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)

2
votes
0answers
30 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 ...
3
votes
2answers
66 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 ...
2
votes
0answers
70 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
votes
1answer
54 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
vote
0answers
59 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 ...
-5
votes
0answers
51 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
votes
2answers
494 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
votes
2answers
82 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 ...
9
votes
2answers
470 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
votes
1answer
113 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
vote
2answers
118 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
votes
1answer
47 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
votes
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
votes
1answer
186 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
votes
0answers
94 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
vote
1answer
73 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
219 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
138 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
184 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
votes
1answer
162 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
votes
1answer
235 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
votes
0answers
10 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 ...
1
vote
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 ...
-3
votes
0answers
38 views

To find the minimum no of games from the given average average may be ither integer or in decimal form [closed]

suppose a player plays no of games and he scores some score in his each games . And his average score is given we have to find how many minimum games he will play to score given average score . And ...
-5
votes
0answers
31 views

There are ten chairs in a row [closed]

There are ten chairs in a row. In how many of these will two people be sitting in adjacent chairs? I count 16. P1 and P2 must sit beside each other, adjacent, so there are only 8 chairs where that ...
4
votes
1answer
130 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
votes
4answers
211 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
votes
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
votes
2answers
272 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
votes
0answers
131 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
votes
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
votes
0answers
50 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
votes
1answer
211 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
votes
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
votes
0answers
108 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
votes
1answer
403 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 ...
-3
votes
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?
0
votes
0answers
43 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
106 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 ...
1
vote
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
124 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, ...
2
votes
1answer
79 views

Orthogonal embeddings and edge lengths

I'm interested in orthogonal embeddings of graphs into the 2-dimensional, i.e where vertices are placed at integer co-ordinates and edges are routed along the grid lines and are not allowed to ...
2
votes
0answers
107 views

Formulating shortest path as submodular minimization

I'm curious about the general question of whether any combinatorial optimization problem with polynomial time solution can necessarily be reformulated as minimizing a submodular function. The answer ...
2
votes
1answer
229 views

Estimate of the number of rabbit integers with a given congruence

Consider the Fibonacci words $B_n$: $B_1 = 1$ $B_2 = 10$ $B_3 = 101$ $B_4 = 10110$ $B_5 = 10110101$ (start with $B_1=1$, and go from $B_n$ to $B_{n+1}$ by replacing every occurence of $1$ in ...
2
votes
0answers
136 views

Polynomials representing locally constant functions

Let $K$ be a finite field with $p$ elements. (a) Let $f\in K\lbrack x\rbrack$ be such that (i) $\deg(f)<p$ and (ii) $f(2x) = f(x)$ for $\geq (1-\epsilon) p$ values of $x$ in $K$. What can we say ...
5
votes
0answers
178 views

Factorization of permutations into two factors with fixed number of cycles, plus a placement condition

In my recent work I have been led to consider the following type of permutation factorizations. Let $\pi_1$ and $\pi_2$ be two fixed permutations, with disjoint support, i.e. $\pi_1$ acts on ...
6
votes
3answers
310 views

Does a planar triangulation always contain a Hamiltonian path?

What about a Hamiltonian path in a triangulation of an n-gon? If not, how long is the longest path?