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12
votes
1answer
812 views

Two to the power of a triangular number: bijections

The numbers $2^{n(n+1)/2}$ come up in various enumerative contexts. In addition to the trivial example (bit-strings of length $n(n+1)/2$) and the old example of domino tilings of Aztec diamonds (...
3
votes
0answers
128 views

Counting perfect matchings with integrals

Has anyone used the Joni-Rota-Godsil integral formula (see details below) to count perfect matchings of square-grid graphs, Aztec diamond graphs, hexagon-honeycomb graphs, etc.? (Or even just to ...
8
votes
1answer
147 views

Exact enumerations from two-dimensional stat mech models

Exact enumerations corresponding to the dimer model on a hexagonal grid, the dimer model on a square grid, and the four-vertex (aka square ice) model on a square grid are known, namely: lozenge ...
2
votes
2answers
438 views

Enumerating solutions to an underdetermined non-homogenous linear system of Diophantine equations

I have a large, under-determined system (60 equations and 116 unknowns) of linear Diophantine equations. I am aware of the algorithms typically used to solve these systems, which is not my question. ...
5
votes
3answers
526 views

Combinatorial counting with symmetry

Let $A$ be a set of objects where $|A|=n$. We want to count all the possible ways that we can arrange these objects into $n$ bags with exactly $n$ objects in each. We can reuse any object, however, no ...
3
votes
2answers
55 views

Enumerating Tri-vertex transitive polyhedra n > 3 faces

How many unique vertex transitive polyhedra exist where each vertex has 3 incident edges for polyhedra with n (= # faces) > 3 ?
7
votes
1answer
412 views

More asymptotics for trees

This is a follow up to my recent question on the asymptotics of A003238. Lucia gave a fine answer to that question, but as I hinted the 'real' problem I have in mind is slightly different, and I've ...
13
votes
1answer
540 views

Are the asymptotics of A003238 known?

Sequence A003238 of the OEIS counts ``rooted trees with $n$ vertices in which vertices at the same level have the same degree.'' The sequence, $a$, begins 1, 1, 2, 3, 5, 6, 10, 11, 16, ... and it is ...
1
vote
1answer
139 views

Truncated sums of symmetric polynomials; reference request for an algebraic derivation

EDIT: This is a case of being too wrapped up in a formulation ($e_j,p_i,$ and the like) to try something simple. It did not occur to me to pull exp to the outside in the weeks I have stared at this. ...
3
votes
0answers
102 views

Enumerating simple algebraic groups and their irreducible representations

Motivation Everything is over an algebraically closed field. Given a faithful representation $G \to \textrm{GL}(V)$, one may try to pin down what the group $G$ exactly is (e.g., the in the case of ...
5
votes
2answers
261 views

Number of unlabelled planar graphs

What are the best known bounds on the number of non-isomorphic (unlabelled) planar graphs on $n$ vertices? Is there a simple proof that this number is at most exponential in $n$?
3
votes
1answer
222 views

Are all symmetric idempotent Latin squares known?

Are all symmetric idempotent Latin squares known? There is such a square of order $n$ if and only if $n$ is odd. However, is there a classification of all of them? (The motivation for the question ...
14
votes
0answers
574 views

“Special” meanders

One of the open problems in combinatorics is enumeration of meanders. Here on MO I only could find them under the heading not-especially-famous-long-open-problems-which-anyone-can-understand. Since ...
4
votes
2answers
314 views

Estimate size of graph by taking random walks

Let $G$ be a connected, finite graph and let $v_0$ be a vertex of $G$. I'm interested in methods of estimating the number of vertices in $G$, based on local exploration only. What I have in mind is: ...
2
votes
2answers
521 views

Enumeration of labeled connected bipartite graphs given partite sets

What would be the closed-form expression defining number of all possible labelled connected bipartite graphs given $\mid X \mid = m, \mid Y \mid = n - m $?
3
votes
2answers
382 views

Counting polynomials with same coefficient sum and a given value at a point

Call an univariate polynomial $f(x) = \sum_{i=0}^{n}a_{i}x^{i} \in \Bbb{Z}[x]$ symmetric if $a_{i} = a_{n-i}$ and $a_{0} = a_{n} > 0$. For a given $\sum_{i=0}^{n}a_{i}$ and $a_{i} \geq 0$, how ...
0
votes
2answers
408 views

