2024/04/19 by Amrita Acharyya, Acharyya, Amrita
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Mathematical functions and polynomials
paper · pdf · doi:10.48550/arxiv.2404.14439
openalex publication_date 2024/04/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work, we study type B set partitions for a given specific positive integer k defined over ⟨ n⟩=\-n, -(n-1),⋯ -1,0,1,⋯ n-1,n\. We found a few generating functions of type B analogue for some of the set partition statistics defined by Wachs, White and Steingrimsson for partitions over positive integers [n] =\1,2,⋯ n\, both for standard and ordered set partitions respectively. We extended the idea of restricted growth functions utilized by Wachs and White for set partitions over [n], in the scenario of ⟨ n⟩ and called the analogue as Signed Restricted Growth Function (SRGF). We discussed analogues of major index for type B partitions in terms of SRGF. We found an analogue of Foata bijection and reduced matrix for type B set partitions as done by Sagan for set partitions of [n] with sepcific number of blocks k. We conclude with some open questions regarding the type B analogue of some well known results already done in case of set partitions of [n].