Enumeration in the lattice of $q$-decreasing words

TL;DR

This paper establishes that the poset of q-decreasing words forms a lattice, enumerates key elements, and analyzes their combinatorial properties and asymptotic behavior for various q values.

Contribution

It proves the lattice structure of q-decreasing words, enumerates join-irreducible elements, and characterizes the structure and asymptotics of coverings and intervals for rational q.

Findings

The poset of q-decreasing words is a lattice.

Enumeration of join-irreducible elements for all q>0.

Asymptotic analysis of coverings, intervals, and meet-irreducible elements.

Abstract

We prove that the poset of $q$-decreasing words equipped with the componentwise order forms a lattice. We enumerate the join-irreducible elements for arbitrary $q>0$, and for any positive rational number $q$, we determine the number of coverings, intervals and meet-irreducible elements. The latter present the same structure as words over an alphabet of $2\lceil q\rceil+1$ letters avoiding $\lceil q\rceil^2+2\lceil q\rceil-1$ consecutive patterns of length 2. Furthermore, we analyze the asymptotic behavior of several of these quantities.