Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [co.combinatorics]

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

15
votes
0answers
2k views

Covers of $Z^k$

This is a question related to covers of $Z^\infty$. Is it possible to cover $Z^k$, $k>1$, with the $l_1$-metric by a constant (not depending on $k$) number of collections of subsets $U^0,...,U^c$ ...
15
votes
0answers
858 views

Polynomials with presumably positive coefficients

After seeing that some positivity problems get their solutions on MO, I am quite enthusiastic of posing my (and not only) problem of positive flavour. In order to state it, I have to introduce the ...
15
votes
0answers
866 views

Is every k-edge-connected graph also k-trail-ordered?

This is an old question of Aradhana Narula-Tam and Philip Lin that I think deserves wider circulation. It appeared in Discrete Math. 257 (2002), page 613, but not many people have looked at it and it ...
15
votes
0answers
875 views

Optimal Monotone Families for the Discrete Isoperimetric Inequality

Background: the Discrete Isoperimetric Inequality Start with a set X={1,2,...,n} of n elements and the family $2^X$ of all subsets of X. For a real number p between zero and one, we consider a ...
14
votes
0answers
190 views

A symmetry of lattice paths

The number of $n$-step NSEW lattice paths from $(0,0)$ to $(a,b)$ that intersect the line $y=k$ precisely $t$ times is independent of $k$, for $0\leq k\leq b$, where we assume $b\geq0$ for simplicity. ...
14
votes
0answers
229 views

References on Discrete field theory vs Discrete differential geometry vs Combinatorial topology

Let me ask several related questions on discretization of classical field theory: In topological folklore, it is known that cochains are "discrete analogues" of differential forms, and coboundary ...
14
votes
0answers
348 views

The rank of a “triangle-free” matrix

This is a version of the question I asked recently, but the assumptions got now strengthened substantially. Suppose that $A=(a_{ij})_{1\le i,j\le n}$ is a square matrix with all elements in $\{0,\...
14
votes
0answers
709 views

A Combinatorial Game: the Snake and the Hunter

The Snake and the Hunter is a game for two players who play in two rounds interchanging the roles of snake and hunter. The game is played in a rectangular grid of points, say $6 \times 6$. In both ...
14
votes
0answers
224 views

Is combinatorial automorphism of symmetric convex polytope always antipodal?

The question is formulated in the title. More precisely, if $P$ is an origin-symmetric convex polytope in $\mathbb{R}^d$, and $f$ is a bijective transform of the set of the vertices of $P$, which ...
14
votes
0answers
432 views

Grothendieck on polyhedra over finite fields

In Grothendieck's Sketch of a Programme he spends a few pages discussing polyhedra over arbitrary rings and concludes with some intriguing remarks on specializing polyhedra over their "most singular ...
14
votes
0answers
367 views

Monotone embedding of complete binary tree in hypercube

Embedding different graphs, especially binary trees, in the hypercube has a huge literature. However, I could not find anything if we restrict the embedding to be monotone. So I would like to ...
14
votes
0answers
363 views

Do the coefficients of these irreducible polynomials always become periodic?

Fix $n\in\mathbb N$ and a starting polynomial (or seed) $p_n=a_0+a_1x+\dots+a_nx^n$ with $a_k\in\mathbb Z\ \forall k$ and $a_0a_n\ne0$. Define $p_{n+1},p_{n+2},\dots$ recursively by $p_r = p_{r-1}+...
14
votes
0answers
396 views

Splay trees and Thompson's group $F$

( I apologize for only indicating some easy to find references, but new users are not allowed to link more than five). This is very speculative, but: Question: Is there a reformulation of the Dynamic ...
14
votes
0answers
726 views

A Conjecture About Directed Graphs that are the Union of Two Trees

Let D=(V,E) be a directed graph that is the union of two edge-disjoint directed spanning trees. Suppose that There no subset X of vertices so that there is precisely one directed edge from X ...
14
votes
0answers
667 views

Capelli determinant = Duflo ( determinant) - was it known ?

Question briefly. Was this fact known: Capelli determinant = Duflo (determinant) ? (This is an equality of the two central elements in universal enveloping of Lie algebra $gl_n$). I googled a lot ...
13
votes
0answers
187 views

Is the set of power matrices decidable?

Let $\text{Mat}(n\times n,\mathbb{Z})$ denote the collection of integer $n\times n$ matrices. We say $M\in \text{Mat}(n\times n,\mathbb{Z})$ is a power matrix if there is an integer $k>1$ and a ...
13
votes
0answers
144 views

Generalization of Newton's identities to Schur functions

In some recent work, I've stumbled across the following identity for $\lambda \vdash n$: $$ n s_\lambda = \sum_{k=1}^n p_k \sum_{\mu \nearrow_k \lambda} (-1)^{\mathrm{ht}(\lambda/\mu)} s_\mu. $$ Here, ...
13
votes
0answers
2k views

Identifying poisoned wines, with a twist

(This is a joint musing with Andrew Gordon and Wyatt Mackey) There is a classic, elementary riddle, discussed before on MO and math.SE: suppose you have 1000 bottles of wine, and one is poisoned. The ...
13
votes
0answers
411 views

What is the geometric intuition behind Wilf-Zeilberger theory?

