An inequality for Kruskal-Macaulay functions
| dc.creator | Ábrego, Bernardo M. | |
| dc.creator | Fernández-Merchant, Silvia | |
| dc.creator | Llano, Bernardo | |
| dc.date | 2008-09-21 | |
| dc.date | 2009-04-25 | |
| dc.date.accessioned | 2026-07-07T13:08:16Z | |
| dc.date.available | 2026-07-07T13:08:16Z | |
| dc.description | Given integers $k\geq1$ and $n\geq0$, there is a unique way of writing $n$ as $n=\binom{n_{k}}{k}+\binom{n_{k-1}}{k-1}+...+\binom{n_{1}}{1}$ so that $0\leq n_{1}<...<n_{k-1}<n_{k}$. Using this representation, the \emph{Kruskal-Macaulay function of}$n$ is defined as $\partial^{k}(n) =\binom{n_{k}-1}{k-1}+\binom{n_{k-1}-1}{k-2}+...+\binom{n_{1}-1}% {0}.$ We show that if $a\geq0$ and $a<\partial^{k+1}(n) $, then $\partial^{k}(a) +\partial^{k+1}(n-a) \geq \partial^{k+1}(n) .$ As a corollary, we obtain a short proof of Macaulay's Theorem. Other previously known results are obtained as direct consequences. | |
| dc.description | February 9th, 2009 version. The introduction was improved. Theorem 1 now establishes equality for some $n$. Corollary 2 (Björner and Vrećica Theorem) was added. Acknowledgements were added | |
| dc.identifier | https://arxiv.org/abs/0809.3549 | |
| dc.identifier | http://arxiv.org/abs/0809.3549 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/228366 | |
| dc.subject | Combinatorics | |
| dc.title | An inequality for Kruskal-Macaulay functions | |
| dc.type | text |