  are  isomorphic.
P3.3.       If relations (“non-edges”) eij, ei*j*, … have in graph G the same binary(–)position WRn– then corresponding smallest supergraphs (Gij=GÈeij) @ (Gi*j*=GÈei*j*) @....  are isomorphic.
 
Before elaboration of the multiplicative identification way was elaborated a heuristic way.
Propositions 4. Heuristic identification:
P4.1.       Fix an element i and form its neighborhood Ni, where the elements, connected with i divide according to distance d to entries Cd.
P4.2.       Fix an element j and fix its neighborhood Nj by condition P4.1.
P4.3.       Fix the intersection Ni ÇNj as a binary graph gij, and fix the distance –d between i and j (in case of adjacency collateral distance +d), the number n of elements (nodes) in gij, number q of adjacencies (edges). Fixing the heuristic binary sign ±d.n.q.ij of obtained graph gij.
P4.4.       Realize P4.1 to P4.3 for each pair i,jÎ[1, |V|]. Obtained preliminary heuristic structure model SMH.
P4.5.       Fixing for each row i its frequency vector ui. Transpose the preliminary model SM by frequency vectors ui lexicographically to partial models SMk.
P4.6.       In the framework of SMk transpose the rows and columns lexicographically by position vectors si to complementary partial models. Repeat P4.6 up to complementary transposing no arises.
Explanation: Heuristic binary signs differentiate the binary signs and characterize these.
Example 5. On the Example 3 presented differently enumerated graphs GA and GB, their heuristic binary signs and structure models SMA and SMB with their common product E3:
ui
 3  4| 1  4  5| 2|   iA
 1  2| 1  2  5| 6|        iB    0 2 3 6 7 9 10  k
 6  7|10 10 10| 3|   3    3    0 0 1 1 1 0 3   1
    6|10 10 10| 3|   4    6    0 0 1 1 1 0 3   1
E3         | 2  2  2| 9|   1    1    0 3 0 0 0 1 2   2
…..  |    2  2| 9|   2    4    0 3 0 0 0 1 2   2
     |       2| 9|   5    5    0 3 0 0 0 1 2   2
     |        | 0|   6    2    1 0 2 0 0 3 0   3
Explanations:
”Diverse” graphs GA and GB have equivalent heuristic structure models SMA » SMB and the same multiplicative model E3. This means that structures are equivalent and all on the examples 2 and 4 presented graphs GA and GB are isomorphic GA @ GB.
The binaries are divided to five binary positions WRn, where the “adjacent pairs” or “edges” divided to three binary(+)positions (full line, a dotted, dashed-line) that coincide with heuristic binary signs C, D, E and corresponding multiplicy binary signs 10, 7, 2, and with two binary(–)positions with signs –A and –B and multiplicative signs 9 and 3. In base of these divide the structural elements to three positions WVk.
The column ui constitutes frequency vectors, where each element i characterize its relationships with other elements. On the base of frequency vectors ui obtained the positions of elements WVk.
The column si constitutes position vectors that represent the connecting of i with elements on the position k.
A principal theoretical algorithm of isomorphism recognition exists really – it consists in rearranging (transposing) the rows and columns of adjacency matrices EA of graph GA as yet these coincides with the EB of GB. But this has an essential lacking – it is too complicated, the number of steps can be up to factorial n!
Propositions 5. On the relationships between isomorphism and structural equivalence:
P5.1.    Isomorphism GA@GB is a such one-to-one correspondence, a bijection j: VA®VB, between elements what retains the structure GS of graphs GA and GB.
P5.2.    Isomorphism recognition does not recognize the structure GS and its properties (positions etc.), but the structure models SM and En recognize the structure and its properties with exactness up to isomorphism.
P5.3.    Structural equivalence SMA»SMB and EnA»EnB is a coincidence or bijection j: WA®WB on the level of binary positions WRn and positions of nodes (elements) WVk.
