We introduce pretty clean modules, extending the notion of clean modules by Dress, and show that pretty clean modules are sequentially CohenMacaulay. We also extend a theorem of Dress on shellable simplicial complexes to multicomplexes.

The tropical semiring (R,min,+) has enjoyed a recent renaissance, owing to its connections to mathematical biology as well as optimization and algebraic geometry. In this paper, we investigate the space of n points on a tropical line in d-space, which is interpretable as the space of n-species phylogenetic trees. This is equivalent to the space of n × d matrices of tropical rank two, a simplici...

The set of all permutations, ordered by pattern containment, is a poset. We present an order isomorphism from the poset of permutations with a fixed number of descents to a certain poset of words with subword order. We use this bijection to show that intervals of permutations with a fixed number of descents are shellable, and we present a formula for the Möbius function of these intervals. We p...

The purpose of this work is to initiate a combinatorial study of the Bruhat-Chevalley ordering on certain sets of permutations obtained by omitting the parentheses from their standard cyclic notation. In particular, we show that these sets form bounded, graded, unimodal, rank-symmetric and EL-shellable posets. Moreover, we determine the homotopy types of the associated order complexes. Résumé. ...