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

learn more… | top users | synonyms (1)

0
votes
0answers
2 views

Maximum connected components $0-1$ matrix

Let the notion of connected matrix be as in here Connected components $0-1$ matrices Here a matrix has one component means we can traverse from a matrix entry $(i,j)$ which is $1$ to any other one by ...
2
votes
2answers
47 views

Removing subtrees

Let $T$ be a complete infinite rooted binary tree. Is it possible to remove (infinitely many) subtrees of $T$ and get a subgraph $G$ such that: $G$ has no complete subtrees (the graph below any ...
1
vote
1answer
36 views

Combinations Question about the construction of some special sets

Let n and k be two given numbers. The goal is to choose n subsets from {1,2,…,n} such that the union of any k of these subsets is the set {1,2,…,n} and the union of any m < k of ...
3
votes
1answer
198 views

Open problems in compressed sensing

What are the main open problems in compressed sensing? I am interested in theoretical as well as in numerical point of view.
1
vote
1answer
189 views

How many k-subsets of the integers {1,…,n} sum to N?

Given the set of integers $S = \{1,..n\}$, how many subsets of $S$ with $k$ elements sum to $N\in \mathbb Z$?
0
votes
0answers
125 views

On matrix rank inequality

Let $A$ be a $\{0,1\}$ square matrix. Let $J$ be all $1$ matrix. Let $\bar{A}=J-A$. Is it possible for $rk_+(A)\geq c\cdot rk_+(\bar{A})^d-1$ and $rk_+(\bar{A})\geq c\cdot rk_+({A})^d-1$ for some ...
2
votes
1answer
88 views

Hopf structures on “pictorial” descriptions of permutations

There is a well-known Hopf structure on permutations, due to Malvenuto and Reutenauer. Here, the F basis elements are permutations in one-line notation, the product of two permutations is their ...
1
vote
0answers
38 views

Reducing enumeration of reduced words in general Coxeter groups to another #P-complete problem

There is a polynomial time algorithm that takes as inputs a Coxeter system $(W,S)$ with $S$ finite (but with $W$ not necessarily finite), say encoded as a Coxeter matrix, as well as two reduced words ...
3
votes
1answer
94 views

Balanced binary code that “resists” local decoding?

I am looking for a construction that can be stated as the following coding problem: a binary code with good distance ($d = \Omega(n)$ where codeword length is $n$) that "resists local decoding" in the ...
8
votes
1answer
307 views

Connected components $0-1$ matrices

Let $M$ be a $0-1$ matrix. Here a matrix has one component means we can traverse from a matrix entry $(i,j)$ which is $1$ to any other one by moving step of $(i\pm1,j),(i,j\pm1),(i\pm1,j\pm1)$ where ...
-4
votes
0answers
51 views

Derangement,recursion and circular permutation [closed]

N people are invited to a dinner party and they are sitting on a round table. Each person is sitting on a chair there are exactly N chairs. So each person has exactly two neighboring chairs, one on ...
7
votes
2answers
300 views

Numbers with all N-digit prefixes divisible by N

In base 10, the number 3816547290 contains every digit exactly once. When I take the first N digits, that substring is divisible by N. For example, 381 is divisible by 3, 38165 is divisible by 5, etc. ...
4
votes
1answer
236 views

Endomorphisms and almost all graphs

Is it known what fraction (almost all?) of graphs have a trivial endomorphism monoid? I can't seem to find any reference to the question. Maybe it's related to the question: what fraction of graphs ...
1
vote
2answers
150 views

Generalization of Bracketing (or one of its many equivalences)

I asked the following question on MathStackExchange, but I have not received any answers after almost 3 days. Although it may not be a research level question, I thought I could ask it here. *"Is ...
11
votes
2answers
280 views

What is the number of equitriangulations of the n-cube?

I wonder if this question has been considered before and if anything is known. My search attempts have failed so far. Let's consider the n-dimesnional cube, $[0,1]^n$, and let's call a simplex with ...
10
votes
2answers
483 views

Conjecture on maximum of symmetric combinatoric function

A curious symmetric function crossed my way in some quantum mechanics calculations, and I'm interested its maximum value (for which I do have a conjecture). (The question was first asked at math.SE, ...
3
votes
0answers
245 views

On a positivity property of Hall-Littlewood polynomials

Here's the new, more thought through version. Consider a sequence of nonnegative integers $\lambda=(\lambda_1,\ldots,\lambda_n)$ with $\lambda_i\ge \lambda_{i+1}+2$ (the weight $\lambda-2\rho$ is ...
4
votes
1answer
92 views

Determining the number of hamiltonian paths of $K_n-C_n$

I would like to know information regarding the function $h(n)$ where $h(n)$ is the number of hamiltonian cycles the graph $K_n$ has after removing the edges that make up a hamiltonian cycle of $K_n$. ...
11
votes
1answer
577 views

Lots of combinatorial interpretations of Catalan numbers

During a lecture I gave on Catalan numbers, I pointed out that that it is possible to give a continuum number of combinatorial interpretations of these numbers. See the solution to (f$^5$) on page 54 ...
2
votes
1answer
152 views

Edge density of triangle-free graphs

Let $G$ be a finite, simple, loopless graph with $|V(G)|=n$. We define its edge density as $$ed(G) := \frac{|E(G)|}{n \choose 2}.$$ Moreover we set $$d_n := \text{max}\big\{ed(G): G \text{ is a ...
13
votes
1answer
965 views

How to prove that the following double sum is always an integer?

