Intervals of Permutations with a Fixed Number of Descents are Shellable

TL;DR

This paper demonstrates that intervals of permutations with a fixed number of descents are shellable by establishing an order isomorphism to a word poset and provides a formula for their Möbius function, advancing understanding of permutation posets.

Contribution

It introduces a novel bijection linking permutation intervals to word posets, proving shellability for fixed descent intervals and deriving a Möbius function formula, with new proofs and conjectures.

Findings

Intervals of permutations with fixed descents are shellable.

A formula for the Möbius function of these intervals is provided.

Intervals with one descent avoiding certain patterns have no nontrivial disconnected subintervals.

Abstract

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 M\"obius function of these intervals. We present an alternative proof for a result on the M\"obius function of intervals $[1,\pi]$ such that $\pi$ has exactly one descent. We prove that if $\pi$ has exactly one descent and avoids 456123 and 356124, then the intervals $[1,\pi]$ have no nontrivial disconnected subintervals; we conjecture that these intervals are shellable.