This problem is somehow inspired by a bunch of impressive posts of combinatorial identities by T. Amdeberhan. Earlier this month I learnt from computer scientists that they have a generic algorithmic ...
13
votes
0answers
319 views

A hard Lefschetz theorem for nilCoxeter algebras

Let $W$ be a finite Coxeter group and $\mathcal{N}(W)$ its nilCoxeter algebra (over the reals, say), as defined at https://en.wikipedia.org/wiki/Nil-Coxeter_algebra. $\mathcal{N}(W)$ has a natural ...
13
votes
0answers
535 views

Special configurations on a circle from a homological algebra problem

Here is the short version of the combinatorial problem: Given a positive integer $n \geq 2$. Draw a circle with $2n$ points indexed by the numbers from $\mathbb{Z}/ 2n \mathbb{Z}$. We colour the ...
13
votes
0answers
513 views

Chasing a 1950s thesis from the University of Dhaka on block designs

On behalf of a friend I am searching for a thesis on block designs from the 1950s. The details are below. Author: Qazi Motahar Husein (Sometimes Husain or Hussein). Title of the Thesis: Symmetrical ...
13
votes
0answers
248 views

Some $q-$analogues of $ \sum\limits_{j = - k}^k {{{( - 1)}^{ j}}}\binom{n}{k-j}\binom{n}{k+j}=\binom{n}{k}.$

Let ${\left( {a;q} \right)_n}=\prod\limits_{j = 0}^{n - 1} {(1-{q^j}a} )$ and let $ {{n}\brack{k}}_q$ denote a $q-$binomial coefficient. I am interested in $q-$analogues of the identity $ \sum\...
13
votes
0answers
382 views

How large must $A$ be if $\{1, \ldots, N\} \subseteq A-A$?

Given a positive integer $N$, what is the size of the smallest set of integers $A$ such that, for any integer $1 \leq k \leq N$, we can find two integers $x, y \in A$ such that $x - y = k$? (An ...
13
votes
0answers
223 views

Polytopes with few vertices and few facets

I recently realized that, for fixed $\alpha$ and $\beta$, the number of (combinatorial types of) $d$-polytopes with $\leq d+1+\alpha$ vertices and $\leq d+1+\beta$ facets is bounded by a constant that ...
13
votes
0answers
192 views

Computing exact or asymptotics for number of strings over an alphabet of size $n$ that have no non-trivial substrings that appear more than once

I ran across a seemingly relatively simple combinatorics problem that appears open. For an alphabet of size $n$, let $A(n)$ be the number of strings over the alphabet that have no substring of length $...
13
votes
0answers
335 views

Are the zeros of Tutte polynomials dense in $\mathbb C^2$?

For the chromatic polynomials of graphs we have two nice theorems which describe the behavior of their zeros: Thomassen proved that the set of real zeros of all chromatic polynomials is the union of $\...
13
votes
0answers
583 views

$\epsilon$-nets with respect to the cut norm

The cut norm $||A||\_C$ of a real matrix $A = (a_{i,j}) \in \mathcal{R}^{n\times n}$ is the maximum over all $I \subseteq [n], J \subseteq [n]$ of the quantity $\left|\sum_{i \in I, j \in J}a_{i,j}\...
13
votes
0answers
3k views

Minimum tiling of a rectangle by squares

Given the $n\times m$ rectangle, I want to compute the minimum number of integer-sided squares needed to tile it (possibly of different sizes). Is there an efficient way to calculate this?
13
votes
0answers
402 views

How much must deleting a spanning tree reduce edge-connectivity?

Suppose you have a 100-edge connected graph (e.g. an infrastructure network). You want to delete the edges of a spanning tree, any spanning tree you choose (e.g. to sell a connected subnetwork). What ...
13
votes
0answers
1k views

Finding a chromatic polynomial by polynomial fitting

I would like to find the chromatic polynomial χ for the n by m rook's graph Gn,m for as many values of n and m possible. The rooks graph is also (a) the line graph of the complete bipartite graph ...
12
votes
0answers
273 views

Escaping from a centralizer

Let $G = Sym(n)$, $n$ even. Let $H<G$ be the stabilizer of the partition $\{\{1,2\},\{3,4\},\dotsc,\{n-1,n\}\}$, or, what is the same, the centralizer of $(1\;2) \dotsc (n-1\; n)$. By Stirling's ...
12
votes
0answers
175 views

Non-Boolean Eulerian interval of finite groups

An Eulerian subgroup lattice is Boolean (see here), so it is natural to wonder whether it is also true for an interval of finite groups. The smallest non-Boolean Eulerian lattice is the following: It ...
12
votes
0answers
302 views

The number of labeled pairs of edge disjoint trees and related questions

I wonder what is known on the following: 1) What is the number $T_k(n)$ of $k$-tuples of (pairwise) edge-disjoint trees $(T_1,T_2,\dots, T_k)$ with $n$ labelled vertices? 2) (harder, it seems) What ...
12
votes
0answers
354 views

What is this sequence?

