100 research outputs found
Permutation Statistics on the Alternating Group
Let denote the alternating and the symmetric groups on
. MacMahaon's theorem, about the equi-distribution of the length and
the major indices in , has received far reaching refinements and
generalizations, by Foata, Carlitz, Foata-Schutzenberger, Garsia-Gessel and
followers. Our main goal is to find analogous statistics and identities for the
alternating group . A new statistic for , {\it the delent number},
is introduced. This new statistic is involved with new equi-distribution
identities, refining some of the results of Foata-Schutzenberger and
Garsia-Gessel. By a certain covering map , such
identities are `lifted' to , yielding the corresponding
equi-distribution identities.Comment: 45 page
On Degrees in the Hasse Diagram of the Strong Bruhat Order
For a permutation in the symmetric group let the {\it total
degree} be its valency in the Hasse diagram of the strong Bruhat order on
, and let the {\it down degree} be the number of permutations which are
covered by in the strong Bruhat order. The maxima of the total degree and
the down degree and their values at a random permutation are computed. Proofs
involve variants of a classical theorem of Tur\'an from extremal graph theory.Comment: 14 pages, minor corrections; to appear in S\'em. Lothar. Combi
Equidistribution and Sign-Balance on 321-Avoiding Permutations
Let be the set of 321-avoiding permutations of order . Two
properties of are proved: (1) The {\em last descent} and {\em last index
minus one} statistics are equidistributed over , and also over subsets of
permutations whose inverse has an (almost) prescribed descent set. An analogous
result holds for Dyck paths. (2) The sign-and-last-descent enumerators for
and are essentially equal to the last-descent enumerator
for . The proofs use a recursion formula for an appropriate multivariate
generating function.Comment: 17 pages; to appear in S\'em. Lothar. Combi
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