# Combinatorics

1
Can we have any logical definition of infinite matroids and related defined codes?

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?

Hi Hossein.

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.

4
Do you know the number of all spanning trees of a given class?

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.

Dear Robert A. Beeler,

thank you for your comment. I am expecting to hear from you good news.

Shiozawa

12
Can we use discrete mathematics in modeling of control and dynamical systems?

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

Thanks. (:

9
Is there a bijective proof of : the rational number q_{m,n} = \frac {(2m)! (2n)!} {m! n! (m+n)!}, where m,n are positive integers, is an integer ?

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.

Thank you, Anna,

but I look for a possible "bijective" proof, I mean :
this number is an integer because it counts something than we can describe.

Sincerly, Jean

2
Re: research
dear friends. Presently I am doing research in automata theory. What is the latest research going on automata theory. friends, I am now in confusion. how to start and how it grow because I am basically mathematician. I want to know more ideas. also Is there any relavant materials available net. actually I started with buchi automaton in automata theory. but I do not know proceed that I have seen many materials. please help me & suggest me good ideas.
thanks & regards
O.V.SHANMUGA SUNDARAM
ovs3662@gmail.com

we completed ph.d thesis work. now preparing synopsis. for this we need external foreign examiner for evaluation of my ph.d thesis. probably we submit thesis before December 2015.. Our topic is related to automata theory using graph theory concept in the core chapter. we request you kindly know or interested please send your latest profile... thank you for considering my obligation....to my email-id is ovs3662@gmail.com

14
Is there any Barnette Graph with 2k (even number greater than 84) vertices?

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?

Increasing dimension of the graph or network. like Hexagonal network.

7
What are the applications of additive combinatorics in the field of engineering and architecture?

I would like to know some practical applications of additive combinatorics in the field of engineering and architecture. Please suggest some useful reference.

To enable agility by additive manufacturing (complexity, agility, efficiency) implementing concurrent, hybrid processes, considering the design space with topology organization,
production constraints, and optimization of variables and materials. Innovation is better through additive manufacturing, such as using Additive Topology Optimized Manufacturing (ATOM) which helped place brackets on the Black Hawk helicopter, and drastically reduces wasted material.

3
Baker Hausdorff decoupling formula
I have a two mode squeezed state ,
exp(ab-a^+b^+)|0>-a|0>_b . I would like to show entangled form which it will give. In this case [A,[A,B]] !=0 & [B,[A,B]] !=0 (!= means not equal) .therefore I can not use well known Glauber (Baker Hausdorff) decoupling formula directly. Does someone has experience on this problem or good suggestion for me. I think, I can do Taylor expansion then try to find some way to decouple this, I hope it will work ...

I think I misunderstood your question.

4
What are the major real life or practical applications of intersection graphs?

What are the major real life or practical applications of intersection graphs? Please suggest some good reference materials.

Clique Graphs, a special case of intersection graphs, have been used in Loop Quantum Gravity:

M. Requardt.
(Quantum) spacetime as a statistical geometry of lumps in random networks.
Classical Quantum Gravity 17 (2000) 2029--2057.

M. Requardt.
Space-time as an order-parameter manifold in random networks and the emergence of physical points.
In Quantum theory and symmetries (Goslar, 1999), pages
555--561. World Sci. Publ., River Edge, NJ, 2000.

M. Requardt.
A geometric renormalization group in discrete quantum space-time.
J. Math. Phys. {\bf 44} (2003) 5588--5615.

8
Knapsack Packing Problem. Anyone have experience with this?

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

Thank you Artiom you have been very helpful! (I have only just seen your previous post now, I must have posted mine at the same time!).. I will keep working on this problem

6
Just where do we draw the distinctive line between traditional data analysis and present day (Big) data analytics?

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

The size of what people mean when they say big data changes (and varies by discipline), so 20 years ago it would have been a lot smaller than now. It might be what you can't store (or do simple analysis of) on a new desktop. Here is a quote from a White House report:

There are many definitions of “big data” which may differ depend ing on whether you are a computer scientist, a financial analyst, or an entrepreneur pitch
ing an idea to a venture capitalist. Most definitions reflect the growing technological ability to capture, aggregate, and process an ever-greater volume, velocity, and variety of data.

https://www.whitehouse.gov/sites/default/files/docs/big_data_privacy_report_may_1_2014.pdf

So, imo, there is no line (and if someone drew one today, its wrong by tomorrow).

10
Where can I find useful literature on graph theoretical applications to biological networks?

Please provide me information regarding the recent developments in the mathematical, especially graph theoretical, studies on biological networks. Please give some good reference too..

Sudev

Thank you so much Professor Ljubomir Jacić, I will go through that.

2
What can be said about the $q$-analogue of the Sheffer identity?

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.

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.

9
What is the number of subsets of a finite set of non-negative integers which are neither the sumsets or the summands of other subsets of $X$?

