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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20 views

Has anyone seen these binary trees (Catalan-type related to the Gegenbauer polynomials and Motzkin paths)?

The OEIS entry A121448 enumerates binary trees with $n$ edges and $k$ vertices with outdegree 1. Has anyone seen these trees? The o.g.f. for this entry, $G(x,t)$, is essentially a discriminant ...
0
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0answers
47 views

Additional condition to the Bollobas theorem in extremal set theory

The Bollobas'1965 theorem is the following: If $A_1,...,A_n$ and $B_1,...,B_n$ are two sequences of subsets of $X=\{1,...,r\}$ such that $A_i\cap B_j = \emptyset$ if and only if $i=j$, then ...
1
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0answers
106 views

universality for large deviations?

This is a question about universality in probability theory, with combinatorics in mind. Consider a sequence of polynomials $P_n$ in one variable, with positive coefficients. Combinatorics is a large ...
8
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0answers
103 views

A seemingly simple combinatorial object that must have an easy generating function

One more question related to my earlier "Special" meanders. I am trying to isolate simplest problems related to it. Here is one. For a composition (i. e. a tuple of natural numbers) ...
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0answers
34 views

Arrange numbers? [on hold]

hello the question i would like to ask is very difficult as english is not my native language so here it goes...I need a program or exel chart that arrange numbers in a group in order to get the ...
1
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0answers
52 views

Sampling efficiently conditioned on linear constraints modulo both $\mathbb{F}_p$ and $\mathbb{F}_2$

Given a prime $p$ and positive integer $t \ll \log p$ (say $t = \sqrt{\log p}$), is there an algorithm that is polynomial time in $\log p$ to sample uniform $X, Y \in \mathbb{F}_p$ conditioned on the ...
6
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1answer
302 views

Does $|A+A|$ concentrate near its mean?

Fix $N$ to be a large prime. Let $A \subset \mathbb{Z}/N\mathbb{Z}$ be a random subset defined by $\mathbb{P}(a \in A) = p$, where $p = N^{-2/3 + \epsilon}$ for some fixed $\epsilon > 0$. My ...
2
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1answer
129 views

How many times does a simple symmetric random walk of length n return to the origin?

Consider the simple symmetric random walk on the integers starting from the origin of length $n$. More precisely, I will denote an $n$ step random walk $w$ as $$ w:= \omega_0 \omega_1 \ldots ...
7
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0answers
63 views

Factoring a multiset into a product of two multisets

Given a multiset $S$ of $mn$ numbers, how hard is to find multisets $A$ and $B$, of $m$ and $n$ numbers respectively, such that $$ S = \{ xy \mid x\in A, y\in B \}~~~~\text{(multiset sense)}, $$ or ...
6
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3answers
197 views

Numerical invariants for a graph or its complement that are bounded by some constant

I'm looking for numerical graph invariants that are bounded by a constant either for a graph $G$ or its complement $\bar{G}$. (The complement graph $\bar{G}$ has the same set of vertices as $G$ but ...
3
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0answers
141 views

A generalization of Rogers-Ramanujan identity

The generalized Rogers-Ramanujan identity has the following form $$\sum_{k_1\geq\cdots\geq k_r\geq 0}\frac{x^{k_1^2+\cdots +k_r^2+k_i+\cdots +k_r}}{(x)_{k_1-k_2}\cdots ...
0
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1answer
62 views

writing an integer as particular summation [on hold]

I think my question is an elementary question. Thanks for any help or comment. Is there any formula for the number of writting a natural number $n$ in a summation as follows, $n=a_1+\dots+a_k$, ...
1
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0answers
83 views

Optimization on Binomial coefficients

Suppose we are given integers $1\leq r\leq N$, we want to study the following $$ \max_{m_0+m_1=N,m_0,m_1\geq 1}\max_j \tbinom {m_0}{j}\tbinom {m_1}{r-j}. $$ $N$ is very large, for instance $N\geq ...
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0answers
52 views

How many ways are there to order the numbers from 1 to 25 so that no primes occur consecutively? [on hold]

I just had this on an exam and was wondering if I answered it correctly. My solution was to first order the 16 composite numbers in this range, giving 17 spots to insert the 9 primes. So we have ...
5
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2answers
513 views

mod 5 partition identity proof

I am looking for a proof that: $$\prod_\limits{m=0}^\infty \dfrac{1}{(1-x^{5m+1})}=\sum_\limits{i=0}^\infty \dfrac{x^i}{\prod_\limits{j=1}^i (1-x^{5j})}$$ The left hand side expands into: ...
8
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1answer
91 views

How many uniquely colored degree two vertices in 3-coloring of subcubic graph?

Is there a graph with maximum degree three that has 3 degree two vertices that must get the same (resp. different) color in every 3-coloring of the graph? I'm interested in any similar results as ...
10
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2answers
566 views

What is the probability that two random permutations have the same order?

