# Questions tagged [enumerative-combinatorics]

The tag has no usage guidance.

478 questions
Filter by
Sorted by
Tagged with
231 views

### Applying $\sum_i \partial_{x_i}$, $\sum_i x_i \partial_{x_i}$ and $\sum_i x_i^2 \partial_{x_i}$ to Schur polynomials

The operators $L_k=\sum_i x_i^k\frac{\partial}{\partial x_i}$, with integer $k$, take symmetric polynomials into symmetric polynomials. Is it known how to write the result of the application of $L_0$, ...
• 1,397
55 views

### How many rigid 4-regular graphs are there?

I am interested in any formulas for the number of globally rigid 4-regular graphs, or Laman graphs of degree at most 4, on $n$ vertices. The bound can be for graphs with labeled or unlabeled vertices.
• 18.2k
121 views

• 22.4k
1 vote
62 views

• 335
342 views

• 261
1k views

### On the finite sum of reciprocal Fibonacci sequences

I want to check if $$\left\lfloor \left( \sum_{k=n}^{2n}{\frac{1}{F_{2k}}} \right)^{-1} \right\rfloor =F_{2n-1}~~(n\ge 3) \tag{*}$$ where $\lfloor x \rfloor$ is th floor function. The Fibonacci ...
• 65
294 views

### Number of partitions of $n$ and number of different integers in 1-avoiding partitions

Consider the number of integer partitions of $n$, usually denoted by $p(n)$ and generated by $$\sum_{n\geq0}p(n)x^n=\prod_{k\geq1}\frac1{1-x^k}.$$ I have encountered an interesting enumeration. Take ...
• 41.7k
210 views

• 33
424 views

### Plane partitions as sums of determinants

Consider the Vandermonde's determinant computed by $$V(x_1,\dots,x_m):=\det(x_j^{i-1})_{i,j=1}^m=\prod_{1\leq i<j\leq m}(x_i-x_j).$$ The number of plane partitions in an $n\times m\times m$ box (...
• 41.7k
226 views

### Proof of a binomial identity

Computations with Maple suggest the following binomial identity \begin{equation*} \forall{p,j}: \sum_{k=j+1}^{p+1} (-1)^j \dfrac{1}{k}\binom{k-1}{j} = \sum_{k=j+1}^{p+1} (-1)^{k-1} \dfrac{1}{...
• 980
1 vote
90 views

### Existence of some lattice path connecting all given lattice paths

My daily work concerns analysis on metric spaces and some time ago it turned out that the problem I am dealing with boils down to a certain combinatorial problem. I've checked a lot of examples and it ...
• 127
211 views

This is a follow up on my earlier MO post. Let's recall the sets $$\mathbf{K}_n=\{\mathbf{k}\in\mathbb{Z}^n: k_i\geq0, k_1+\cdots+ k_n=n, k_1+\cdots k_i\leq i, \text{for all 1\leq i\leq n}\}$$ and $\... • 41.7k 5 votes 1 answer 389 views ### Catalan sequences vs composition sequences In the paper, A polytope related to empirical distributions, plane trees, parking functions, and the associahedron, Pitman and Stanley studied the$n$-dimensional polytope $$\Pi_n(\mathbf x)=\{y\in\... • 41.7k 0 votes 2 answers 173 views ### Asymptotic approximation of a convolution of binomial coefficients I would like to find the following limit which is somewhat similar to the usual Vandermonde's convolution for binomial coefficients. Define L as follows.$$ L \triangleq \lim_{N\to\infty} \frac{1}{2^... • 155 0 votes 0 answers 39 views ### Enumerating directed cacti by the number of vertices and edges Let's say that a directed cactus is a labeled directed graph, such that each vertex belongs to at most one simple cycle. In other words, it is a directed graph such that all its strongly connected ... • 1,131 1 vote 0 answers 38 views ### Probability of generating the same DAG after picking a topological ordering 3 times in a row Consider the following process: Choose a random permutation$p$of$\{1, 2, \dots, n\}$out of$n!$options. Choose a random directed acyclic graph$G$that has$p$as a topological ordering out of$...
• 1,131
A sequence $\{a_n\}_{n \geq 0}$ is said to be P-recursive if there exist polynomials $p_0(n), p_1(n), \dots , p_k(n)$ such that $$\sum_{i=0}^k p_i(n) a_{n+i}=0$$ for all $n \in \mathbb N$. Let $P$ ...