Generating Functions for Statistics on Ck wreath Sn

Anthony Mendes ; Jeffrey Remmel

Séminaire lotharingien de combinatoire, 54A (2005-2007)

Résumé

Foata and Han [Adv. in Appl. Math. 18 (1997), 489-509; Electron. J. Combin. 4 (1997), Article #R9] proved some remarkable generating functions for statistics on the hyperoctahedral group B_n. These generating functions can be specialized to give a large number of generating functions for permutation statistics appearing in the literature. In this paper, we give a new proof of Foata and Han's result by defining a homomorphism on the ring of symmetric functions and applying it to a simple symmetric function identity. Our methods easily extend to derive several natural extensions of Foata and Han's generating functions. In particular, we show that there exists a natural family of generating functions for permutation statistics over wreath products C_k wreath S_n of cyclic groups C_k with the symmetric group S_n which can be viewed as generalizations of the Foata-Han generating functions. We also prove some new generating functions for the Foata-Han statistics for tuples of permutations of B_n or C_k wreath S_n whose common descent set contain a final sequence of at least s or whose common descent set contain a final sequence of exactly size s.