Morse matching with 0-cells and (n-1)-cells

Suppose one has a Morse matching (acyclic matching) of a poset $P$ of rank $n$ in which the only unmatched cells are 0-cells and $(n-1)$-cells, the same number $k$ of each one. If $P$ is connected ...
3
votes
1answer
294 views

bounded partitions and bounded signed partitions of integers

Define a bounded signed partition of length $m$ and of bounded height $h$ of an integer $n$ by a relation: $$n = \pm a_{1} \pm a_{2} \pm a_{3} \pm \dots \pm a_{m}$$ where each $a_{i}$ is a integer in ...
4
votes
1answer
189 views

Statistics on partitions equidistributed with number of even parts

Fix a positive integer $n$. For a partition $λ$ of $n$, let $e(λ)$ be the number of even parts in $λ$. Using bijections, we can show the statistic $e(λ)$ is equidistributed on the set of partitions of ...
7
votes
1answer
337 views

Non-enumerative proof that there are many simple permutations?

Terence Tao asked for a non-enumerative proof that a positive proportion of permutations are derangements and got a great answer. Inspired by this, I'd like to ask about another family of permutations....
18
votes
2answers
998 views

Is this similar to a known combinatorial identity?

(Apologies if this is too obscure.) In joint work with Izzet Coskun we came across the following kind of combinatorial identity, but we weren't able to prove it, or to identify what kind of identity ...
13
votes
2answers
765 views

How many triangulations of the genus $g$ surface on $n$ vertices?

By "a triangulation of $X$", I mean a simplicial complex whose geometric realization is homeomorphic to $X$. Tutte showed that the number of combinatorially distinct triangulations $t(n)$ of the $2$-...
3
votes
1answer
192 views

Number of isomorphism classes of triangulations of a convex polygon

The number of triangulations of a convex $n$-gon is $C_{n-2}$ the $n-2$nd Catalan number. What I am wondering, is if there is a way to enumerate the isomorphism types of these as graphs? I am ...
1
vote
1answer
314 views

Enumerating unlabeled trees with degree at most 3

Does anyone know if there is currently any research or any potential bounds on the number of trees on $n$ vertices with degree at most $3$? One can bound this above by $C_{n}$ the $n$th Catalan number,...
4
votes
1answer
204 views

Reference request: enumeration under group action

Is there a reference for the following lemma (which is useful in counting unlabeled k-trees)? It seems to me that it should be known, but I haven't been able to find it anywhere. Let $G$ be a finite ...
3
votes
2answers
161 views

Enumerating 0-1 finite boxes without null rays.

Here rays are called lines. Call $M(a_1,a_2)$ the number for matrices of length $a_1$ and height $a_2$, made of $0$ and $1$, having neither null vector nor null co-vector. In other words any line (row ...
24
votes
3answers
1k views

What can be proved about the Ramanujan conjecture using elementary means?

The Ramanujan conjecture states that the coefficients $\tau(n)$ in the identity $$q\prod_{m=1}^\infty(1-q^m)^{24}=\sum_{n=1}^\infty\tau(n)q^n$$ satisfy the inequality $|\tau(n)|\leq d(n)n^{11/2}$, ...
2
votes
1answer
165 views

Statistics on Lehmer codes

I am looking at words $\alpha_1 \ldots \alpha_n$, where $\alpha_j \in \{ 1, \ldots, j \}$. Thinking of $\alpha_j$ as a height, these words can be interpreted as left-to-right paths on the positive ...
18
votes
1answer
517 views

Salié permutations and fair permutations

In October 2010, I published a Monthly problem that introduced the concept of a fair permutation, which is a permutation $\pi$ such that for every $i$, either $\pi(i) > i$ and $\pi^{-1}(i) > i$, ...
4
votes
0answers
174 views

When can we determine an $f$-vector or rank-generating function from its unordered list of coefficients?

Let $f_i$ be the number of $i$-dimensional faces in a $d$-dimensional simplicial complex $\Delta $. Recall that the $f$-vector of $\Delta $ is the vector $(f_{-1},f_0,f_1,\dots ,f_d)$ where $f_{-1}=...
10
votes
6answers
647 views

max # of words with restricted total content

This is the sort of problems in combinatorics with a rather innocent look that turn out to be quite challenging - at least for a bunch of physicists! :) Suppose we have a multiset $\mathbf{M}$ on a ...
0
votes
1answer
96 views

Enumeration of quadrangulations with a boundary and simple faces.

I wish to enumerate all quadrangulations of a $2p$ gon with $n$ internal vertices. Quadrangles are required to have simple faces. Simple face means all four vertices of each quadrangle are distinct. ...
5
votes
0answers
178 views

How does the number of self-avoiding paths between two points scale, in a square/cubic lattice?

