In this paper, we study Forman's discrete Morse theory in the context of weighted homology.
We develop weighted versions of classical theorems in discrete Morse theory. A key …

This is a brief survey of some open problems on permutation patterns, with an emphasis on
subjects not covered in the recent book by Kitaev, Patterns in Permutations and words. I first …

The set of all permutations, ordered by pattern containment, forms a poset. This paper
presents the first explicit major results on the topology of intervals in this poset. We show that …

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 …

We study the values of the Möbius function μ of intervals in the containment poset of
permutations. We construct a sequence of permutations π n of size 2 n− 2 for which μ (1, π …

We introduce a formal definition of a pattern poset which encompasses several previously
studied posets in the literature. Using this definition we present some general results on the …

We characterize the downsets of integer partitions (ordered by containment of Ferrers
diagrams) and compositions (ordered by the generalized subword order) which have finite …

We use discrete Morse theory to determine the Möbius function of generalized factor order.
Ordinary factor order on the Kleene closure A∗ of a set A is the partial order defined by …

A permutation is an ordering of the letters 1,..., n. A permutation σ occurs as a pattern in a
permutation π if there is a subsequence of π whose letters appear in the same relative order …

Crystal bases were introduced by Kashiwara when studying modules of quantum groups.
These crystals are combinatorial structures that mirror representations of Lie algebras. Each …