P5.4.    In the case of large symmetric graphs recognizes the products En the binary positions more exact than heuristic models SM, where need to use the binary signs of higher degree. That why it is necessary to treat both in together, bearing in mind also that the heuristic binary signs characterize the essence of relationship itself.
P5.5.    Recognition of the positions by the structure model is more effective than detecting the orbits on the base of the group AutG.
Example 6. To the recognition on the Example 1 represented structure of isobutane suffice use the heuristic model SM:
Explanation: Decomposing the elements C and H to four positions corresponds to actuality. The positions are visually appreciable also on the Example 1.
3. List of tasks that solving based on the identified graphs (structure)
To conclusion it should be emphasized that the recognition of graph’s structure (organizing) is based on the identification (distinction) of binary relations between elements. Binary relation can be measured as a “relation of the distance”, “circle relation”, “clique relation”, etc. Binary relation is recognizable by the corresponding binary sign.
The complex of tasks that are based to recognizing structures is broad, various and novel (differ from up to now set up) [3 - 8]. We list here some.
1.      The relations between structural positions, automorphismsm and group-theoretical orbits.
2.      Structural classification the symmetry properties of graphs.
3.      Measurement the symmetry of graphs.
4.      Analyzing different situations of structural equivalency and graphs isomorphism.
5.      Positional structures that open the “hidden sides” of graphs.
6.      Unknown sides of well-known graphs.
7.      Adjacent structures and reconstruction problem. It is connected with general solving the notorious Ulam’s Conjecture.
8.      Sequences of adjacent structures and their associations – the systems of graph structures.
9.      Probabilistic characteristics of graph’s systems.
10.    The relations of graph systems with classical attributes.
References
1.       Euler. L. Solutio problematis ad geometriam situs pertinentis. – Comment. Academiae Sci. I. Petropolitanae 8 (1736), 128-140.
2.       Mayer, J. Developments recents de la theorie des graphes. – Historia Mathematica 3 (1976) 55-62.
3.       Tevet, J.-T. Semiotic testing of the graphs: a constructive approach and development. S.E.R.R., Tallinn, 2001.
4.                     Hidden sides of the graphs. S.E.R.R. Talinn, 2010.
5.                     Semiotic modeling of the structure. ISBN 9781503367456, Amazon Books. 2014.
6.                     Süsteem. ISBN 9789949388844. S.E.R.R., Tallinn, 2016.
7.                     Systematizing of graphs with n nodes. ISBN 9789949812592. S.E.R.R., Tallinn, 2016.
8.                     What is a graph and how it to study. ISBN 9789949817559. S.E.R.R., Tallinn, 2017.
9.       Weisfeiler, B. On Construction and Identification of Graphs. Springer Lect. Notes Math., 558, 1976 (last issue 2006).
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It is easy to prove that (quote wikipedia):
If A is an adjacency matrix of the directed or undirected graph G, then the matrix An (i.e., the matrix product of n copies of A) has an interesting interpretation: the element (i, j) gives the number of (directed or undirected) walks of length n from vertex i to vertex j. If n is the smallest nonnegative integer, such that for some i, j, the element (i, j) of An is positive, then n is the distance between vertex i and vertex j.
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I want to build a kind of guess game. I do not know the right name but the concept of the game is: person-1(P-1) thinks a name(of anything) and person-2 will have to predict that name by asking as less questions as possible. For example:
p1: thinks something(Steve jobs)
p2: Are you talking about man?
p1: yes.
p2: Is he/she media guy?
p1: No
P2: is he tech personality?
p1: yes
p2: steve jobs.
p1: yes.
So p2 has to ask 4 questions. It could be even more as number of predictors are infinite. Now I want to model this scenario. My Goal is to reduce the number of question. Note that the number of predictors are limited. So situation is not that broad.
I can think of decision tree. But question is, how can I decide where to split so that length of the brunch will be small.
Any suggestion/reference will be appreciated.
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Maximize the entropy, the information gain
for every question.
Regards,
Joachim
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Assume we have a class of graphs. Now what does this sentence mean?
