Science topics: Mathematical SciencesCombinatorics

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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).

Questions related to Combinatorics

Dear Researchers,

Do researchers/universities value students/researchers having published sequences to the OEIS?

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).

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

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)

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

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.

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.

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.

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.

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.

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?

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.

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?

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.

I am a researcher in combinatorics and Semigroup in Ahmadu Bello University, Zaria seeking to know the binding line between both.

Thank you.

Where to find the answer?

Bell polynomials of the second kind B

*(*_{n,k}*x*_{1,}*x*_{2},...,*x*_{n}_{-}_{k}_{+1}) are also called the partial Bell polynomials, where*n*and*k*are positive integers. It is known that B*(1,1,...,1) equals Stirling numbers of the second kind*_{n,k}*S*(*n,k*).What are the values of the special Bell polynomials of the second kind B

*(0,1,0,1,0,1,0,...) and B*_{n,k}*(1,0,1,0,1,0,...)? Where can I find answers to B*_{n,k}*(0,1,0,1,0,1,0,...) and B*_{n,k}*(1,0,1,0,1,0,...)? Do they exist somewhere?*_{n,k}Dear Researchers,

Kindly let me know about 5 top most research problems in number theory which are concerning to

Double series and / or combinatorics

How to prove or where to find a proof of the lower Hessenberg determinant showed by two pictures uploaded here?

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.

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.

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?

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

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.

for k=1, 2, 3 we get 7, 13, 19 as primes.

for k=5, 6, 7 we get 31, 37, 43, as primes.

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!

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?

Is it a variant of Vandermonde convolution formula for falling factorials? What is the answer? See the picture.

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.

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).

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.

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".

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?

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.

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 $WM(i)$?

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) .

Or generally:

\sum _{i=1}^n \sum _{j=1}^i \left\lfloor \frac{-j+n+1}{2 i}\right\rfloor

what are the classes of posets closed under taking ordinal sum of posets?

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?

Let the set of all repunits (repeated units) i.e., 1,11,111,... is

**X**.=>

**X = { R**_{n}: n belongs to postive integer }(Repunit of n defined by R

_{n.}1 is R_{!. }11 is R_{2, }etc....)Generally

**X**is not satisfy closed property.- Under what condition or when
**X**satisfy closed property? - For which n, R
_{2n+1}is prime?

Let q be an odd positive integer, and let N

_{q}denote the number of integers a such that 0 < a < q/4 and gcd(a, q) = 1. How do I see that N_{q}is odd if and only if q is of the form p^{k}with k a positive integer and p a prime congruent to 5 or 7 modulo 8?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.

Is there any provision to apply combinatorial analysis on image fusion?

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.

When a bipartite complete graph

*K*,_{m}*is given, two subgraphs of*_{n}*K*,_{m}*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.*_{n}Any spanning tree in

*K*,_{m}*has*_{n}*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*M*

^{N}^{-1}・

*N*

^{M}^{-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.

We usually use differential equations, ordinary and partial, difference and delayed. But, could dynamics be captured using discrete mathematics structures, or combinatorics?

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.

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?

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

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.

Some authorities like Davenport have already explained that traditional (Small?) data relates to corporate operations while Big data relates to corporate products and services.

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?

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(x^{2})+2A(x^{3})]/6It 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?

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

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.

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

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.

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!

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 $\bigl(-\frac\pi2,\frac\pi2\bigr)$.

I would like to list out all simple cycles in a connected graph.

What is the best software package available for carrying out computations related to algebraic coding theory, especially codes over rings.

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.

If A and B Are non-singular then we have determinant of (A^2-B^2)=-determinant of AB , Am I correct? If so, then the problem is If A and B are singular, how can we prove?

Would like to know if (6677,333,166) could be cyclical on the torus.

I am asking for a combinatorial interpretation of a formula for Bell numbers in terms of Kummer confluent hypergeometric functions and Stirling numbers of the second kind. See the formula (8) and Theorem 1 in the attached PDF file or http://arxiv.org/abs/1402.2361. Could you please help me? Thank a lot.

Symmetric designs=combinatorial 2-designs with as many points as blocks.

There are many approaches using different matrices and eigenvectors to solve the min-cut problem. What is the best theoretical result providing a good approximation from a spectral cut to the solution?

I always thought that the sum of angular momentum should be zero. I am wondering if this applies on a sub-atomic scale as well as a macroscopic scale.

It can either be related to Key Distribution or any other application, just an overview needed.