Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.

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10
votes
6answers
934 views

Algorithms for calculating R(5,5) and R(6,6)

Calculating the Ramsey numbers R(5,5) and R(6,6) is a notoriously difficult problem. Indeed Erdős once said: Suppose aliens invade the earth and threaten to obliterate it in a year's time unless ...
2
votes
2answers
211 views

binomial/factorial identity mod p

In trying to determine the spectrum of a well-known ergodic transformation, I came up with the following useful (for me) result. Let $p$ be a prime and $a$ a positive integer. Then for $M$ a positive ...
3
votes
1answer
84 views

$P_3$-factors for 3-regular, 3-connected cubic graphs

Suppose that $G=(V,E)$ is a simple graph. We know if $G$ is 3-regular, 3-connected and $|V|=4k$ for some $k\in \mathbb{N}$, then $G$ has a $P_4$-factor. Question. Let $G=(V,E)$ be 3-regular, ...
2
votes
2answers
178 views

Simultaneous lcms

Suppose that we have some finite number of $k$-tuples then we define the lcm of two of these tuples to be the tuple of lcms of the co-ordinates. E.g. $[(9, 10), (5, 18)] = ([9, 5], [10, 18]) = (45, ...
16
votes
0answers
291 views

Need explicit formula for certain “$q$-numbers” involving gcd's

The question is motivated by yet another possible approach to a combinatorial problem formulated previously in "Special" meanders. I'm not giving details of the connection as I believe the ...
7
votes
2answers
270 views

When is a sequence the sum of two Beatty sequences?

In other words, given a sequence $(s_n)$, how can we tell if there exist irrationals $u>1$ and $v>1$ such that $$s_n = \lfloor un\rfloor + \lfloor vn\rfloor$$ for every positive integer $n$? ...
1
vote
1answer
181 views

Counting matrices of special types

How many symmetric and non-symmetric $n\times n$ matrices with $0/1$ entries are there such that every row is distinct and every column is distinct? (I am looking for a proof as well). If only every ...
5
votes
2answers
237 views

Four Dimensional Rook Domination

Let $\gamma(G)$ denote the domination number of a graph, and $G\,\square\,H$ denote the cartesian product of two graphs. Then $K_8\,\square\, K_8$ is the rook graph, whose vertices are the squares of ...
3
votes
1answer
179 views

Largest symmetric matrix given rank

Let $\mathscr{M}[n,d]$ be collection of $n\times n$ symmetric matrices with real entries from $\{0,1\}$ such that every row/column is distinct with sum of every row/column being $d$. What is minimum ...
2
votes
1answer
43 views

Vanishing Restricted Isometric Constant

In compressed sensing, we are interested in the restricted isometry property. Suppose the design matrix is $n$ by $p$, consisting of $np$ iid $\mathcal{N}(0, 1/n)$ entries. Assume both $n$ and $p$ are ...
3
votes
1answer
153 views

higher dimensional analogue of EGZ theorem

The EGZ theorem states that any multiset of $2n-1$ integers has a subset of size $n$ the sum of whose elements is a multiple of $n$. Kemnitz-Reiher theorem is a 2-dimensional analogue of EGZ. Here is ...
9
votes
0answers
106 views

Is there a nice formula for the “non-crossing substitution” of linear combinatorial species?

Background A linear species is a functor $$F : \mathrm{Lin} \to \mathrm{FinSet},$$ where $\mathrm{Lin}$ is the category of totally ordered sets and bijections and $\mathrm{FinSet}$ is the category ...
4
votes
0answers
137 views

inequality in a shape of inclusion exclusion formula

I have two inequalities to show, both of which describe some probabilities. First I know how to handle, and it follows from applying arithmetic-harmonic mean inequality: consider 9 numbers ...
5
votes
0answers
130 views

Extrapolation between longest increasing and longest alternating subsequences

The question When should we expect Tracy-Widom? motivated me to post the following question, in which I have been interested for a while. Let $f(n)$ be a function from the positive integers to ...
4
votes
0answers
99 views

Generalization of Sprague-Grundy Theorem

In my research on Combinatorial Game Theory, I used a certain theorem that is essentially a generalization of the Sprague-Grundy theorem. Because the result hinges too much on the work of others to be ...
8
votes
3answers
652 views

Why does the bitxor function appear in Nim?

