On Degrees in the Hasse Diagram of the Strong Bruhat Order

For a permutation $\pi$ in the symmetric group $S_n$ let the {\it total degree} be its valency in the Hasse diagram of the strong Bruhat order on $S_n$, and let the {\it down degree} be the number of permutations which are covered by $\pi$ 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 $T_n$ be the set of 321-avoiding permutations of order $n$. Two properties of $T_n$ are proved: (1) The {\em last descent} and {\em last index minus one} statistics are equidistributed over $T_n$, 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 $T_{2n}$ and $T_{2n+1}$ are essentially equal to the last-descent enumerator for $T_n$. 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

Hecke Algebra Actions on the Coinvariant Algebra

Two actions of the Hecke algebra of type A on the corresponding polynomial ring are studied. Both are deformations of the natural action of the symmetric group on polynomials, and keep symmetric functions invariant. We give an explicit description of these actions, and deduce a combinatorial formula for the resulting graded characters on the coinvariant algebra.Comment: 21 pages; final form, to appear in the Journal of Algebr

Combinatorial Gelfand Models

A combinatorial construction of a Gelafand model for the symmetric group and its Iwahori-Hecke algebra is presented.Comment: 15 pages, revised version, to appear in J. Algebr