"each of the graphs in the class, monotonically should make no difference". 
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@Nazanin "A graph peoperty is monoton if every subgraph of a graph with property P, also has the property P"  is called "hereditary property".  
As for the original (part of a) sentence: if we join it to another domain of mathematics, it makes sense such as it is, despite imprecise wording: e.g. in a class of exponential functions with a basis a>1... 
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If there are k sets of vertices in a graph, with the condition that each vertex in a set should be connected to at least one vertex from each of the other sets, then what is the least number of complete sub-graphs $K_{k}$ in this graph?
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Here you have another counterexample, is this what you mean? 
I don't understand your question about being $\chi$-colorable. Usually, $\chi(G)$ stands for the chromatic number of G... In the case of the graph I sent, it's 2-colorable.
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A NSWE-path is  a path consisting of North, South, East and West steps of length 1 in the plane. Define a weight w for the paths by  w(N)=w(E)=1 and w(S)=w(W)=t. Define the height of a path as the y-coordinate of the endpoint. For example the path NEENWSSSEENN has length 12, height 1 and its weight is t^4.
Let B(n,k) be the weight of all non-negative NSEW-paths of length n (i.e. those which never cross the x-axis) with endpoint on height k.
With generating functions it can be shown that for each n the identity
(*)            B(n,0)+(1+t)B(n,1)+…+(1+t+…+t^n)B(n,n)=(2+2t)^n
holds. The right-hand side is the weight of all paths of length n.
Is there a combinatorial proof of this identity?
For example B(2,0)=1+3t+t^2 because the non-negative paths of length 2 with height 0 are EE with weight 1, EW+WE+NS with weight 3t, and WW with weight t^2.
B(2,1)=2+2t because the non-negative paths are NE+EN with weight 2 and NW+WN with weight 2t. And  B(2,2)=1 because w(NN)=1.
In this case we get the identity
B(2,0)+(1+t)B(2,1)+(1+t+t^2)B(2,2)=(2+2t)^2.
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In the meantime I have found a combinatorial proof in the literature: Naiomi T. Cameron and Asamoah Nkwanta, On some (pseudo) involutions in the Riordan group,  J. Integer Sequences   8 (2005), Article 05.3.7, proof of identity 1. https://www.researchgate.net/profile/Asamoah_Nkwanta/publications?sorting=newest
Instead of NSWE-paths they use bicolored Motzkin paths, but their proof can easily be translated to the situation of NSWE-paths. So my question has been answered.
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Let $n\ge 4$ and $W=\{w_1,w_2,\cdots,w_{n-1}\}$ be a given sequence of positive numbers. Suppose that $w_i$ corresponds the weight of the edge $e_i$ of the weighted path $P_n$ for $i=1,2,\ldots, n-1$. Let also $M_i$ be an $i$-matching of $P_n$.
Denote by $W(M_i)$ the weight of such an matching, {\it i.e.,} the product of all weights of such an $i$-matching, and by $WM(i)$ the sum of all weights of $i$-matchings of $P_n$.
{\bf Question:} Who know the applications, including combinatorics and physics, of the
parameter $WM(i)$?
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Read 《Integer Flows and Cycle Covers of Graphs》 and 《Circuit Double Covers of Graphs》, the author is Prof. Cunquan Zhang. Another field is circular flow theory, you may read Prof. Genghua Fan's papers.  Hope it is useful.
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Can someone suggest me a good software for drawing graphs? The suggested software should be one which can be used to draw the attached graph (together with mathematical symbols) .
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Thanks everyone for your kind suggestions. I tried most of the suggested softwares and found ipe the best one. Special thanks to Seyyed Aliasghar Hosseini,  Imran Khaliq and Darren Strash. 
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Or generally:
\sum _{i=1}^n \sum _{j=1}^i \left\lfloor \frac{-j+n+1}{2 i}\right\rfloor
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I'm assuming from your question that you're looking for a closed form solution.
