Bell Numbers and Stirling Numbers of the Mycielskian of Trees
J. Allagan, G. Morgan, D. Sinclair
TL;DR
The paper addresses the enumeration of graph partitions via graphical Bell numbers and graphical Stirling numbers for key graph families, notably complete multipartite graphs, complete bipartite graphs with a removed perfect matching, and Mycielskian trees. It develops explicit formulas, including B(G;k) and B(G) for multipartite graphs, and novel closed forms for K_{n,n}-M and Mycielskian stars, with B(K_{n,n}-M)=\sum_{s=0}^{n} \binom{n}{s} B_{n-s}^2 and B(M(St_n);3)=2^n+1, B(M(St_n);2n)=2n^2-3n+3. The results are embedded in a broader combinatorial context through OEIS identifications (e.g., A000051, A096376) and validated computationally, linking partition enumerations to classical sequences and pattern-avoidance phenomena. The work illuminates how graph topology governs partition counts and opens avenues for asymptotic analysis, spectral connections, and extensions to iterative Mycielskians with potential applications in clustering and combinatorial optimization."
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.