# 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.

7,207 questions
Filter by
Sorted by
Tagged with
191 views

### Sum of multi-index factorials

Fix $d\in\mathbb{N}\setminus\{0\}$. For $j\in\mathbb{N}\setminus\{0\}$, let \begin{align*} [j] = \Big\{\alpha\in \mathbb{N}^d: \sum^d_{i=1}\alpha_i=j\Big\}. \end{align*} For $\alpha\in[j]$, define ...
97 views

### Number of sequences satisfying termination conditions

This question was originally posed on math.SE but seems to require research-level mathematics expertise: Two players play each other in a match of games of chess where the match winner is the first ...
71 views

### Getzler's stable graphs for modular operads

In The semi-classical approximation for modular operads, Getzler displays a table at the bottom of page two enumerating certain stable graphs. (This is related to the MO-Q "Stable graphs: Feynman ...
135 views

### A diagonal generating function for Fibonacci: Part II

In my earlier MO question, I mentioned although we have for the Fibonacci numbers that $$F_n=[x^n]\left(\frac1{1-x-x^2}\right),$$ is there a function $F(x)$ such that $F_n=[x^n]\left(F(x)\right)^n$? ...
105 views

### Number of spanning trees in third power of cycle

The following page https://oeis.org/A005822 gives the number of spanning trees in third power of cycle, for example, $1, 1, 2, 4, 11, 16, 49, 72, 214, 319, 947, 1408,\ldots$. It is unclear for ...
99 views

### Bijections of Littlewood-Richardson coefficients

Let $c^{\lambda}_{\mu\nu}$ be the Littlewood-Richardson coefficients, where $\lambda,\mu,\nu$ are partitions. We know that $c^{\lambda}_{\mu\nu}= c^{\lambda}_{\nu\mu}$. Up to now, what are the ...
69 views

55 views

### What is the minimal $m$ for which the independence graph is $n$-universal?

Suppose, an $m$ sided die is rolled. Let's define the independence graph $I_m$ as a graph with the set of all possible events as vertices, and edges between two events iff they are independent. ...
149 views

125 views

### Explicit upper bound on the number of simple rooted directed graphs on 𝑛 vertices?

Harary mentioned this problem in "The number of linear, directed, rooted, and connected graphs" on p. 455, l. 3–5, but a short and crisp upper bound is missing. I believe that someone must have ...
215 views

### Sum from combinatorics on nonnegative integer numbers

Let $n_1,n_2,\ldots,n_k\in\{0,1,2,\ldots\}$. Can you calculate the sum $$\sum_{n_1,n_2,\ldots,n_k\geqslant0}\mathbb{1}_\left\{n_1+\frac{n_2}{2}+\ldots+\frac{n_k}{k}<1\right\}?$$ If it's helpful, ...
75 views

82 views

92 views

### submodules of the exterior algebra

Let $A_{n,q}$ be the exterior algebra of a vector space of dimension $n$ over the finite field $F_q$. Let $a_{n,q}$ be the number of submodules of $A_{n,q}$ (meaning submodules of the $A_{n,q}$-...
Let's start with a warm up problem. Suppose I am given two binary vectors $x, y \in \{0, 1\}^n$ of length $n$ that differ in exactly $r$ places, i.e. $||x-y||_0=r$, i.e. their hamming distance is $r$...