2023/05/27 by X. Gao, Frank Z. K. Li, Gao, X. +5
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #05A05 #05A19 #Advanced Combinatorial Mathematics #Algorithms and Data Compression #Combinatorics (math.CO) #FOS: Mathematics #Genome Rearrangement Algorithms
paper · pdf · doi:10.48550/arxiv.2305.17426
openalex publication_date 2023/05/27 · openalex created_date 2023/05/31 · openalex updated_date 2026/07/28
The elements in the hyperoctahedral group \mathfrakBn can be treated as signed permutations with the natural order ⋯<-2<-1<0<1<2<⋯, or as colored permutations with the r-order -1<r-2<r⋯<r0<r1<r2<r⋯. For any π∈\mathfrakBn, let desB(π) and idesB(π) be the number of descents and inverse descents in π under the natural order, and let desB(π) and idesB(π) be the number of descents and inverse descents in π under the r-order. In this paper, by investigating signed permutation grids under both the natural order and the r-order, we give combinatorial proofs for six recurrence formulas of the joint distribution of descents and inverse descents over the hyperoctahedral group \mathfrakBn, the set in involutions of \mathfrakBn denoted by InB, and the set of fixed-point free involutions in \mathfrakBn denoted by JnB, respectively. Some of these six formulas are new, and some reveal the combinatorial essences of the results obtained by Visontai, Moustakas and Cao-Liu through algebraic approaches such as quasisymmetric functions. Furthermore, from these formulas, we conclude that (desB,idesB) and (desB,idesB) are equidistributed over both \mathfrakBn and InB, but not on JnB.