Let q = Floor[n/2i] and r = n - q*2i (quotient and remainder when n is divided by 2i). Therefore, 0 <= r < 2i.  Since i <= n,
     sum_{j=1}^i Floor[(n-j+1)/2i] = iq - d,
where d=0 if i-1 <= r < 2i, and d=i-1-r if 0 <= r < i-1.
This gives a closed form for the inner summation.  Obtaining a closed form over both summations looks like a harder problem, but maybe possible.
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what are the classes of posets closed under taking ordinal sum of posets?
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Let X and Y be two posets on the disjoint sets P and Q. The  disjoint union P+Q is defined as:
(1)if (x 'less than, or equal to' y) is in P, for x and y in P, then x <=  y in P+Q and
(2)if  (x 'less than, or equal to' y) is in Q, for x and y in Q, then x <= y in P+Q. 
The ordinal sum P*Q is defined as the partial order on P U Q, that satisfy (1) and (2), and the additional condition: (3) (x 'less than, or equal to' y) in P*Q for x in P, and y in Q.
The disjoint union + is commutative, but the ordinal sum '*' is not. 
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Since a uniform permutation has a probability of 1/n!, only distances of O(1/n^n) can be regarded as negligible. Section 2.2 of the attached paper gives a network with distance O(1/n^n) but the depth is O(n). Is there a permutation network with the same distance but an asymptotically smaller depth?
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Please attach the file because the link is not redirecting to appropriate web page of MIT.  
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Let the set of all repunits (repeated units) i.e., 1,11,111,... is X.
=> X = { Rn : n belongs to postive integer }
(Repunit of n defined by Rn. 1 is R!. 11 is R2, etc....)
Generally is not satisfy closed property.
  1. Under what condition or when X satisfy closed property?
  2. For which n, R2n+1 is prime?
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Yes dear Srinivasan, put the second "word" just next the first one. In French we call it juxtaposition, I think the terminology is the same in English .
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Let q be an odd positive integer, and let Nq denote the number of integers a such that 0 < a < q/4 and gcd(a, q) = 1. How do I see that Nq is odd if and only if q is of the form pk with k a positive integer and p a prime congruent to 5 or 7 modulo 8?
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I see that you, too, follow the Putnam Exam.
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Assuming we have a planar Graph G=(V,E). There is a function f(G) which returns a positive real number and we wish to assign either 0 or 1 to each node in order to minimize the function f(G). Trying all different combinations of assigning 0 or 1 to each node to find the combination which returns the minimum value of f(G) would be computational intractable. Hence, the question is whether there exists a algorithm to find (or approximate) the minimum for such a function.
Note that the function f has the properties that a solution exists and is unique. Further, it is known that the Graph that minimizes the function has 0 in the vast majority of nodes and the set where the nodes are 1 is connected. 
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Thilo,
One heuristic you might try, given that you have the "sum of squares of distances is small" phenomenon, is to start by looking at all vertices at distance <= k from a fixed vertex.  These sets of vertices, essentially closed balls, will often have small sums of squares of distances, I'd guess (I'm sure that there's an isoperimetric result out there of which I'm unaware...)   Then looking at local changes to these sets might enable you to find better --- and perhaps even optimal --- solutions.  At the very least they'd give you a good choice of upper bounds.
Neil
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Is there any provision to apply combinatorial analysis on image fusion?
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The exercise I'm dealing with asks me to show that by adding S = K to the usual reduction rules for the SKI-calculus, one obtains an inconsistent equivalence. This must be done without using Böhm's theorem.
Now, I've found two terms (not combinators) M and N with the following properties:
M = x
N = y
M can be obtained from N by replacing one or more occurrences of S with K
From the rule S = K we thereby get that M = N, and hence that x = y. Which means that any term can be proved equal to any other.
Do you think such an answer could work? Actually, Böhm's theorem (as shown in the book I am studying) establishes that, for distinct combinators G and H, there is a combinator D such that DxyG = x and DxyH=y. So, I feel it would have been more appropriate to find a combinator D such that Dx1x2S = x_i and Dx1x2K = x_j with i , j = 1 , 2 and i ≠ j. But I have HUGE problems in finding combinators, so I have only found terms respectively reducing to one of the variables, and obtainable one from another by replacing S with K or vice versa.
