421 research outputs found
Rees products and lexicographic shellability
We use the theory of lexicographic shellability to provide various examples
in which the rank of the homology of a Rees product of two partially ordered
sets enumerates some set of combinatorial objects, perhaps according to some
natural statistic on the set. Many of these examples generalize a result of J.
Jonsson, which says that the rank of the unique nontrivial homology group of
the Rees product of a truncated Boolean algebra of degree and a chain of
length is the number of derangements in .\Comment: 31 pages; 1 figure; part of this paper was originally part of the
longer paper arXiv:0805.2416v1, which has been split into three paper
Operads of compatible structures and weighted partitions
In this paper we describe operads encoding two different kinds of
compatibility of algebraic structures. We show that there exist decompositions
of these in terms of black and white products and we prove that they are Koszul
for a large class of algebraic structures by using the poset method of B.
Vallette. In particular we show that this is true for the operads of compatible
Lie, associative and pre-Lie algebras.Comment: 16 pages, main result about Koszulness generalized to a large class
of compatible structure
Some applications of Rees products of posets to equivariant gamma-positivity
The Rees product of partially ordered sets was introduced by Bj\"orner and
Welker. Using the theory of lexicographic shellability, Linusson, Shareshian
and Wachs proved formulas, of significance in the theory of gamma-positivity,
for the dimension of the homology of the Rees product of a graded poset
with a certain -analogue of the chain of the same length as . Equivariant
generalizations of these formulas are proven in this paper, when a group of
automorphisms acts on , and are applied to establish the Schur
gamma-positivity of certain symmetric functions arising in algebraic and
geometric combinatorics.Comment: Final version, with a section on type B Coxeter complexes added; to
appear in Algebraic Combinatoric
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