Permutation statistics and weak {B}ruhat order in permutation tableaux of type $B$

TL;DR

This paper explores the relationships between permutation tableaux of type B and various permutation statistics, including alignments, crossings, inversions, and cycles, and characterizes the weak Bruhat order in this context.

Contribution

It introduces new connections between permutation tableaux of type B and permutation statistics, and characterizes the weak Bruhat order using these tableaux.

Findings

Alignments, crossings, and inversions are realized in permutation tableaux of type B.

Cycles of signed permutations are understood through bare tableaux of type B.

The covering relation in weak Bruhat order is characterized via permutation tableaux of type B.

Abstract

Many important statistics of signed permutations are realized in the corresponding permutation tableaux or bare tableaux of type $B$: Alignments, crossings and inversions of signed permutations are realized in the corresponding permutation tableaux of type $B$, and the cycles of signed permutations are understood in the corresponding bare tableaux of type $B$. This leads us to relate the number of alignments and crossings with other statistics of signed permutations and also to characterize the covering relation in weak Bruhat order on Coxeter system of type $B$ in terms of permutation tableaux of type $B$.