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Yeah I see now there's a much simpler solution. But I wondered if I had "well done" the exercise. Actually, it seems to me it works. I found two terms, M and N, the first one reducing to an arbitrary x and the second one to an arbitrary y. Moreover, M can be obtained from N by replacing an occurrence of S with K and therefore, as soon as one admits S = K as a rule, it follows that M = N. Since M = x and N = y, by transitivity one should get x = y for every x and y.
Actually, M and N, as I built them, contain other terms which are in the end not relevant to respective reducibility to x and y. But that's should not be a problem. It should be like to have two terms with variables (x, y, z, w), possibly with z = w, respectively reducing to x and y.
That's how I did the exercise...
PS: inconsistent in the sense that one can trivialize the equality relation.
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recently, some researchers work on infinite structure of matroids.
independent axiom of matroids  have a first role in definition of codes on GF(q) that we can see this point in representable matroids. now , can we have any logical definition  of this point for related codes of infinite matroids?
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Hi Hossein.
I haven't thought much about this issue, but I found the attached paper on the archive.
Section 2.6 seems to touch on the issue of infinite matroids being representable over a field.
If that field is GF(q), one is close to your question, I assume.
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When a bipartite complete graph Km,n is given, two subgraphs of Km,n are in the same class when the degree of each right vertex coincides. I want to know the number of all spanning trees in a given class.   
Any spanning tree in Km,n has M+N-1 edges. A class whose right degrees do not sum up to M+N-1 does not contain any spanning tree. The number of classes with total degree M+N-1  is the repeated combination of country labels taken N -1 times. Thus the number take the form
                 (M + N - 2)! / (M - 1)! (N -1)! . 
From Scoins' formula the number of all spanning trees in Km,n is
                   MN-1NM-1 .
As a consequence, there are in general many spanning trees in a class in which the right degrees sum up to M+N-1.  I want to know an explicit formula that gives the number of all spanning trees for a given class with degree sum M + N -1.
This question is derived in the course of Ricardian trade theory study.  
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Dear Robert A. Beeler,
thank you for your comment. I am expecting to hear from you good news. 
Shiozawa
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We usually use differential equations, ordinary and partial, difference and delayed. But, could dynamics be captured using discrete mathematics structures, or combinatorics?
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If you look applications of tools provided by discrete mathematics in dynamical systems then this book can be good place to start:
Lind, Douglas; Marcus, Brian. An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge, 1995. xvi+495 pp. ISBN: 0-521-55124-2; 0-521-55900-6
It provides a very smooth introduction into symbolic dynamics from point of view of discrete structures (graphs, formal languages, Perron-Frobenius theory, etc.) and then reveal connections with topology, mixing, topological entropy, etc.
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I found this problem without comments in a french exercise book of 1982, now out of print : {\em 1932 exercices de mathématiques} by Luc Moisotte, ISBN 2-04-015483-3.
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 I send my proof as a file attachment.pdf . Please, answer me. Your sincerely, Anna Tomova. 
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In 1969 , David Barnette conjectured that 3 regular , 3-connected , bipartite , planar graph is Hamiltonian. . I am interested  to generate Barnette  graph for given  even number of vertices. There are countably finite number of Barnette graphs available in the literature.
Consider a grid graph with 4 vertices (cycle C 4) which is Hamiltonian. Increasing dimension in one direction , we see that the resulting graph is always Hamiltonian but not Barnette. Can one generate  countably infinite number of  Barnette graphs from one small Barnette graph?     
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It has been shown that fullerenes are hamiltonian [1] and they form an infinite family of cubic, 3-connected, planar graphs with faces of size at most 6. There is at least one fullerene on 2k vertices for each k greater than 11. Of course these are not bipartite since they contain pentagons, so I guess you are mainly interested in the other Barnette's conjecture. But, the thesis by Jan Goedgebeur also talks about this second conjecture and not the one about bipartite graphs.
