Type B Set partitions, an analogue of restricted growth functions

Abstract

In this work, we study type B set partitions for a given specific positive integer $k$ defined over $⟨n⟩=\{-n,-(n-1),\cdots -1,0,1,\cdots n-1,n\}$. 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 $[n]=\{1,2,\cdots n\}$, 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 $[n]$.