Type B (p, q)-Stirling Numbers via Signed Restricted Growth Functions and Rook Theory

Stirling numbers are among the most classical objects in enumerative combinatorics, counting set partitions and permutations. In this paper, we study their (p,q)-analogues in type B from a rook-theoretic point of view. We introduce a type B Ferrers board and establish a bijection between signed restricted growth functions and type B rook placements. In addition, we defined the weighted statistics levLBB(w) and levLSB(w) over the set of signed restricted growth functions. The associated statistics yield a weighted enumeration that recovers the (p,q)-Stirling polynomials of type B, their recurrence relations and generating functions. We then introduce type B Laguerre boards and prove that their rook numbers coincide with the Lah numbers of type B.