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I am trying to calculate the most compact way of grouping a set of pixels together. Does anyone have a readable guide on how to do this?
My initial results are given below for clusters of up to 10 pixels. Results are expressed in terms of the sum of unique interpixel distances for a given cluster (e.g. for a 3 pixel cluster it is the sum of the the distances ab, ac, bc).
1 = 0
2 = 1
3 = 3.4
4 = 6.8
5 = 13.5
6 = 21.1
7 = 31.4
8 = 44.1
9 = 58.9
10 = 78.5
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I looked at the paper. So at least for L1 distances, we do not have uniqueness, the incremental approach partially fails (some optimal solutions do not extend to +1 optimal solutions, while some optimal solutions are not extension of -1 optimal solutions), and 4x4 square would yield L1 sum of 320, while the optimal result of 318 is displayed in the paper. Good luck.
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In \cite{Roman} page 25 we read that,  a sequence $s_n(x)$ is Sheffer for $(g(t), f(t))$, for some invertible $g(t)$, if and only if
$$ s_n(x+y)=\sum\limits_{k=0}^{\infty}\binom nk p_k(y) s_{n-k}(x),$$
for all $y$ in complex numbers,  where $p_n(x)$ is associated to $f(t)$. 
Noting to the fact that $e_q(x+y) \neq e_q(x)e_q(y)$, leads to conclude that $ s_{n,q}(x+y) \neq e_q(yt)s_{n,q}(x)$, and ,therefore, we do not have the $q$-analogue of the identity above directly. Is it possible to express the $q$-analogue of the above mentioned identity in any other way, or should we neglect such an identity for $q$-Sheffer sequences at all?
Any contribution is appreciated in advance. 
\bibitem{Roman}Roman S., Rota G. The umbral calculus. Advances Math. 1978;27:95–188.
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Dear Dr. Waldemar W Koczkodaj
Thanks for your response. I faced with this question, for the first time, while I was studying the sequence of $q$-Appell polynomials. Since that time, this question has been in my mind for a long time and so far I have not been able to make myself convinced by a perfect answer to it. The reason to ask this question here, actually, is to consult with the experts of this field and read about their different ideas which are originated, clearly, from different points of view. Although I am enthusiastic for learning more and more and go forward through the science, in case that I found a good answer which helps me to publish my studies, I make you sure that I will definitely obey the publication rules and humanity. 
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Some authorities like Davenport have already explained that traditional (Small?) data relates to corporate operations while Big data relates to corporate products and services.
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In these papers, I have argued that small data analysis based on Gaussian statistics, while BIG data based on power law statistics, and small data analysis based on Euclidean geometry, while BIG data based on fractal geometry.
Jiang B. (2015b), Geospatial analysis requires a different way of thinking: The problem of spatial heterogeneity, GeoJournal, 80(1), 1-13, Preprint: http://arxiv.org/ftp/arxiv/papers/1401/1401.5889.pdf
Jiang B. (2015a), Head/tail breaks for visualization of city structure and dynamics, Cities, 43, 69-77, Preprint: http://arxiv.org/ftp/arxiv/papers/1501/1501.03046.pdf
Jiang B. and Miao Y. (2014), The evolution of natural cities from the perspective of location-based social media, The Professional Geographer, xx(xx), xx-xx, DOI: 10.1080/00330124.2014.968886, Preprint: http://arxiv.org/abs/1401.6756
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Last year, Dr. Yitang Zhang has published a paper for the upper bound of twin primes, which is 7*10^7. Does anyone has idea to achieve lower bounded gap?
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Terence Tao and his collaborators did tons of work on this in Polymath 8 (mentioned ambiguously in the above comment). You can read part of what was done here:
Tao's complete corpus of work on that project is:
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In Maple 16,  how can we with the software combstruct,  to give the sentence about the recurrence formula,
A(x)=1+x[A(x)3+3A(x)A(x2)+2A(x3)]/6
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Very strange, first the gf then the object to count!
