Séminaire Lotharingien de Combinatoire, B54At (2007), 40 pp.
Anthony Mendes and Jeffrey Remmel
Generating Functions for Statistics on
Ck wreath Sn
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 Bn. 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
Ck wreath Sn
of cyclic groups Ck with the symmetric group Sn
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
Bn or Ck wreath Sn
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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