Repeatable patterns and the maximum multiplicity of a generator in a reduced word

TL;DR

This paper investigates the maximum multiplicity of a simple transposition in reduced words for the longest permutation, revealing periodic optimal patterns and providing bounds and exact values for these multiplicities.

Contribution

It introduces the concept of repeatable patterns to analyze maximum multiplicities and characterizes the asymptotic behavior using periodic functions for fixed k and large n.

Findings

Maximum multiplicity is asymptotically linear in n with a rational coefficient.

Optimal patterns are periodic and can be explicitly constructed.

Provides bounds and exact values for the multiplicity coefficient c_k.

Abstract

We study the maximum multiplicity $\mathcal{M}(k,n)$ of a simple transposition $s_k=(k \: k+1)$ in a reduced word for the longest permutation $w_0=n \: n-1 \: \cdots \: 2 \: 1$, a problem closely related to much previous work on sorting networks and on the "$k$-set" problem. After reinterpreting the problem in terms of monotone weakly separated paths, we show that, for fixed $k$ and sufficiently large $n$, the optimal density is realized by paths which are periodic in a precise sense, so that \[ \mathcal{M}(k,n)=c_k n + p_k(n) \] for a periodic function $p_k$ and constant $c_k$. In fact we show that $c_k$ is always rational, and compute several bounds and exact values for this quantity with "repeatable patterns", which we introduce.