I have verified the following double sum is always an integer for $s$ up to $1000$ via Maple. But I can not prove it. Proofs, hints, or references are all welcome. Thanks! ...
7
votes
1answer
154 views

A polytope with a bound on the sum of any $k$ variables

Let $2\le k\le n-1$ and define the polytope $$P_k(n) = \lbrace (x_1,\ldots,x_n) \in \mathbb{R}^n : -1\le x_{i_1}+\cdots +x_{i_k} \le 1 \text{ for all } 1\le i_1\lt\cdots\lt i_k\le n\rbrace.$$ There ...
10
votes
1answer
200 views

Partitions of $\mathbb{R}^+$ into subset closed by sum and product

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a "non trivial" solution of the Cauchy functional equation, i.e. $f$ is not of the form $f(x)=cx$ for any $c\in\mathbb{R}$ and satisfies the relation ...
4
votes
2answers
218 views

Marked chain polytope, has this been studied?

Fix $n$ and consider the polytope given by the inequalities $$x_i\leq x_j, \text{ and } 0 \leq x_i \leq a_i \text{ for all } 1\leq i<j \leq n,$$ where $a_i \leq a_i\leq \dots \leq a_n$ are fixed ...
7
votes
1answer
158 views

Universal graph homomorphisms

By a graph I mean a pair $G = (V, E)$ where $V$ is a set and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\}: a\neq b \in V\}$. A graph homomorphism between graphs $G, H$ is a map $f:V(G)\to V(H)$ such ...
0
votes
0answers
51 views

Colorful version of Fisher's inequality for block designs

Is there such a thing? I am thinking of Karatheodory and Tverberg analogues here.
3
votes
1answer
121 views

How to generate $n$ FP32 rationals s.t. no two distinct k-el. subsets have same sum?

First some Background: I have lots and lots of integer matrices, whose rows are $k$-combinations (without repetitions and sorted) of numbers from the set $S:=\{1,...,n\}$ and needed to be compared ...
2
votes
1answer
101 views

a question about points and line segments in the plane

Let $P=\{P_1,P_2\cdots P_n\}$ be a set of $n\geq 4$ points in the plane and $P_iP_j$ be the line segment connecting $P_i$ and $P_j$ that satisfy: $(1)$Any three points of $P$ are not on a line; ...
1
vote
1answer
48 views

Number of different normalized inner products

Let $u,v\in\{0,1\}^n$ be $0-1$ vectors with $n$ components. Let $I=\langle u,v \rangle$. Clearly $I$ can take values in $\{0,1,\dots,n-1,n\}$. How many different values can $$I'=\frac{\langle u,v ...
2
votes
0answers
144 views

Dehn-Sommerville relations for $\Delta$-complexes

Let $M$ be a closed, triangulated manifold of dimension $m$ and $K(M)$ be its triangulation. Let $f_i$ denote the number of $i$-simplices of $K(M)$. As proved by Klee the face numbers satisfy the ...
0
votes
0answers
59 views

Upper bound on number of cells created by varieties of co-dimension 1

Say I have polynomials $p_1,p_2,\dots,p_m$ in $\mathbb{R}^n$ (ie. over $n$ variables), each of degree $d$. Is there an upper bound on the number of "regions" created by the surfaces $p_i = 0$? Let's ...
0
votes
0answers
70 views

Explicit basis/weight vectors for irreducibles inside the plethysm $Sym^m(\bigwedge^p \mathbf(V))$

This is a follow up to this question about finding the multiplicities of irreducible representations restricted to Young diagrams of 2-columns or less, inside the plethysm $Sym^m(\bigwedge^p ...
9
votes
1answer
247 views

Cauchy-Schwarz proof of Sidorenko for 3-edge path (Blakley-Roy inequality)

Is there a "Cauchy-Schwarz proof" of the following inequality? Theorem. Given $f \colon [0,1]^2 \to [0,1]$, one has $$ \int_{[0,1]^4} f(x,y)f(z,y)f(z,w) \, dxdydzdw \geq \left(\int_{[0,1]^2} f(x,y) ...
11
votes
1answer
156 views

Fastest algorithm to compute the width of a poset

An colleague recently came to me with a problem concerning the scheduling of tasks in the presence of constraints (of the kind: task $x$ can't begin until task $y$ has been completed). It turned out ...
0
votes
0answers
63 views

