Bell Numbers and Stirling Numbers of the Mycielskian of Trees

TL;DR

This paper derives explicit formulas for Bell numbers and Stirling numbers related to specific graph classes, including multipartite graphs and Mycielskian trees, revealing deep combinatorial connections and extending known results.

Contribution

It provides new explicit formulas for graphical Bell and Stirling numbers for classes of graphs like multipartite and Mycielskian trees, extending previous combinatorial results.

Findings

Explicit formulas for Bell numbers of complete multipartite graphs

Closed-form expressions for Mycielskian star graphs

Connections to OEIS sequences and combinatorial pattern avoidance

Abstract

We establish explicit formulas for Bell numbers and graphical Stirling numbers of complete multipartite graphs, complete bipartite graphs with removed perfect matchings, and Mycielskian trees. For complete multipartite graphs $K(n_1,\ldots,n_\ell)$, we provide a simplified proof that $B(G) = \prod_{i=1}^\ell \bell{n_i}$. We derive $B(K_{n,n} - M) = \sum_{k=0}^{n} \binom{n}{k} \bell{k}^2$ for removed perfect matching $M$, and for Mycielskian star graphs, $B(M(St_n); 3) = 2^n + 1$ and $B(M(St_n); 2n) = 2n^2 - 3n + 3$. Results extend to Mycielskians of arbitrary trees. Our computational verifications establish links between graphical Bell numbers and fundamental sequences in combinatorics and pattern avoidance, including identification of several OEIS entries: A000051, A096376, A116735, A384980, A384981, A384988, A385432, and A385437.