Positivity for Special Cases of $(q,t)$-Kostka Coefficients and Standard Tableaux Statistics

Positivity for special cases of (q, t)-Kostka coefficients and standard tableaux statistics

We present two symmetric function operators H 3 and H qt 4 that have the property H mH(2a1b)(X; q, t) = H(m2a1b)(X; q, t). These operators are generalizations of the analogous operator H 2 and also have expressions in terms of Hall-Littlewood vertex operators. We also discuss statistics, aμ(T ) and bμ(T ), on standard tableaux such that the q, t Kostka polynomials are given by the sum over stan...

0 A ug 1 99 9 Positivity for special cases of ( q , t ) - Kostka coefficients and standard tableaux statistics

We present two symmetric function operators H qt 3 and H qt 4 that have the property H qt 3 H (2 a 1 b) [X; q, t] = H (32 a 1 b) [X; q, t] and H qt 4 H (2 a 1 b) [X; q, t] = H (42 a 1 b) [X; q, t]. These operators are generalizations of the analogous operator H qt 2 and also have expressions in terms of Hall-Littlewood vertex operators. We also discuss statistics, a µ (T) and b µ (T), on standa...

Degrees of Stretched Kostka Coefficients

Given a partition λ and a composition β, the stretched Kostka coefficient Kλβ(n) is the map n 7→ Knλ,nβ sending each positive integer n to the Kostka coefficient indexed by nλ and nβ. Derksen and Weyman [DW02] have shown that stretched Kostka coefficients are polynomial functions of n. King, Tollu, and Toumazet have conjectured that these polynomials always have nonnegative coefficients [KTT04]...

A Macdonald Vertex Operator and Standard Tableaux Statistics

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...

An Elementary Method for Computing the Kostka Coefficients