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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 …

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 …

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 …

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).

For $3\le d\le n-4$ the group size is almost always 1. The next most likely group size is 2, which most probably occurs due to a transposition (I don't know where this is proved formally). There is no …

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 …

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 …

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 …

In my comment I was misreading the question, sorry. The situation for the real question is as follows. For very low degrees (say, at most 3) it isn't hard to get the exact number as a single or doubl …

(A comment rather than an answer.) Here is a plot of $a_n/n^5$ (red) and $b_n/n^5$ (blue). It might not go far enough to show the asymptotic behaviour, but a possibility is that $a_n$ and $b_n$ are as …

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 …

It is a $k\times n$ latin rectangle: write the permutations one per row. This paper has a nice summary of theoretical and practical methods. The sum of the permutation matrices can be interpreted as …

If you search for "ranking $k$-ary trees" or "ranking $t$-ary trees" you will find several published papers on this. For example: This This

If you are looking for a simple formula, you are out of luck except for small $n$ or $T$. As Sam mentions in a comment, these are the integer points in a dilated Birkhoff polytope. Since the vertices …

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 …