I am interested in the orders of random permutations. Since the law of the logarithm of the order of a permutation converges to a normal law (for instance Erdös-Turan Statistical group theory III), ...
0
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1answer
120 views

Frankl's union-closed sets conjecture for infinite families

This question is motivated by Frankl's union-closet sets conjecture. Let $X$ be a non-empty set. We say that a family ${\cal A} \subseteq {\cal P}(X)$ is union-closed if $\emptyset\notin{\cal A}$ and ...
1
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2answers
119 views

question about literature in the field of Ramsey's theory [on hold]

i am searching for an on- line paper or a book, or maybe just a paper or a book which consists a proof of finite Ramsey's theorem for sets (not for graphs). i need a combinatorial proof which is not ...
2
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1answer
49 views

Induced matching of cycle

Definition: A graph $G$ is chordal if every induced cycle in $G$ has length 3, and is co-chordal if the complement graph $G^c$ is chordal.The co-chordal cover number, denoted $cochord (G)$, is the ...
2
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1answer
73 views

Discrepancy of elements in minimal members of a union-closed set

This question is motivated by Frankl's union-closet sets conjecture. Let $n\in\mathbb{N}$ and set $[n] = \{0,1,\ldots,n\}$. We say that a family ${\cal A} \subseteq {\cal P}([n])$ is union-closed if ...
0
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1answer
189 views

Order-Perserving Bijection $f:A\to A^*$?

Let $A$ be a well-quasi-ordered infnite set. Does there exist an order-preserving bijection $f:A\to A^*$, where $A^*$ is the free monoid over $A$ under the subword ordering? Would this subword ...
0
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0answers
100 views

Ramanujan graphs and stable real polynomials

I have a problem with the lemma 6.5 and the theorems 5.1, 6.6 in the paper of Adam Marcus, Daniel Spielman and Nikhil Srivastava. http://arxiv.org/pdf/1304.4132.pdf I do not understand how they use ...
1
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1answer
79 views

Vectors which average to zero over any graph neighborhood

Given an undirected connected graph on $n$ nodes, let $S$ be the subspace of vectors $x \in \mathbb{R}^n$ which satisfy $$\sum_{j \in N(i)} x_j = 0,$$ for all $i=1, \ldots, n$. Here $N(i)$ is the set ...
7
votes
2answers
255 views

Even parking functions and spanning trees of complete bipartite graphs

Set $\mathbb{N} := \{0,1,2,\ldots\}$. A parking function of length $n$ is a sequence $(\alpha_1,\ldots,\alpha_n) \in \mathbb{N}^n$ whose weakly increasing rearrangement $\alpha_{i_1} \leq \alpha_{i_2} ...
2
votes
2answers
142 views

Number of ways of tiling a $2 \times n$ rectangle using rectangles with integer sides

How many ways are there of tiling a $2 \times n$ rectangles using rectangular tiles with positive integer side lengths? I've done some work on this and have found a way of calculating this that's ...
5
votes
1answer
191 views

Does this notion related to species/operads/FI-modules have a name?

Let $B$ be the symmetric monoidal category of finite sets and bijections with disjoint union. Let $C$ be a symmetric monoidal category. Is there a standard name for a lax monoidal functor $F:B \to C$? ...
5
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1answer
163 views

How often can subsets of a universe intersect exactly once?

My question is inspired by the following observation: Claim: It is not possible to choose $n$ subsets of the universe $[n]$, each of size $\Omega(n)$, such that for each subset $S$ and each element ...
11
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1answer
241 views

A sum over characters of the symmetric group

Let $C_\mu$ be the size of the conjugacy class in $S_n$ of permutations whose cycletype is the partition $\mu\vdash n$. Let $\chi$ be the characters of the irreducible representations of $S_n$. Let ...
0
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0answers
11 views

Approximation preserving reductions

I've seen in the following document https://hal.archives-ouvertes.fr/hal-00958028/document A definition of the $\leq_{S}$ reduction defined specifically for minimisation problems at the bottom of ...
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0answers
44 views

Directed graph Laplacian with exactly one negative eigenvalue

Let $G$ be a digraph with adjacency matrix $A =(A_{ij})$ where $A_{ij}=1$ if and only if there is a directed edge $i \to j$ and $A_{ij}=0$ otherwise. Let $D= (D_{ij})$ be the degree matrix with ...
4
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1answer
132 views

Probability bound for perfect matching

Let $p<1$ be a constant. Consider two sets $A,B$, each with $n$ vertices. For each pair $(a,b)\in A\times B$, the edge between $a$ and $b$ appears with probability $p$, independently of the ...
3
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4answers
212 views

How small can a set system containing a large subset of every set be?

