Work

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

Year: 2024

Type: article

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.Mathematics Subject Classifications: 05A05, 05A16, 05E99Keywords: Reduced words, permutations, \(k\)-sets, wiring diagram, weakly separated

Source: Combinatorial Theory

DOI: 10.5070/c64264252

Authors: Christian Gaetz, Yibo Gao, Pakawut Jiradilok, Gleb Nenashev, Alexander Postnikov

Institutions: University of California, Berkeley, Peking University, St Petersburg University, Massachusetts Institute of Technology

Language: English

Topic: Advanced Combinatorial Mathematics

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funders: National Science Foundation, Knut och Alice Wallenbergs Stiftelse, Office of Naval Research