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It is well known that the majority rule may not be transitive for some configurations of individual preferences. Domain restrictions are possible ways out. But what is known about maximal such domains (i) with respect to the cardinality? (ii) via set inclusion?
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Thank you very much. I also like the job done on mutiple issues.
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I and another three colleagues have an ongoing paper about the application of sociometry to multiple human resource allocation to multiple projects. The problem is mathematically quite challenging but we are on a dead end concerning one of the outcomes we are dealing with.
Particularly, we have been applying meta-heuristics and evolutionary algorithms to solve the problem of allocating groups of people so as to maximize cohesion among them. The research is quite interesting and we think it is going to open multiple and very interesting research and industry application in the near future.
However, we are stuck in one part. We are trying to calculate the number of viable combinations of people who can work either full-time, part-time or not work at all in several simultaneous projects and we need a person with advanced knowledge in combinatorics to give us a hand. We are willing to pay or to put his/her name as co-author on the paper.
Getting straight to the point, the problem statement is as follows:
There are N people (i=1...N) who can be selected to work in P simultaneous projects (j=1...P). Each person can have a dedication of work full-time (1), half-time (0.5) or not work (0), that is, three possible allocations (0, 0.5, 1).
Now, we know that each project j requires Rj people. How many different and viable combinations are there?
Numerical Example:
People available  Project j=1     Project j=2      Project j=3=P
i=1                       0 or 0.5 or 1   0 or 0.5 or 1    0 or 0.5 or 1     Row sum =<1
i=2                       0 or 0.5 or 1   0 or 0.5 or 1    0 or 0.5 or 1     Row sum =<1
i=3                       0 or 0.5 or 1   0 or 0.5 or 1    0 or 0.5 or 1     Row sum =<1
i=4                       0 or 0.5 or 1   0 or 0.5 or 1    0 or 0.5 or 1     Row sum =<1
i=5                       0 or 0.5 or 1   0 or 0.5 or 1    0 or 0.5 or 1     Row sum =<1
Requirements      R_1=2            R_2=1             R_3=1              Row sum=4
,that is, we need 2+1+1 people working in these 3 projects and 1 out of the five available people is not used. We can use each person totally (1), half-time (0.5) or not use him/her, but the combinations must be feasible, i.e., each person cannot be assigned over 1 (full-time) and the requirements have to be fulfilled.
This problem is easier when the people can only work full-time or not work, but with half-times is far more complicated.
If anyone think that he/she is able to solve it, or just want further details, please  contact me.
Thanks
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Thanks Peter. We really appreciate the time you have spend solving this problem. Your contribution will be duly ackowledged as we agreed.
Thanks too for all you laborious explanations trying to make that someone like me understood the mathematical and combinatorics vision of this problem.
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Skew room squares exist for all odd values greater than 5. If n is prime it is a simple matter to generate a skew starter. But 667 is not prime. 667=23*29 which means a computer search has to be done in order to generate one. I would be satisfied with the skew room square of side 667, even though we can show it exists we can't seem to construct it. Any suggestions on this particular problem are appreciated.
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The reference provided 10 minutes ago may not be appropriate.
But, one may try some direct product construction as follows.
Start from skew starter S_1 in Z_{23} and skew starter S_2 in Z_{29}.
Find a permutation P=(p_0,p_1,...,p_{28}) such that
P-I = (p_0-0, p_1-1, ..., p_{28}-28)
And
P+I = (p_0+0, p_1+1, ..., p_{28}+28)
Are both permutations in Z_{29}. For example, take p_s=2s.
For each pair {x, y} in S_1, construct 29 pairs {(x,p_s), (y,s)}, s=0,1,…,28.
This generates 11x29 pairs.
Add 14 more pairs {(0,u),(0v)} such that {u,v} is from S_2.
It may be easy to verify if the 11x29+14 pairs form a skew starter in Z_{23}xZ_{29}.
If it still does not work, please let me know.