Fix $1>c>0$. Consider the set $[n]=\{1,2,\ldots,n\}$ and the set of all subsets of this set which we'll denote as $2^{[n]}$. Let $S \subseteq 2^{[n]}$ be a set system such that for every ...
2
votes
1answer
150 views

Extracting a full rank matrix from a 0-1 matrix

If $A$ is a $n\!\times\!n$ $0$-$1$ matrix of rank $k<n$. If ever possible, what would be an efficient way of extracting a full rank $k\!\times\!k$ sub-matrix of $A$ by removing columns and rows of ...
3
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1answer
69 views

Generalizations of the Triangle Removal Lemma to smaller exponents

The Triangle Removal Lemma states: For all $\epsilon > 0$, there is a $\delta > 0$ such that any graph on $n$ vertices with at most $\delta n^3$ triangles may be made triangle-free by ...
10
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2answers
275 views

Blocking sets in three dimensional finite affine spaces

What is the smallest possible size of a set of points in $\mathbb{F}_q^3$ which intersects (blocks) every line? Clearly the union of three affine hyperplanes that intersect in a singleton, say $x = ...
2
votes
2answers
365 views

Closed formula for the generating function of the sequence of powers

Does anyone know of a closed formula for the function $f_k(x)=\sum_{n=1}^{\infty}{n^k x^n}$ ? That is, the generating function of the sequence $1^k,2^k,3^k...$. It is not hard to see that ...
0
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0answers
133 views

Shortest distance on a combinatorial chess board

Consider a random $n\times n$ combinatorial $0/1$ square matrix over field $\Bbb F$ of rank $r$ with every row distinct and every column distinct as a chess board. Definition of combinatorial ...
0
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0answers
41 views

Inapproximability of Combinatorial Optimization Problems

I've been reading the "Inapproximability of Combinatorial Optimization Problems" by Luca Trevisan (see: link). On pages 3-4 it mentions that a polynomial time algorithm for 3SAT would exist if there ...
0
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0answers
52 views

Question on abstract polytopes

Let $(P,\le)$ be an abstract $n$-polytope, with $n\ge 2$. Let $H,H',K$ be $m$-faces, with $0\le m \le n-2$. Is it true that there is a sequence $\{H_0=H,H_1,...,H_{r-1},H_r=H' \} \subseteq P$ so that ...
2
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1answer
217 views

Generalization of the Bollobas theorem in extremal set theory

The Bollobas'1965 theorem is the following: If $A_1,...,A_n$ and $B_1,...,B_n$ are two sequences of subsets of $X=\{1,...,r\}$ such that $A_i\cap B_j = \emptyset$ if and only if $i=j$, then ...
4
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0answers
85 views

For which divisors $a$ and $b$ of $n$ does there exist a Latin square of order $n$ that can be partitioned into $a \times b$ subrectangles?

There exists a Latin square of order $8$ which can be partitioned into $2 \times 4$ subrectangles: $$ \begin{bmatrix} \color{red} 1 & \color{red} 2 & \color{red} 3 & \color{red} 4 & ...
4
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0answers
112 views

Reference for the notion of polyhedra “degenerations”

Let $P$ be a convex polyhedron and let $P(t)$ be a continuous deformation thereof, such that: a) $P(0)=P$; b) for all $t\in[0;1)$ the polyhedron $P(t)$ is strongly combinatorially equivalent to $P$ ...
2
votes
2answers
176 views

Batched Coupon Collector Problem

The batched coupon collector problem is a generalization of the coupon collector problem. In this problem, there is a total of $n$ different coupons. The coupon collector gets a random batch of $b$ ...
11
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1answer
407 views

Plugging $1-x$ into Schur polynomials

I have a symmetric Laurent polynomial $f$ in $k$ variables expressed as a linear combination of Schur polynomials. I'd like to know what happens when I make the substitution $p(x_1,\ldots,x_k)\mapsto ...
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2answers
131 views

Does every 3-regular bridgeless graph have a perfect matching? [closed]

Let $G$ be a simple $3$-regular (every vertex has degree $3$) $2$-edge connected graph. Does $G$ contain a perfect matching?
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44 views

Co-chordal graph properties

Definition: A graph $G$ is chordal if every induced cycle in $G$ has length 3, and is co-chordal if the complement graph $G^c$ is chordal. Suppose $G$ is co-chordal subgraph of $G_1$, where ...
3
votes
1answer
107 views

How different can characters be for a sum of modular forms to still be in Gamma_0?

I have a modular form I am constructing out of sums and products of various dissected divisor-sum series, namely forms of the type $$f_i = \sum_{j=0}^\infty \sigma_1(36j+i) q^{36j+i}.$$ Each of these ...
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0answers
23 views

Some confusion regarding the definition of NPO reduction

I've seen the following definition in a paper on approximation preserving reductions. Definition:Let $\pi_{1}$ and $\pi_{2}$ be two NPO maximization problems. Then we say that $\pi_{1} \leq_{R} ...
5
votes
2answers
126 views

Generalized cycle index polynomial for the symmetric group

The answer to a particular calculation in quantum information theory gives me the following expression: Given $M$ specific elements of the symmetric group $S_n$, define the polynomial $$Z_n(\pi_1, ...