Suppose that X is a finite set of positive integers. The sumset of two subsets A and B of X is defined as A+B={a+b:a\in A, b\in B}. Then, what is the number of subsets of X which are neither the non-trivial sumsets of any two other subsets of X nor the non-trivial summands of any other subsets of X?  Also, please suggest some useful references in this area.

Sudev

I'm still trying to understand the question.

Suppose X = {1,2,3,5}.

Then X has 2^4 possible subsets.  But {2,3} is out because it is the sumset of {1,2} and {1}.  Then are {1,2} and {1] out because they are summands of {2,3}?

If so, we're left with just two subsets - {1,3,5} and {1, 2, 3, 5}.

9
What are the graphs whose total graphs are complete graphs?

The total graph T(G) of a given graph G is a graph such that the vertex set of T corresponds to the vertices and edges of G and two vertices are adjacent in T if their corresponding elements are either adjacent or incident in G. Can we have non-trivial graphs whose total graphs are complete?

I checked  response of Prof. Singaram Dharmalingam now, quick proof :)

Thank you for the question though.

Good Luck!

4
How can we relate set theory with networks theory?

Would you please help me to identify some applications of set theory, in particular sumset theory, in networks such as  in social and biological networks? Please suggest some useful reading too/

I will suggest you this link

Connected Dominating Set: Theory and Applications - Springer
http://www.springer.com/mathematics/applications/book/978-1-4614-5241-6

6
Can anybody suggest any references for combinatorial studies on perturbation theory?

Are there any combinatorial studies on perturbation theory? Can we relate graph theory to that area? If so, please suggest some useful references for a beginner like me.

Thank you Prof. Sudev.

8
Is it right to take the sumset A + ∅ = ∅?

Is the concept A + ∅ = ∅ correct? A is any set of non-negative integers. I
think that if it is so, it contradicts the condition on the cardinality of sum sets
that |A| + |B| − 1 ≤ |A + B| ≤ |A| |B|. Please give your expert opinions.

I guess, Kemperman's theorem is for non-empty sets.

18
Is there any possibility for the sumset of two sets of integers, which are not arithmetic progressions, to be an arithmetic progression?

The notion of sumset of two sets is defined as A+B={a+b:a in A, b in B}. If the elements of A and B are not in AP,  then can the elements of A+B be an arithmetic progression? If so, what are the conditions required for that? Please can you recommend me some good references.

Don't know if that was already pointed out, but the example by Alex Ravsky is as "bad" as that neither A nor B contain a an arithmetic progression of length 3.

4
Why does the four colorability of planar graphs not ensure the non-biplanarity of K_9?

My earlier expectation regarding the sufficiency of 4CT to adress this issue, is not correct.

• Source
##### Article: An Analytic Proof of Four Color Problem
[Hide abstract]
ABSTRACT: Abstract – An analytical proof of the Four Color Conjecture has been described in this article. Kempe’s chain argument and Heawood’s technique to prove Five color theorem' has been exploited. Success has come through the searching of special triangles, around a vertex of degree 5, for three recursions. This proof stands on the principle of mathematical induction, so requires no computer assistance.

+ 1 more attachment

My latest publication in http://www.ijeijournal.com/pages/v4i6.html may help to give some insight on this issue.

4
Compute the number of path from a node to a different one in a weighted undirected connected 3-regular graph.
The graph is a weighted undirected connected 3-regular graph. The number of nodes is N.
For each node there is one loop with weight $= \frac{1}{N}, and two other edges which goes from the node to its two `nearest neighburs with weight$= \frac{\epsilon}{N}, where \^epsilon is a small parameter.
Therefore, we would like to know for a given value of $N$ how many different paths of lenght $2T$ are possible to go from one node to another passing through $k$ edges with weight $= \frac{\epsilon}{N}$ ?

I agree with Prof. Patrick Solé.  Maple is a very useful software for any kind of mathematical operations.

The number of paths of length n on the pair of given vertices (vi, vj) is the (i, j)th entry of An where A is the adjacency matrix of the given graph G.

You may refer refer the following link.

B Roberts, D P Kroese, Estimating the Number of s-t Paths in a Graph, http://jgaa.info/accepted/2007/RobertsKroese2007.11.1.pdf

Also visit the following page for a recursive algorithm to find all paths between two given vertices of a given graph.  http://www.technical-recipes.com/2011/a-recursive-algorithm-to-find-all-paths-between-two-given-nodes/

3
Are there any specific studies on the maximal bipartite sub-graphs of different products of two regular graphs?

Could you suggest me some of the articles on the maximal bipartite sub-graphs of different products of graphs, especially two regular graphs?

Dear Prof. Vitaly Voloshin, Thank you for the links...

4
Where can I find some real life or scientific applications of the sumsets of two sets of real numbers?

Please provide me some practical/real life applications for the sumsets of the sets of integers or real numbers. Please suggest me some useful references too..

