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 …

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 g …

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. …

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-exc …

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. …

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 iss …

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 befor …

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 …

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 …

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 …

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} ( …

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}{\m …

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, …

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 w …

Edited - some comments may now be out-of-date. I thought I had a complete set of solutions to this: Cut a square into identical pieces so that they all touch the center point. …