Counting path generating sentences in a specific formal language

Given a formal grammar of a language or an Turing machine of the language, can we count the path that generating sentences of the language? For example, we know that if the grammar is context-free ...
1
vote
2answers
135 views

Proving a random bipartite graph contains a perfect matching

I have the following problem consider a random bipartite with vertex classes $A$ and $B$ of size $|A|=|B|=\mathrm{log}^{2}(n)$ graph in which every possible edge is chosen independently with ...
28
votes
8answers
2k views

Generalizations of the Four-Color theorem

The four color theorem asserts that every planar graph can be properly colored by four colors. The purpose of this question is to collect generalizations and strengthenings of the four color theorem ...
1
vote
1answer
61 views

The number of orderings of elements with order-of-appearance constraints

Consider $(q_1,q_2,...) \in Q$ non-intersecting sets of distinct elements $(e_{(i,1)},e_{(i,2)},...)\in q_i$. How many ways can one write down an ordering of all of the $\sum_j |q_j|$ elements s.t.: ...
6
votes
3answers
473 views

A question about symmetric matrix

Let $A= (a_{ij})_{ij}, 1 \leq i, j \leq n$ be a symmetric $n \times n$ matrix. Suppose (1) $a_{ij} \geq 0$ are real numbers; (2) The sum of each row $\sum_{j=1}^{n} a_{ij} = 1$ for $1 \leq i \leq ...
1
vote
0answers
73 views

A natural sum over multisets (expectation over multinomial)

I think this is a natural question but am not sure where to find resources. Consider the possible multisets arising from choosing $n$ times an item from one of $k$ categories. We can represent one ...
0
votes
0answers
48 views

Computation on Random Bipartite graphs

I'm looking at a random bipartite graph $K_{\omega(n)}*K_{\omega(n)}$ where $\mathrm{log}(n)\leq \omega(n) \leq n^{1/2}$, in which each of the $\omega(n)^{2}$ edges is placed randomly with probability ...
4
votes
2answers
133 views

Matroids relaxations of a given matroid

Let $\mathcal{M}$ be a rank-$d$ matroid on $[n]$. Say a matroid $\mathcal{N}$ is a relaxation of $\mathcal{M}$ if $\mathrm{rank}(\mathcal{N})=d$, $\mathrm{groundset}(\mathcal{N})=[n]$, and every ...
1
vote
0answers
57 views

Matchings in random bipartite graphs

I was wondering if anyone could point me in the direction of a text or paper which would help deal with the following problem Suppose i am given a $K_{\mathrm{log}(n)} \times K_{\mathrm{log}(n)}$ ...
6
votes
0answers
171 views

Minimal “basis” in $n$ dimensional unit cube

Let's $$ B^n=\{\bar\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_n)|\alpha_i\in \{0,1\}\};~~~~n=1,2,\ldots $$ and let's $$ C\subseteq B^n, $$ $$ S(C)=\{\bar\alpha\oplus\bar\beta\ | \bar\alpha,\bar\beta\in ...
2
votes
2answers
120 views

Number of *distinct* dot products of an integer vector by elements of a hyper-rectangle

Imagine a vector $\boldsymbol{v}$ composed of integers, and the set $S$ of all integer vectors within a hyper-rectange, with one corner at the origin and other at $\boldsymbol{m}$. In other words: $S ...
3
votes
1answer
87 views

Sum over integer compositions of $m$ with $n$ parts of a fixed monomial in the parts

Is there an explicit formula for the following quantity? $$f_m(a_1,\ldots,a_n):=\sum_{\substack{k_1+\ldots+k_n=m \\ k_1,\ldots,k_n\in \mathbb{N}}} k_1^{a_1}\ldots k_n^{a_n}\ ,\hspace{1cm} ...
0
votes
1answer
79 views

The weighting function for the infinite product of necklaces

Let us consider the limit $\lim_{n\to \infty}\prod_{p=1}^n N(p,a)$ where $N(n,a)$ is the number of fixed necklaces of length $n$ composed of $a$ types of beads. Let's rewrite the product in a way ...
2
votes
2answers
138 views

“Homomorphism fingerprint” for graphs

Let $G, H$ be simple, undirected graphs without loops. We say that $G, H$ have the same homomorphism fingerprint if $|\text{Hom}(X, G)| = |\text{Hom}(X, H)|$ for all graphs $X$. (By graph ...
1
vote
0answers
81 views

Properties and name of some polynomials

I have encountered in a problem some polynomials given by $P_k(x) = \prod_{j=0}^{k-2} (kx-j)$. I need to understand if these polynomials are known, and if they have certain special properties, as ...
1
vote
2answers
219 views

On the number of monic polynomials

Let $a,b,c,d\in\Bbb N$ with $c<b$. Let $N_+(a,b,c,d)$ be the number of monic polynomials $f\in \Bbb Z[x]$of degree $d$ with non-negative coefficients such that $$f(a)=b$$ $$f(0)=c$$ What is the ...