The sum is sometimes a direct sum, like in coordinate axes. R2 = (1,0) R + (0,1) R and this fact has a lot of applications (!) Another interesting fact is that R = Z + [0,1] and the decomposition is again unique. This leads to the functions integer part and fractional part. Also interesting, the sum Z + sqrt(2) Z is dense in R. These are the most easy (trivial) examples, but there are a lot of nice sumsets of reals!

1
Is it possible to find an infinite arithmetic progression of value of k such that we can get primes of the form 6k+1 successively?

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

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

Even if true this is likely beyond current technology. It would imply Green Tao theorem:

http://en.wikipedia.org/wiki/Primes_in_arithmetic_progression

2
How to decrease the bounded gaps between primes?

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?

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:

http://terrytao.wordpress.com/2014/09/30/the-bounded-gaps-between-primes-polymath-project-a-retrospective/

Tao's complete corpus of work on that project is:

http://terrytao.wordpress.com/tag/polymath8/

9
Up to now, what is known on (the maximal) domains that guarantee the transitivity of the majority rule?
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?

Thank you very much. I also like the job done on mutiple issues.

1
How can we express the recurrence formula about the labeled tree in Maple16?

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

Very strange, first the gf then the object to count!

9
Are there any statisticians to collaborate on a combinatorics challenge for a paper?

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

1
How do I use the Ptolemy inequality to study the geodesic angle?
here, we try to use the ptolemy inequality to study the geodesic angle. and we introduce 2 preliminary results :

1. the ptolemy theorem in euclid spherical geometry
by studying the matrix of 4 quadruple points on the euclid spherical, we find the fact that, since there are 6 lines which connect these 4 points, so if 5 of the 6 lines are equal. then we will get that the sixth line's length is double as the other equally 5 lines;this result imply that the angle between the lines is included in $\frac{\sqrt{a}}{2r}$$\in(2k\pi+\pi/5,2k\pi+5\pi/6) ; then we substitute this result into the discriminant, and we get an inequality about the radius in n+1 dimension: op_{n+1}^2$$\ge$$\frac{r^2}{t^2}(1+\frac{r^2}{n})-\frac{r^2}{n}$

2. ptolemy inequality in minkowski geometry
here ,we use the centroid method to study the n-polygon problem in minkowski geometry. firstly, we introduce a well known problem such that:
if every angle of a polygon is equal, and the sidelines are :
$1^2,2^2,......,N^2$, then $\sum{n_{s}}e^{isa}=0$
by this theorem, we can factorize the mass on the vertex into pairs and the number of pairs is primes $N/2=\Pi{p_{i}^{a_{i}}}$ ,the weight of each pair is $\sum{4k-1}$,then we can divide these pairs into groups and every group has prime points too(page 21 in[1]); consequently we can divide the sidelines into 2 parts: $1^2,3^2,......$ and $2^2,4^2,......$;

the next step is to construct regular n-polygon and use the ptolemy inequality to make a regulation for the average of the sum of the mass in different group, and we can rearrange these groups of mass to ensure the first part $1^2,3^2,......$ is larger than the average and the second part $2^2,4^2,......$ is less than it.

therefore we can apply this average of sum to the distance formula in minkowski geometry in polar coordinate (page 24 in [1]).
here we also use combinatorics method (result we get in step 2) to study the natural logorithm in distance formula of minkowski geometry(page 25).

our goal is to represent the polar angle in minkowski geometry as the product of the mass lie on different vertex (page 26 in [1]).
so our question is how to apply the 2 results above to the geodesic angle?

by the inequality for the radius in $n+1$ dimension euclid spherical, we can ensure $v_{n+1}\ge0$; consequently we can substitute the representation of the polar angle in step 2 to spherical equation , which imply that we can also restrict the range of
$cos^2{\varphi}$ that is $[\sqrt{33}-4l,3]$

lastly,we apply the property of ptolemy space to get our estimate for geodesic angle,the bounde is :
$1-4e^2d/3+\frac{2}{3-4/3e^{2d}}$

is this method feasible? for more detail, you can refer to :

application of the ptolemy theorem (3) (page 15-27)
the analysis techniques for convexity: CAT-spaces (3)

This is a good question with lots of interesting paths to consider in looking for answers.    This question leads to a consideration of metric spaces that are Ptolemaic and Buseman convex spaces, stemming from

H. Busemann, The Geometry of Geodesics, 1955

A good place to start is

T. Foertsch, A. Lytchak, V. Schroeder, Non-positive curvature and the Ptolemy inequality:

http://carma.newcastle.edu.au/jon/Preprints/Papers/CAT(0)/Papers/fls-ptolemy.pdf

These authors prove that a metric space is CAT(0) if and only it is Ptolemy and Busemann convex, p. 11.  Ultrarays assocated with geodesics and their enclosed angles are considered, starting on page 8.