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Given a permutation A on the set [n], is there a way to determine the maximum number of disjoint cycles of AC where C ranges over all n-cycles on [n]? For which class of permutations A, this problem has been studied before?  Thanks
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Hamming distance between two permutations a and b is defined to be the minimum number of transpositions needed to bring a to b. If you express a^{-1} b as a one-line sequence, then Hamming distance is n - the number of fixed points. 
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What is the best software package available for carrying out computations related to algebraic coding theory, especially codes over rings.
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I need a good reason why improper uniform prior could be use as a prior in Serial Numbered Population (SNP) problem. Or maybe someone can tell me about improper uniform prior itself. An pdf link might be very helpful for me.
I attached the journal.
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I'm not sure if Jochen's answer is entirely correct; "An improper prior can be used when the resulting posterior is proper."  I'd modify it to say that an improper prior can be used when the resulting posterior can be achieved with a *proper* prior and the limits of the distribution, after the posterior is calculated, is taken to infinity and the result is shown to be the same.  For example, you can use a proper, bounded uniform prior and then, after you calculate the posterior, take the limits as the bounds grow to infinity to show that if you used the improper prior in the first place it would give the same answer.
Seen this way, the improper prior is a short-cut to make the analysis cleaner.  For example, I personally find E T Jaynes' analyses very clear because he uses improper priors (when he can!), and warns that one has to be careful to do it properly if there is even a hint of trouble.  I find a paper like Bretthorst's "Difference of Means" paper to be a much harder read because he uses proper priors throughout, even in cases where I think the improper prior would work.
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Could anyone suggest me some references about the uniqueness of the addresses of an Iterated Function System? E.g. points of Cantor set can be coded by a unique address, how general is this property?
More precisely, I am interested in the sufficient conditions to have a one-to-one relation between the shift space of the Iterated Function Systems and  its attractor. 
Thank you in advance!
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Dear Anna,
There are three kinds of attractors of iterated function systems, totally disconnected, just touching and overlapping. Totally disconnected attractors have metrically equivalent structure to the Cantor set. This can be found in Barnley's book Fractals Everywhere. The remaining cases may be treated with the help of lifted IFS, which is also explained in the book. 
Best regards
Miroslav
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It is well known that the secant $\sec z$ may be expanded at $z=0$ into the power series
\begin{equation}\label{secant-Series}
\sec z=\sum_{n=0}^\infty(-1)^nE_{2n}\frac{z^{2n}}{(2n)!}
\end{equation}
for $|z|<\frac\pi2$, where $E_n$ for $n\ge0$ stand for the Euler numbers which are integers and may be defined by
\begin{equation}
\frac{2e^z}{e^{2z}+1}=\sum_{n=0}^\infty\frac{E_n}{n!}z^n =\sum_{n=0}^\infty E_{2n}\frac{z^{2n}}{(2n)!}, \quad |z|<\pi.
\end{equation}
What is the power series expansion at $0$ of the secant to the power of $3$? In other words, what are coefficients in the following power series?
\begin{equation}
\sec^3z=\sum_{n=0}^\infty A_{2n}\frac{z^{2n}}{(2n)!}, \quad |z|<\frac\pi2.
\end{equation}
It is clear that the secant to the third power $\sec^3z$ is even on the interval $\bigl(-\frac\pi2,\frac\pi2\bigr)$.
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Dear Dr. Feng,
Take a look at:
Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1
in particular section: 7.1 Multinomial Euler numbers
The results in section 7.1 are for powers of sech, so you'll need to substitute t -> it.
The above paper is freely available from the Journal's website, and also from my ResearchGate pages.
Best wishes,
Ghislain Franssens
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I would like to list out all simple cycles in a connected graph.
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You need to give additional details about your problem. If you know adjacency matrix of your graph and want to count number of simple cycles of fixed length look at slides and papers in my profile.
Sergey Perepechko
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Please help me with a derivation of "amortized cost of a splay tree operation" . I am waiting for that please consider it fast and let me know.
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