# 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.

1,586 questions
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### 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 $U^0,...,U^c$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...