Sequential partitions of nonpure simplicial complexes
A \(d\)-dimensional simplicial complex \(\Gamma\) is partitionable when each facet \(F\) of \(\Gamma\) has a face \(\psi(F)\) such that \(\Gamma\) is the disjoint union of the intervals \([\psi(F),F]\). When \(\Gamma\) is shellable, any of its shellings naturally induces a partition. It is a result of \textit{A. Björner} and \textit{M. L. Wachs} [Trans. Am. Math. Soc. 349, No. 10, 3945--3975 (1997; Zbl 0886.05126)] that a partition of \(\Gamma\) induced by a shelling is such that the number of \(k\)-dimensional facets of \(\Gamma\) for which \(\psi(F)\) is \(i\)-dimensional is the term \(h_{i+1,k+1}\) of row \(i+1\) and column \(k+1\) in the \(h\)-triangle of \(\Gamma\), a refinement of the \(h\)-vector of \(\Gamma\) that encodes non-trivial information when \(\Gamma\) is non-pure. In addition, also in the case of a shelling-induced partition of \(\Gamma\), the intervals \([\psi(F),F]\), where \(F\) ranges over the facets of \(\Gamma\) of dimension at least \(r\) form a partition of the subcomplex \(\Gamma^{r,d}\) induced by the facets of \(\Gamma\) of dimension at least \(r\). In this article, related notions are studied but in the case of partitionable, non necessarily shellable simplicial complexes. In particular, the author calls a partition of \(\Gamma\) layer compatible when the intervals \([\psi(F),F]\) corresponding to the facets \(F\) of dimension at least \(r\) form a partition of \(\Gamma^{r,d}\) for all \(r\), and \(h\)-compatible when the number of \(k\)-dimensional facets of \(\Gamma\) such that \(\psi(F)\) is \(i\)-dimensional is equal to \(h_{i+1,k+1}\) for all \(i\) and \(k\). It is shown that \par (i) \(\Gamma\) has a layer-compatible partition if and only if it has an \(h\)-compatible partition, \par (ii) the partitionability of \(\Gamma^{r,d}\) for all \(r\) does not imply the existence of a layer compatible partition, \par (iii) the partitionability of \(\Gamma\) and, for all \(r\), of the pure sub-complexes of \(\Gamma\) induced by the \(r\)-dimensional faces does not imply the partitionability, for all \(r\), of \(\Gamma^{r,d}\), and \par (iv) the partitionability of \(\Gamma\) does not imply the partitionability of its sub-complexes induced by the \(r\)-dimensional faces.
scientific article; zbMATH DE number 7433386
scientific article