Descent sets of cyclic permutations in types B and D

2025/08/04 by Kevin Liu, Liu, Kevin
Mathematics · Biochemistry, Genetics and Molecular Biology · #Advanced Combinatorial Mathematics #Genome Rearrangement Algorithms #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.2508.02432

Abstract

Elizalde constructed a bijection ϕ from the cyclic permutations π∈ S_n+1 to the symmetric group S_n satisfying Des(π)∩ \1,2,…,n-1\=Des(ϕ(π)). We give a corresponding result on the signed symmetric group B_n by constructing a function Φ from the cyclic signed permutations π∈ B_n+1 to B_n satisfying Des(π)∩ \0,1,…,n-1\=Des(Φ(π)). Moreover, letting D_n+1⊆ B_n+1 be the subgroup consisting of signed permutations with an even number of sign changes, we show that the restriction of Φ to the cyclic signed permutations in D_n+1 or its complement is a bijection. Our function Φ reduces to Elizalde's original bijection ϕ under the natural identification of the symmetric groups as subgroups of the signed symmetric groups. One application of our results is asymptotic normality of the descent and flag major index statistics on the cyclic signed permutations in B_n and D_n.

Citations

• Descents and flag major index on conjugacy classes of colored permutation groups without short cycles
• Moments of Colored Permutation Statistics on Conjugacy Classes
• Carries, Shuffling and An Amazing Matrix
• The Flag Major Index and Group Actions on Polynomial Rings
• Descent identities, Hessenberg varieties, and the Weil conjectures
• q-Eulerian Polynomials Arising from Coxeter Groups

Related

• Number of permutations with same peak set for signed permutations
• Canonical reduced words and signed descent length enumeration in Coxeter groups
• Combinatorial equivalence of separable elements in types A and B
• The Sorting Index and Permutation Codes
• Descent set distribution for permutations with cycles of only odd or only even lengths