Young’s lattice is a partial order on integer partitions whose saturated chains correspond to standard Young tableaux, one type of combinatorial object that generates the Schur basis for symmetric functions. Generalizing lattice, we introduce new weak compositions call key poset. Saturated in this poset objects generate polynomials, nonsymmetric polynomial generalization basis. skew functions, ...

We identify a subalgebra Ĥ + n of the extended affine Hecke algebra Ĥn of type A. The subalgebra Ĥ + n is a u-analogue of the monoid algebra of Sn n Z≥0 and inherits a canonical basis from that of Ĥn. We show that its left cells are naturally labeled by tableaux filled with positive integer entries having distinct residues mod n, which we term positive affine tableaux (PAT). We then exhibit a c...

For a semisimple Lie algebra g the orbit method attempts to assign representations of g to (coadjoint) orbits in g∗. Orbital varieties are particular Lagrangian subvarieties of such orbits leading to highest weight representations of g. In sln orbital varieties are described by Young tableaux. Inclusion relation on orbital variety closures defines a partial order on Young tableaux. Our aim is t...

The two parameter family of coefficients Kλμ(q, t) introduced by Macdonald are conjectured to (q, t) count the standard tableaux of shape λ. If this conjecture is correct, then there exist statistics aμ(T ) and bμ(T ) such that the family of symmetric functions Hμ[X; q, t] = ∑ λKλμ(q, t)sλ[X] are generating functions for the standard tableaux of size |μ| in the sense that Hμ[X; q, t] = ∑ T q aμ...

We derive combinatorial identities, involving the Bernoulli and Euler numbers, for the numbers of standard Young tableaux of certain skew shapes. This generalizes the classical formulas of D. André on the number of up-down permutations. The analysis uses a transfer operator approach extending the method of Elkies, combined with an identity expressing the volume of a certain polytope in terms of...

In this paper the number of standard Young tableaux (SYT) is evaluated by the methods of multiple integrals and combinatorial summations. We obtain the product formulas of the numbers of skew SYT of certain truncated shapes, including the skew SYT ((n + k)r+1, nm−1)/(n − 1)r truncated by a rectangle or nearly a rectangle, the skew SYT of truncated shape ((n+ 1)3, nm−2)/(n− 2)\ (22), and the SYT...

This paper contains two results on the number fσ/τ of standard skew Young tableaux of shape σ/τ . The first concerns generating functions for certain classes of “periodic” shapes related to work of Gessel-Viennot and Baryshnikov-Romik. The second result gives an evaluation of the skew Schur function sλ/μ(x) at x = (1, 1/22k , 1/32k , . . . ) for k = 1, 2, 3 in terms of fσ/τ for a certain skew s...