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Séminaire Lotharingien de Combinatoire, B54At (2007), 40 pp.

# Anthony Mendes and Jeffrey Remmel

# Generating Functions for Statistics on
*C*_{k} wreath *S*_{n}

**Abstract.**
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*.

Received: December 1, 2005.
Accepted: September 13, 2006.
Final Version: January 5, 2007.

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