This is again a question that I asked at Stack Exchange, but got no answer so far, so I am trying here. Let: $$ a_n=\sum_{k\ge0}(k+1) {n+2\brack k+2}(n+2)^kB_k$$ $B_k$ is the Bernoulli number. ${n\...
12
votes
0answers
315 views

The threshold for a perfect matching in a random subgraph of a regular bipartite graph?

The following question seems very natural. It is a well known consequence of Hall's Theorem that every regular bipartite graph has a perfect matching. Another classical result states that the ...
12
votes
0answers
567 views

Pointwise (Hadamard) matrix product and the rank

$\DeclareMathOperator{\rk}{rk}$ Suppose that $A$ is a square matrix of order $n$. If, for any polynomials $P$ and $Q$ with $\deg P+\deg Q\le 2$, we have $$ P(A)\circ Q(A^t) = P(1)Q(1)\, I_n \tag{$\...
12
votes
0answers
246 views

Number of updown sequences of $1,1,2,2,\cdots,n,n$

I would like to count the updown sequences of the set $\{1,1,2,2, \cdots, n,n \}.$ Sequence $a_1, a_2, a_3, \ldots$ is an updown sequence if the sequence satisfies the following: $ a_1 \lt a_2 \gt a_3 ...
12
votes
0answers
425 views

Computer searches for the $g$-conjecture

McMullen's $g$-conjecture aims the classify possible $f$-vectors of simplicial $d$-spheres. The $g$-conjecture has been proven for polytopal spheres and for simplicial spheres of dimension $d < 5$. ...
12
votes
0answers
215 views

Reference request: exponential growth rates of subword-closed languages are integers

For a language $L$ over the finite alphabet $\Sigma$, let $L_n$ denote the set of words in $L$ of length $n$. The word $u$ is a subword of $w$ if $u$ can be obtained from $w$ by deleting letters. The ...
12
votes
0answers
188 views

Coefficients of universal Schubert polynomials

Let $e_i^j$ be the elementary symmetric polynomial in $x_1,x_2,\ldots,x_j$. Then the ordinary Schubert polynomial has an expansion of the form $$S_u(x)=\sum_{i_1,i_2,\ldots,i_n}{a^{i_1,i_2,\ldots,i_n}...
12
votes
0answers
187 views

Do the Laver tables converge to the Sierpinski triangle with a line segment sticking out in the hyperspace topology?

Let $(\{1,...,2^{n}\},*_{n})$ denote the $n$-th Laver table. Let $$C_{n}=\{(\frac{x}{2^{n}},\frac{x*_{n}y}{2^{n}})|x,y\in\{1,2,3,...,2^{n}\}\}$$ for all $n\in\mathbb{N}$. Then since $C_{n}$ is a ...
12
votes
0answers
398 views

Transitivity of balanced mass transport in Z

Given two atomic measures $\mu$ and $\nu$ on $\mathbb{Z}$, write $\mu \sim \nu$ iff there exist countable decompositions $\mu = \mu_1 + \mu_2 + \cdots$ and $\nu = \nu_1 + \nu_2 + \cdots$ along with ...
12
votes
0answers
200 views

Why must commuting maps (of an interval) without common fixed points have at least 11 fixed points for the composition?

I've been looking at the examples of commuting functions on a closed interval which have no common fixed points. These were discovered in 1967 by William M Boyce and J Philip Huneke. Earlier work by ...
12
votes
0answers
319 views

On some special spanning trees of grid graphs

I would like to know if there are existing results on the following objects: spanning trees of a grid graph, with no corridor where a corridor is a vertex having exactly two neighbors, on opposite ...
12
votes
0answers
467 views

A $q$-analogue of Foulkes' character related to alternating permutations

My paper "Alternating permutations and symmetric functions" at http://math.mit.edu/~rstan/papers/altenum.pdf enumerates certain classes of alternating permutations, such as those whose inverse is ...
12
votes
0answers
178 views

Does a huge set of random points in the plane almost surely have a checkerboard-triangulation

A set of $n$ points in the plane in generic position (no alignement of three points) has at least $2.012^n$ different triangulations of its convex hull involving only the set of given points.We call ...
12
votes
0answers
192 views

Proving that the set of $\lfloor n/3 \rfloor+1$ partial Latin squares given by Pebody is unavoidable?

Introduction Cutler and Öhman (2006) attribute to Pebody (via personal communication) a construction of a set of $k:=\lfloor n/3 \rfloor+1$ partial Latin squares which are unavoidable (i.e., any ...
12
votes
0answers
516 views

Possible orders of products of 2 involutions which interchange disjoint residue classes of the integers

Definition / Question Definition: Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $...
12
votes
0answers
375 views

Best known constant for parallel sorting

I recently found myself talking about Szemerédi's mathematics, and briefly discussed his famous sorting network, discovered with Ajtai and Komlós. Apparently their algorithm is not practical because ...