# Tagged Questions

**1**

vote

**1**answer

148 views

### A generalisation of Narayana-like numbers (walks on the 2D lattice)

I apologize in advance if this question has a trivial answer. I am pretty sure this kind of problem was already studied and I am mostly asking for good references.
Given integers $0 < k \le n+1,$ ...

**3**

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**0**answers

74 views

### Counting Problems where Labeled is Known but Unlabeled is Not

Cayley's formula states that the number of labeled trees on $n$ vertices is $n^{n-2}$. There are many nice proofs of this compact formula.
To contrast, counting unlabeled trees is considerably ...

**7**

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**2**answers

284 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 ...

**4**

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**1**answer

149 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 ...

**4**

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**3**answers

344 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 ...

**6**

votes

**1**answer

386 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 ...

**2**

votes

**1**answer

197 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 ...

**1**

vote

**2**answers

371 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 $?

**7**

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**1**answer

301 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 ...

**10**

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**2**answers

508 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 ...

**4**

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**1**answer

177 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 ...

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**0**answers

678 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" ...

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**0**answers

250 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 ...

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**0**answers

201 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 ...

**36**

votes

**6**answers

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 ...

**3**

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**1**answer

322 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 ...

**3**

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**1**answer

793 views

### Polya's theory of counting and commutative algebra

Hi. 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 ...

**12**

votes

**1**answer

742 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 ...

**7**

votes

**1**answer

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 ...

**19**

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**1**answer

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 = ...

**2**

votes

**4**answers

7k 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 ...

**13**

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**2**answers

655 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 ...