Bell Numbers and Stirling Numbers of the Mycielskian of Trees
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917132031229952 |
|---|---|
| author | Allagan, J. Morgan, G. Sinclair, D. |
| author_facet | Allagan, J. Morgan, G. Sinclair, D. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_06980 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bell Numbers and Stirling Numbers of the Mycielskian of Trees Allagan, J. Morgan, G. Sinclair, D. Combinatorics Discrete Mathematics 05A18, 05C30, 05C15, 05C76, 68R10 G.2.1; G.2.2 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. |
| title | Bell Numbers and Stirling Numbers of the Mycielskian of Trees |
| topic | Combinatorics Discrete Mathematics 05A18, 05C30, 05C15, 05C76, 68R10 G.2.1; G.2.2 |
| url | https://arxiv.org/abs/2512.06980 |