The tag has no usage guidance.

learn more… | top users | synonyms

37
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 ...
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 ...
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 $|\tau(n)|\leq d(n)n^{11/2}$, ...
20
votes
1answer
803 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 ...
19
votes
1answer
1k views

Fibonacci, compositions, history

There are three basic families of restricted compositions (ordered partitions) that are enumerated by the Fibonacci numbers (with offsets): a) compositions with parts from {1,2} (e.g., 2+2 = 2+1+1 = ...
18
votes
1answer
504 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 $\pi^{-1}(i) > i$, ...
17
votes
2answers
962 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 ...
15
votes
2answers
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 ...
14
votes
2answers
219 views

Generating functions for objects with irrational sizes

A problem I'm investigating concerns a combinatorial class in which the 'atoms' have irrational sizes. It seems likely that this is something that has been considered before, but I haven't been able ...
14
votes
0answers
506 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 ...
13
votes
5answers
903 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 ...
13
votes
1answer
526 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 ...
13
votes
2answers
691 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 ...
13
votes
2answers
717 views

Is there any meaning to a “nice bijective proof?”

From Zeilberger's PCM article on enumerative combinatorics: The reaction of the combinatorial enumeration community to the involution principle was mixed. On the one hand it had the universal ...
12
votes
1answer
770 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 ...
10
votes
1answer
719 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 ...
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 ...
10
votes
1answer
578 views

Are there enough additive permutations?

I am hoping to learn enough about additive permutations to help with a number theory problem. These permutations also have connections to difference sets, orthomorphisms, transversals, and other ...
9
votes
8answers
2k views

Which came first: the Fibonacci Numbers or the Golden Ratio?

I know that the Fibonacci numbers converge to a ratio of .618, and that this ratio is found all throughout nature, etc. I suppose the best way to ask my question is: where was this .618 value first ...
9
votes
2answers
644 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)} ...
9
votes
2answers
288 views

Principal Order Ideals in the Weak Bruhat Order

Let $\sigma\in S_n$ be a permutation on $n$ elements, and $\mathrm{Inv}(\sigma):=\{(i,j) : 1\leq i<j\leq n\text{ and }\sigma(i)>\sigma(j)\}$ be its set of inversions. In the weak order on ...
9
votes
0answers
99 views

Cycles of length $2^n - 2$ in the De Bruijn graph

It is well known that the number of (cyclic) De Bruijn sequences is $2^{2^{n-1}-n}$. This number may also be interpreted as the number of cycles of length $2^n$ in the De Bruijn graph of order $n$. ...
8
votes
1answer
146 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 ...
7
votes
1answer
1k 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 ...
7
votes
2answers
747 views

Has anyone tabulated 2-knots? Would anyone like to try?

I'd love to have a list of 'small' 2-knots, for some sense of small. It's not clear what one should filter by, but there are two obvious candidates Write a movie presentation, and count the frames. ...
7
votes
2answers
364 views

What is the number of noncrossing acyclic digraphs?

A noncrossing graph on $n$ vertices is a graph drawn on $n$ points numbered from $1$ to $n$ in counter-clockwise order on a circle such that the edges lie entirely within the circle and do not cross ...
7
votes
1answer
271 views

Number of median graphs?

What is the number of $n$-vertex median graphs? These graphs generalize hypercubes and trees, and have many applications. It seems unlikely that a closed form expression is known, so I would also be ...
7
votes
1answer
328 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 ...
7
votes
1answer
2k views

Number of graphs with a given number of nodes, edges and triangles

Hi. Does anyone know if it is possible to enumerate the set of labeled/unlabeled graphs (loopless, undirected, only one edge between pairs of nodes) having a given number of nodes, edges and ...
7
votes
1answer
405 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 ...
7
votes
0answers
157 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 ...
6
votes
1answer
279 views

A parity counting problem for subsets over finite fields

Let ${\mathbb F}_p$ be the prime field of $p$ elements and $b$ be an element in ${\mathbb F}_p$. For a subset $T\subseteq {\mathbb F}_p$, define $$Bias(T)=|N_e( {\mathbb F}_p,b)-N_o( {\mathbb ...
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 ...
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: ...
5
votes
2answers
251 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 $n$?
5
votes
1answer
182 views

Set counting problem with a cap on the intersection between the set and a fixed partition

Fix sets $T_1,\ldots T_m$ as a $k$-partition of $[m\cdot k]=\{1,2,\ldots,m\cdot k\}$, so that $|T_i|=k$ and $|T_i\cap T_\ell|=0$. 1) For any $j\le k$, how many sets $C\subset [m\cdot k]$ are there ...
5
votes
0answers
166 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 ...
5
votes
0answers
807 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" ...
4
votes
1answer
114 views

Determining the number of hamiltonian paths of $K_n-C_n$

I would like to know information regarding the function $h(n)$ where $h(n)$ is the number of hamiltonian cycles the graph $K_n$ has after removing the edges that make up a hamiltonian cycle of $K_n$. ...
4
votes
3answers
476 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 ...
4
votes
2answers
289 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: ...
4
votes
1answer
200 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 ...
4
votes
1answer
185 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 ...
4
votes
0answers
172 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 ...
3
votes
4answers
8k views

Non-isomorphic graphs of given order.

It is well discussed in many graph theory texts that it is somewhat hard to distinguish non-isomorphic graphs with large order. But as to the construction of all the non-isomorphic graphs of any given ...
3
votes
2answers
375 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 ...
3
votes
1answer
172 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 ...
3
votes
1answer
278 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 ...
3
votes
3answers
274 views

Are there infinite sets of stellations of polyhedra?

Lists of stellations of polyhedrons are given particular rules like in the book The Fifty Nine Icosahedra which follows "Miller's Rules". There seems to be no "correct" ruleset to use, so more ...
3
votes
2answers
969 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 ...