Type B set partitions, an analogue of restricted growth functions

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 analogues for some of the set partition statistics defined by Wachs, White and Steingrímsson 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 specific 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 \([n]\).