I am conducting research in Combinatorial Game Theory (CGT). Although I have done a considerable amount of reading, I do not completely understand why the bit-xor function also known as the nim-sum ...
0
votes
0answers
60 views

Minimum rank of certain matrices

Let $\mathscr{M}[n]$ be collection of $n\times n$ matrices with real entries from $\{0,1\}$ such that every row is distinct and every column is distinct. What is minimum real rank of matrices in ...
5
votes
1answer
139 views

partition of a convex set into squares

Let $P$ denote the perimeter function. It's not hard to prove that for any rectangle $R$ in $\mathbb{R}^2$, $R$ can be partitioned into a countable collection of squares $\{Q_k\}_{k=1}^{\infty}$ such ...
2
votes
1answer
165 views

Expected value (probability) maximization with binomial distribution

I need to solve an optimization problem that involves an expected value like $$F(n,x) = \sum_{k=0}^n \binom{n}{k} p^k(1 - p)^{n - k} f(k,x).$$ Here $f(k,x)$ is actually a probability coming from a ...
0
votes
0answers
60 views

Classic question on integer partitions (with distinct summands)

I guess that the following was solved sometime in the 18th century, but could not find a reference to it. I am interested in approximations to the following integer partition problem: Denote $R(N,L)$ ...
3
votes
1answer
146 views

Products of relative prime numbers with least sum

Let $P(n)$ be the set of subsets $P$ of $\mathbb{N}$ with the properties All elements of $P$ are relative prime to each other. The product of all $k \in P$ is greater or equal to $n$. Now let ...
10
votes
0answers
240 views

Reference request: a combinatoric result [closed]

When I tried to construct a counterexample in my research, I encountered the following result, which should be true. Let $m=m(n)$ be a function that grows faster than $\sqrt n$, so $m(n) = ...
0
votes
0answers
87 views

A sum on a minimums conjecture

Prove or disprove: $$m(n,k,s)=\sum_{a_1=1}^n \sum_{a_2=1}^n \cdots \sum_{a_k=1}^n \min(a_1, a_2,\cdots, a_k)^s =$$ $$ \sum _{i=0}^{k-1} \frac{(-1)^i}{i!} F(n,i+s) \sum _{j=0}^{k-1} \frac{\partial ...
2
votes
1answer
96 views

Probability of Hamming weight

Given $s,t\in(0,1)$, $c>1$, $n\in\Bbb N$, pick ${n^t}$ random vectors $\{v_i\}_{i=1}^{{n^t}}$ such that each $v_i\in\{x\in\{0,1\}^{2^n}:|x|_{hamming}={2^{n-n^s}}\}$. Denote $v_j\cap v_j$ to be ...
4
votes
1answer
124 views

Motzkin polynomials and enumeration of chord diagrams

On page 12 of the paper Enumeration of chord diagrams on many intervals and their non-orientable analogs" by Alexeev, Andersen, Penner, and Zograf is a list of polynomials which are a refinement of ...
5
votes
0answers
168 views

Unique Nash equilibrium games

Multicast network design game is a special case of a general network design game (http://www.cs.cornell.edu/home/kleinber/focs04-game.pdf) in which there is a target vertex $t$ and $n$ rational ...
2
votes
0answers
54 views

Vanishing of finite difference operators by composition under a cyclic condition

Consider $n$ finite difference operators $D_1, \ldots, D_n$ acting on real-valued functions $f_1 (y), \ldots , f_n (y)$ of a variable $y$, with the following properties: (i) $D_i f_i (y) = 0$ for ...
12
votes
2answers
291 views

Asymptotics of coefficients of implicitely defined generating function

I have two integer sequences $\{a_n\}_{n=0}^\infty$ and $\{b_n\}_{n=0}^\infty$. Explicit formulas for the $a_n$ are known and their asymptotic growth is fully understood. My wish is to also understand ...
10
votes
0answers
316 views

Combinatorial results by Poincaré duality

For the n-dimensional Torus, the k-th homology group (with integer coefficients) is isomorphic to the direct sum of $n \choose k$ copies of $\mathbb{Z}$. Poincaré duality thus gives us a somewhat ...
6
votes
2answers
389 views

Embedding of planar graphs

I've recently come across the following lemma. Lemma (Valiant): A planar graph $G$ with maximum degree $4$ can be embedded in the plane using $O(|V|)$ area in such a way that its vertices are at ...
2
votes
0answers
37 views

Catalogs/numbers/constructions of non-isomorphic conference matrices

I am interested in complete catalogs of non-isomorphic conference matrices, similar to those of Hadamard matrices. Do such catalogs exist? If yes, then where could they be found, and what is an ...
0
votes
1answer
46 views

Extracting path information for a directed acyclic graph

For a research problem I am tackling, I have a directed acyclic graph $G(V,E)$. With every node in $V$, I have a variable $y$ associated. Now, given two nodes $i$ and $j$, I would like to have the sum ...
2
votes
0answers
53 views

Linear intersection number of a product of graphs

A linear hypergraph is a pair $\pi=(X, L)$ where $X\neq \emptyset$ is a set and $L\subseteq {\cal P}(X)$ has the following properties: for $e\in L$ we have $|e|\geq 2$; if $e_1\neq e_2 \in L$ then ...
2
votes
1answer
130 views

Number of lines of symmetry of a set of lattice points

Given some finite $S\subseteq\mathbb R^2$, it is clearly possible for $S$ to have arbitrarily many lines of symmetry. However, it is not very clear if the same is necessarily true for subsets of ...
3
votes
0answers
99 views

“Standard” notation for symmetric functions?

