Freaks City

Definition Stirling numbers are fundamental combinatorial objects categorized into two distinct types: First kind: Counts the number of ways to partition a set of $n$ elements into $m$ circular permutations. Second kind: Counts the number of ways to partition a set of $n$ elemants into $m$ non-empty unordered subsets. Recurrence Relations Bot ...

The problem requires counting unlabeled unrooted colored trees with maximum independent set size falling in a given range, which is an extended variant of classic unlabeled unrooted tree counting, so we can adapt standard techniques for that problem. For unlabeled rooted trees, the generating function $F$ satisfies $F = x\mathcal{E}(F)$, where ...