Type B Set partitions, an analogue of restricted growth functions

TL;DR

This paper introduces and studies type B set partitions, extending concepts like restricted growth functions and major index to this setting, and provides generating functions and bijections for these structures.

Contribution

It develops the concept of Signed Restricted Growth Functions as an analogue for type B partitions and extends classical results like Foata bijection to this new context.

Findings

Derived generating functions for type B partition statistics.

Introduced Signed Restricted Growth Functions (SRGF) for type B partitions.

Extended classical bijections and matrix representations to type B setting.

Abstract

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