Here is what I encountered in the paper "The Optimal Lattice Quantizer in Three dimensions" by Barnes and Sloane. Here is the setup: Let $\Lambda$ be a lattice in $\mathbb{R}^3$. Around each ...
1
vote
1answer
129 views

Selecting columns of a set of boolean matrices with constraint on the ones in each row

I've come up with the following question in my research: Let $S$ be a finite set of $n \times n$ matrices with elements 0 or 1. denote $n_i$ as the total number of 1's in the $i$th row of all matrices ...
8
votes
1answer
196 views

Cheeger Numbers for 3-regular Graphs

A student wanted a challenging Graph Theory programming project and I had him try to determine the maximum value of the Cheeger number (isoperimetric number) among all 3-regular graphs of order $n$, ...
3
votes
2answers
306 views

Geometry, Number Theory and Graph Theory of n-gon, permutation and graph labeling?

Given $n$ and $t$ lengths $ l_i, 1\leq l_1<l_2<\cdots<l_t\leq n-1$, of directed diagonals within an $n$-gon such that $l_1+\cdots+l_t\neq 0 \pmod n)$. Does it exist a directed path within ...
4
votes
1answer
76 views

How to realize any non-crossing matching as $\mathrm{Re}[p(z)]=0$

Asymptotically any polynomial is $p(z) = z^n + O(z^{n-1})$. Therefore $\mathrm{Re}[p(z)]= r^n \cos(2\pi i \theta)$ which vanishes at $\theta = \frac{(k+ \frac{1}{2})\pi}{n}$. Those $2n$ line ...
4
votes
0answers
93 views

Counting vertex-permutations of a finite tree which rip all edges

Given a finite tree $T$ with $n$ vertices labelled $1,\dots,n$, we say that a permutation $\sigma$ of $1,\dots,n$ rips all edges if $\{\sigma(i),\sigma(j)\}$ is never an edge for every edge $\{i,j\}$ ...
2
votes
1answer
93 views

Maximal neighbour-full partition of $\{0,1\}^n$

What is the largest complete minor of the $n$-dimensional hypercube? (which we call $k(n)$) Alternatively, what is the partition of $\{0,1\}^n$ with each set connected and neighboring each other that ...
1
vote
0answers
92 views

Evaluation of Macdonald Polynomials at $x_1=x_2=…=x_n = h$

This question is related to my previous question (here). Let $P_\lambda$(q,t) be the Macdonald polynomials with partition $\lambda$. Let $\Lambda$ denote the ring of symmetric functions over the ...
3
votes
0answers
85 views

A “nice” Orthogonal Basis for Translation Invariant Symmetric Polynomials

It is going to be a rather long question, so I will first state it and then try to explain and motivate it. Take $\Lambda_n $ as the graded ring of symmetric polynomials of a field $F$ in $n$ ...
36
votes
0answers
2k views

How to prove this polynomial always has integer values at all integers?

Let $m$ be any positive integer. $$ P_m(x)=\sum_{i=0}^{m}\sum_{j=0}^{m}{x+j\choose j}{x-1\choose j}{j\choose i}{m\choose i}{i\choose m-j}\frac{3}{(2i-1)(2j+1)(2m-2i-1)}. $$ Question: $P_m(x)$ always ...
10
votes
2answers
937 views

A new generalisation of Fermat's little theorem?

I would like to share a personal result, which I interpret as a "generalisation of Fermat's little theorem". I know that there exist already several generalisations (e.g. Euler's) but I would like to ...
3
votes
0answers
129 views

Algebraic integers with conjugates of fixed norm

Let $n$ be a positive integer and let $Q_n^d$ be the set of algebraic integers $\zeta$ which live in a degree $d$ extension of the rationals and such that any Galois-conjugate $\zeta^\sigma$ of ...
2
votes
1answer
91 views

sub-variety of (P^1)^4

Let $S$ be the sub-algebra generated by the set $\{x_1x_2y_3y_4, x_1x_3y_2y_4, x_1x_4y_2y_3, x_2x_3y_1y_4, x_2x_4y_1y_3, x_3x_4y_1y_2\}$ of homogeneous polynomials of degree $4$. I need to study the ...
4
votes
0answers
210 views

Every integer, and every positive integer?

Construct $A_k = (a_1, …, a_k)$ and $D_k = (d_1, …, d_k)$ inductively from $a_1 = d_1 = 0$, starting with $k = 1$, as follows: let $h$ be the least integer $>-a_k$ such that $h \not\in D_k$ and ...
2
votes
1answer
103 views

Representation numbers of numerical semigroups

I've been playing around with numerical semigroups lately. I'm pretty new to this stuff, so I apologize in advance if my notation is non-standard. Fix positive integers $x_1,\dots,x_r$ with ...
-1
votes
1answer
133 views

Number of Nice Matrices

Let's call a nice matrix a square matrix of size $n$ with elements from $\{1,...,n\}$ such that every row and every column contains all the number $1,...,n$, What is the number of nice matrices ? a ...