Consider two different infinite graphs, whose vertices are drawn from $\mathbb Z^2$ or $\mathbb Z^3$. Let $P_d : \mathbb Z^d \times \mathbb N \to \mathbb N$ for $d \in \{2,3\}$ be the function such ...
13
votes
5answers
908 views

Is the following invariant of rooted trees a complete invariant?

Recall that rooted trees may be generated by starting with a trivial rooted tree (just a vertex), along with the operations of grafting a number of trees (identify their roots) and adding a new vertex ...
5
votes
0answers
845 views

On “The Average Height of Planted Plane Trees” by Knuth, de Bruijn and Rice (1972)

I am trying to derive the classic paper in the title only by elementary means (no generating functions, no complex analysis, no Fourier analysis) although with much less precision. In short, I "only" ...
1
vote
0answers
262 views

An Pure intriguing counting problem of index sets

Hi Guys. The problem here seems like a homework, but I think that it is not that easy.It comes from a theorem I recently proved.The content of the theorem is not important, the issue is that I have no ...
19
votes
3answers
1k views

A generalization of the triangle counting problem for simple weighted graphs

One nice identity is that $$\operatorname{tr}(A^3)/6$$ counts the number of triangles of a graph with adjacency matrix $A$. It also implies that triangle counting in a graph can be performed in sub-...
0
votes
2answers
264 views

Enumeration Result [closed]

Hi I have a very soft question: What exactly is the definition of an enumeration result? Let say I want to enumerate some combinatorial structure and I came up with an equation for a generating ...
6
votes
0answers
217 views

Counting Selections of Entries such having an Extremal Permutation of length n^2+1

Let $S_{n^2+1}$ be permutations of length $n^2+1$. By Erdos-Szekeres Theorem. any $s \in S_{n^2+1}$ would have a monotone subsequence(increasing or decreasing)of length $n+1$. Say a permutation $s$ of ...
20
votes
1answer
819 views

A strange sum over bipartite graphs

While mucking around with some generating functions related to enumeration of regular bipartite graphs, I stumbled across the following cutie. I wonder if anyone has seen it before, and/or if anyone ...
26
votes
6answers
2k views

Combinatorial Morse functions and random permutations

This question has its origin in combinatorial topology. In the 90s R. Forman proposed a discrete counterpart of Morse theory. In his case, a Morse function on a triangulated space is a function ...
39
votes
6answers
3k views

Non-enumerative proof that there are many derangements?

Recall that a derangement is a permutation $\pi: \{1,\ldots,n\} \to \{1,\ldots,n\}$ with no fixed points: $\pi(j) \neq j$ for all $j$. A classical application of the inclusion-exclusion principle ...
9
votes
2answers
656 views

An identity involving an infinite integral with a sinh in the denominator

I recently encountered the rather appealing looking integral, which appears in the theory of random matrices : $$\int_{-\infty}^{\infty}\prod_{j=1}^{p-1}(j^{2}+z^{2})\frac{zdz}{\mathrm{sinh}(2\pi z)} ...
1
vote
1answer
687 views

Cayley's Theorem regarding marked trees

Hello, I have the following proof of Cayley's Theorem: Proof. This proof counts orderings of directed edges of rooted trees in two ways and concludes the number of rooted trees with directed edges ...
4
votes
1answer
435 views

Number of simplicial polytopes with a given f-vector

Plenty of very nice literature is available on the characterization of f-vectors of simplicial complexes of diverse sorts (results by Billera, Bjoerner, Kalai, Stanley, among others). I mention, as an ...
8
votes
1answer
2k views

Counting non-isomorphic graphs with prescribed number of edges and vertices

I'd love your help with this question. Let $n\geq3$ be a fixed integer. How many non-isomorphic graphs with $p$ vertices and $q$ edges are there where $p+q=n$? Thank you very much. Crossposted at ...
9
votes
0answers
166 views

What is known about the number of permissible simplicial complexes given the number of k-cells?

Motivation: I am working on a problem that reduces to finding simplicial complexes given some data (details are unnecessary), but all I have managed to wrangle from my input is the number of cells of ...
3
votes
2answers
984 views

Polya's theory of counting and commutative algebra

Do you know if there exist algebraic studies of the ring of the power series which emerge when using the theory of Polya for enumeration of sets with certain symmetries? For instance if some ideals ...
5
votes
3answers
2k views

Dissecting a square

Edited - some comments may now be out-of-date. I thought I had a complete set of solutions to this: ...
12
votes
1answer
784 views

Succesful applications of algebra in combinatorics

Hi. This may be a very general question. Are there any examples of problems in combinatorics which were open, but which found a solution when stated in algebraic terms? If yes, could somebody ...