# 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.

6,504 questions

**0**

votes

**0**answers

52 views

### Calculating the number of solutions of integer linear equations

Let $N$ be a natural number. Consider the following set of matrices whose entries are non-negative integers:
$$X_N:=\left\{(c_{ij})_{i,j=1}^4\in M_4(\mathbb{Z}_{\geq 0})\bigg| \sum_j c_{1j} = \sum_i ...

**1**

vote

**1**answer

32 views

### Constructing Group Divisible Designs - Algorithms?

I am starting my research on group divisible designs this year and I wonder if there are any algorithms/software that help with constructions.
Thank you

**-4**

votes

**0**answers

60 views

### Expected number of times of choosing a word out of a given vocabulary when words are grouped into non-overlapping bins

Two players (player C and player G) are playing a (modified) word guessing game. Both players share the same vocabulary $V$ and words in $V$ are grouped into $K$ bins, denoted as $b_1$, $b_2$, ..., $...

**0**

votes

**0**answers

40 views

### Infinite products from the fake Laver tables-Now with no set theory

We say that a sequence of algebras $(\{1,\dots,2^{n}\},*_{n})_{n\in\omega}$ is an inverse system of fake Laver tables if for $x\in\{1,\dots,2^{n}\}$, we have
$2^{n}*_{n}x=x$,
$x*_{n}1=x+1\mod 2^{n}$,...

**0**

votes

**0**answers

89 views

### How many solutions to $p_i|a_{i,1} p_1 + \dotsc + a_{i,n} p_n$?

Consider a system of $n$ divisibility conditions on $n$ prime variables:
$$p_i|a_{i,1} p_1 + \dotsc + a_{i,n} p_n,\;\;\;\;\;1\leq i\leq n,$$
where $a_{i,j}$ are bounded integers. How many solutions ...

**5**

votes

**1**answer

88 views

### Sufficient conditions for the coefficients of a generating function to dominate those of its square

Let $f(z)$ be a generating function (so in particular, its power series coefficients are nonnegative). I am interested in conditions which would ensure that for every $n$, the coefficient of $z^n$ in $...

**-6**

votes

**0**answers

24 views

### combinatorics question with a deck of cards [on hold]

I have a standard deck of 52 cards. How do I find the number of hands of 13 cards that contain 4 cards of the same rank? (A,2-10, J,Q,K)
I'm starting off with $${48\choose 9}^{13}$$ but I am unsure ...

**2**

votes

**1**answer

133 views

### Alternating binomial-harmonic sum: evaluation request

Let $H_k=\sum_{j=1}^k\frac1j$ be the harmonic numbers.
QUESTION. Can you find an evaluation of the following sum?
$$\sum_{a=1}^b(-1)^a\binom{n}{b-a}\frac{H_{b-a}}a.$$

**6**

votes

**0**answers

148 views

+200

### On a problem for determinants associated to Cartan matrices of certain algebras

This is a continuation of Classification of algebras of finite global dimension via determinants of certain 0-1-matrices but this time with a concrete conjecture and using the simplification suggested ...

**2**

votes

**0**answers

42 views

### Are contraction-sensitive graphs necessarily vertex-transitive?

We say that a finite, simple, undirected graph $G=(V,E)$ is contraction-sensitive if collapsing any $2$ non-adjacent points increases the Hadwiger number. An example of such a graph is the icosahedron....

**2**

votes

**1**answer

75 views

### Algorithm to compute the convex hull of a set of $m$ possibly intersecting convex polygons in the plane

I am trying to find an algorithm to compute the convex hull of a set of $m$ possibly intersecting convex polygons in the plane, with a total of $n$ vertices. Let $h$ denote the number of vertices on ...

**3**

votes

**1**answer

112 views

### Determinant of an “almost cyclic” matrix

Let $n\geq 3$, let $Z$ be the matrix of the cyclic shift (the companion matrix of $X^n-1$), and for $\mathbf{d}\in \mathbb{C}^n$ let
$\operatorname{diag}(\mathbf{d})$ be the diagonal matrix with $\...

**2**

votes

**2**answers

91 views

### Does any long path in a planar graph contain one of O(n) k-tuple of vertices?

My question is a bit related to both the container method and shallow cell complexity.
Let's start with that the number of length $\ell$ paths (where $\ell$ denotes the number of vertices of the path!)...

**2**

votes

**0**answers

28 views

+50

### Does the bounded branching/log depth dihotomy hold for rooted trees?

Let $T$ be a rooted tree. For any subtree $T' \subset T$ define its leaf-weight $lw$ as the number of leaves of $T'$.
Further, for $T' \subset T$ define the branch-depth of a node $v \in T'$ as the ...

**8**

votes

**0**answers

53 views

### Minimizing union of overlapping rectangles

Believe it or not, this has something to do with making triangle-free graphs bipartite...
We have a collection of $k$ axis parallel rectangles with side lengths $(a_i,b_i)$. We want to arrange them (...