68 research outputs found
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
Triangle-Free Triangulations, Hyperplane Arrangements and Shifted Tableaux
Flips of diagonals in colored triangle-free triangulations of a convex
polygon are interpreted as moves between two adjacent chambers in a certain
graphic hyperplane arrangement. Properties of geodesics in the associated flip
graph are deduced. In particular, it is shown that: (1) every diagonal is
flipped exactly once in a geodesic between distinguished pairs of antipodes;
(2) the number of geodesics between these antipodes is equal to twice the
number of Young tableaux of a truncated shifted staircase shape.Comment: figure added, plus several minor change
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
Shape Avoiding Permutations
Permutations avoiding all patterns of a given shape (in the sense of
Robinson-Schensted-Knuth) are considered. We show that the shapes of all such
permutations are contained in a suitable thick hook, and deduce an exponential
growth rate for their number.Comment: 16 pages; final form, to appear in J. Combin. Theory, Series
Enumeration of Standard Young Tableaux
A survey paper, to appear as a chapter in a forthcoming Handbook on
Enumeration.Comment: 65 pages, small correction
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