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This isn't research mathematics, but since there is an answer already I'll add one. The key issue for efficiency in this type of problem is to not generate partial potential solutions that are not pa …

Write the permutations as the rows of a matrix. For example $[(a,b,c),(a,c,b),(b,c,a)]$ becomes $$\begin{matrix} a & b & c\\ a & c & b\\ b & c & a\end{matrix}~~.$$ So far we just have an $n\times n$ …

Expanding on Igor's answer, if you search for "discrete uniform distribution" you will find lots of articles on it. For example this.

If I understand correctly, these are A001683 if turning over the $n$-gon is not allowed as an isomorphism, and A000207 if it is. In both articles there are formulae. You might also be interested tha …

The complete list of vertex transitive planar graphs was determined in 1979 by Fleischner and Imrich. See here. To quote: Theorem 3. The connected, simple, planar vertex-transitive graphs are the si …

This is about the generation problem, which is different from the theoretical enumeration problem. One method, which is not efficient for large sizes is to use the tools geng and multig from nauty: g …

Define $$ N(m,r,s) = \begin{cases} 0 & \text{if } r+s\gt m, \text{ otherwise} \\\\ 3^m-2 & \text{if } r=s=0 \\\\ 3^{m-s}-1 & \text{if } r=0,s\gt 0, \\\\ 3^{m-r}-1 & \text{if } r\gt 0,s=0, \\\ …

In this question we are counting labelled simple graphs. No concept of isomorphism is involved. Let $G(n,e,t)$ be the number of labelled simple graphs with $n$ vertices, $e$ edges, and $t$ sets of th …

We can give a construction showing that all counts up to $(1+o(1))\binom n3$ can be obtained, confirming Terry's probabilistic approach. Consider graphs consisting, apart from isolated vertices, of a …

There are a number of subtly different definitions out there. The following two papers enumerate types of quadrangulations of a disk, but I didn't check if the type you are interested in is included: …

This is closely related to Lucia's answer. For parameter $t$, let $X_t$ be the random variable with probability generating function $$ p_t(x) = \sum_{i=0}^c \mathrm{P}(X_t=i)\, x^i = \sum_{i=0}^c \fr …

The mean number of fixed points is 1. This is very elementary. Consider the operation of rotating three values around: $p(i)\to p(j)\to p(k)\to p(i)$. Given a permutation with no fixed points, there …

Let me start my answer by noting that this is fundamentally the wrong approach to the problem of reducing a large set of graphs by isomorphism type. The best software (nauty, Bliss, Traces) can put a …

As noted in the OEIS entry, the sequence is bounded below by the number of unlabelled rooted trees, so the radius of convergence can be only equal or less. For unlabelled rooted trees, the radius of c …

This is sequence A002816 at OEIS. You can find a recurrence and asymptotic expansion there. There is a summation for it in this old Stanford research report (end of page 6).