Combinatorial equivalence of separable elements in types A and B

2025/10/14 by Liao, Yong, Yang, Yuping, Yu, Houyi
#Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2510.12046

Abstract

We study the combinatorial equivalence of separable elements in types A and B. A bijection is constructed from the set of separable permutations in the symmetric group S_n+1 to the set of separable signed permutations in the hyperoctahedral group B_n. This bijection preserves descent statistics and induces a poset isomorphism under the left weak order. As a consequence, separable signed permutations are enumerated by the large Schröder numbers, and their descent polynomials are shown to be γ-positive. Building on a recursive characterization of separable signed permutations via direct sum and skew sum operations, we derive explicit product formulas for the rank generating functions of the principal upper and lower ideals of separable signed permutations under the left weak order.

Citations

• Distributions of statistics on separable permutations
• Separable elements and splittings in Weyl groups of Type B
• On the action of the Weyl group on canonical bases
• The weak order on the hyperoctahedral group and the monomial basis for the Hopf algebra of signed permutations
• Permutation classes
• Smoothness of Schubert varieties via patterns in root systems
• Structure of the Malvenuto-Reutenauer Hopf algebra of permutations

Related

• Separable elements and splittings in Weyl groups of Type B
• The Weak Bruhat Order and Separable Permutations
• Descent sets of cyclic permutations in types B and D
• Proof of Stasinski and Voll's Hyperoctahedral Group Conjecture
• A descent-excedance correspondence in colored permutation groups