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(Too long for a comment). Note that there are two different ways to define "all the embeddings" of a graph, both completely natural. I'll illustrate by example. Suppose a graph consists of two triangl …

Given that an upper bound is requested, here is one. By symmetry one can replace $\frac1{k_1}+\cdots+\frac1{k_m}$ by $\frac1{k_1}$ and multiply everything by $m$. Now do the sum over $k_1$ (which I'll …

As $k\to\infty$, $\binom{2k-2}{k-1}\sim 2^{2k-2}/\sqrt{\pi k}$. As $N-k\to\infty$, $\binom{2N-2k}{N-1}\sim 2^{2N-2k}/\sqrt{\pi (N-k)}$. Now approximate the sum by an integral: $$\int_0^N \frac{\ln k} …

(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 …

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 …

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

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 …

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 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 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 …

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 OEIS sequence A181230. The square case $r=c$ is OEIS sequence A088310. See those pages for formulas. As Pat Devlin mentions, the asymptotic problem is trivial if both $r$ and $c$ increase qu …

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 …

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 …

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 …