# 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.

791 questions
177 views

### How distributive are the fake Laver tables?

The Laver table $A_{n}$ is the unique algebra $(\{1,...,2^{n}\},*)$ such that $x*1=x+1$ for $1\leq x<2^{n}$, $2^{n}*1=1$, and $x*(y*z)=(x*y)*(x*z)$. Let's now replace the Laver table $A_{n}$ with ...
265 views

### Is there an efficient algorithm to check whether two matrices are the same up to row and column permutations?

Define $\mathcal M_n$ as the set of all $n\times n$ matrices with each entry either 1 or $x$. Two such matrices are equivalent iff they can be obtained from each other by swapping pairs of rows and ...
317 views

### Bichromatic pencils

A pencil is a collection of some lines through a point, called the center of the pencil. If the points of the plane are colored, then call a pencil bichromatic if there is a color that is present on ...
296 views

### 0-1 matrix combinatorial problem

Let $M \in \{0,1\}^{n \times n}$ and let $r_i$ be its $i$-th row. Given constant $p \in (0,1/2]$, let the number of $1$s in each row be at least $p\,n$. Given constant $c \in (0,1)$, what is the ...
262 views

### hooks and contents: Part I

For a cell $\square$ in the Young diagram of a partition $\lambda$, let $h_{\square}$ and $c_{\square}$ denote the hook length and content of $\square$, respectively. R Stanley proved the following ...
1k views

226 views

### Graphs with constant edge imbalance

The imbalance of an edge $(u,v) \in E(G)$ of a graph $G$ is defined as $|d(u)-d(v)|$ ($d$ being, as usual the degree). (This concept was introduced by Albertson in 1997) I'm interested in the set of ...
223 views

1k views

### Sperner's lemma and Tucker's lemma

In their article "A Borsuk-Ulam Equivalent that Directly Implies Sperner's Lemma" (American Mathematical Monthly, April 2013), Nyman and Su write "[W]e are unaware of a direct proof that Tucker's ...
464 views

### Is every polynomial a factor of a trinomial?

We say that $T(X) \in \mathbb{Q}[X]$ is a trinomial if there exist $A,B,C \in \mathbb{Q}$ such that $T(X) = AX^n + BX^m + C$ for some $n \geq m \in \mathbb{N}$. Is it true that for each irreducible ...
222 views

### Permanent of Nakayama algebras

See https://en.wikipedia.org/wiki/Nakayama_algebra for the definition of Nakayama algebras and define the permanent of such an algebra to be the permanent of its Cartan matrix. (all algebras are ...
350 views

### Rigid Strongly Regular Graphs

I need a few examples of graphs that are strongly regular as well as rigid, i.e., have only the trivial automorphism. Any references to relevant literature would be appreciated. Thanks.
254 views

### Reference for partial Hadamard matrices

Definition. An $m\times n$ matrix is said to be a partial Hadamard matrix (let's say PHM) if its entries are chosen from $\lbrace -1, 1 \rbrace$ such that the dot product of each pair of row vectors ...
168 views

### Strongly minimal covers

Let $H=(V,E)$ be a hypergraph, that is $V$ is a set and $E\subseteq \mathcal{P}(V)$. We say that $C\subseteq E$ is a cover of $H$ if $\bigcup C = V$. A cover $M\subseteq E$ is said to be strongly ...
461 views

### Automorphism groups of indefinite non-unimodular integer lattices

Does anyone know of any papers in which structural aspects of the orthogonal group of some indefinite non-unimodular integral lattice are calculated? The exact lattice isn't so important and they don'...
150 views

### Matrices with only two different entries and maximal determinant

Define $\mathcal M_n$ as the set of all $n\times n$ matrices of full rank with each entry either 1 or $x$. I am interested in how big the determinant of such a matrix can be. For this, we define in a ...
205 views

### Matching with probabilistic edges

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 ...
277 views

### Counting / characterizing the isolated points of a particular algebraic variety

I'm not a professional geometer / topologist, so please thanks for your patience :) Setup The following questions are the first in a series of steps I'm undertaking in an attempt to break down a ...
151 views

168 views

### Covering a finite ring with arithmetic progressions

Let $n\geq 2$ be a positive integer and $k$ be a number between $1$ and $n$. Recently, I came across the following question about $\mathbb Z/n\mathbb Z$ and I wonder if it was studied before. I'd be ...
208 views

### Combinatorial 0-1 vector problem

Let $M \in \{0, 1\}^{n\times n}$. Given a constant integer $c \ge 2$, let the number of $1$s in each row be equal to $\frac{n}{c}$ (assuming $c$ is a divisor of $n$). Let $\mathcal{M}_c$ be the set of ...
181 views

### Is the Erdős–Rényi giant component result applicable here?

Consider a matrix whose elements are independently assigned a value $1$ with probability $p$ and a value $0$ with probability $1-p$. Define a cluster of cells as a maximal connected component in the ...
173 views

### Will this greedy algorithm always work?

Let p(n) be the number of unrestricted partitions of n. p(0) is taken to be 1. Let set 1 and set 2 be two empty sets. Here's an algorithm. Put p(n) into set 1. On each successive step, k=1,2,3,..., ...
109 views

### Does there exist any analogous result for site percolation?

This is a sequel to the question: Proof and interpretation of the following percolation theory result for $n\times n$ square grid In the paper: The Birth of the Infinite Cluster:Finite-Size Scaling ...
104 views