We prove a restriction of an analogue of the Robinson-Schensted-Knuth correspondence for semi-skyline augmented fillings, due to Mason, to multisets of cells of a staircase possibly truncated by a smaller staircase at the upper left end corner, or at the bottom right end corner. The restriction to be imposed on the pairs of semi-skyline augmented fillings is that the pair of shapes, rearrangeme...

We prove a combinatorial formula for the Hilbert series of the Garsia-Haiman bigraded Sn-modules as weighted sums over standard Young tableaux in the hook shape case. This method is based on the combinatorial formula of Haglund, Haiman and Loehr for the Macdonald polynomials and extends the result of A. Garsia and C. Procesi for the Hilbert series when q = 0. Moreover, we construct an associati...

Closed formulas are known for S(k, 0; n), the number of standard Young tableaux of size n and with at most k parts, where 1 ≤ k ≤ 5. Here we study the analogous problem for S(k, ℓ; n), the number of standard Young tableaux of size n which are contained in the (k, ℓ)-hook. We deduce some formulas for the cases k + ℓ ≤ 4.

In the study of Kostka numbers and Catalan numbers, Kirillov posed a unimodality conjecture for the rectangular Narayana polynomials. We prove that the rectangular Narayana polynomials have only real zeros, and thereby confirm Kirillov’s unimodality conjecture. By using an equidistribution property between descent numbers and ascent numbers on ballot paths due to Sulanke and a bijection between...

The Selberg integral is an important integral first evaluated by Selberg in 1944. Stanley found a combinatorial interpretation of the Selberg integral in terms of permutations. In this paper, new combinatorial objects “Young books” are introduced and shown to have a connection with the Selberg integral. This connection gives an enumeration formula for Young books. It is shown that special cases...

A well-known theorem of Frame, Robinson, and, Thrall states that if h is a partition of n, then the number of Standard Young Tableaux of shape h is n! divided by the product of the hook-lengths. We give a new combinatorial proof of this formula by exhibiting a bijection between the set of unsorted Young Tableaux of shape A, and the set of pairs (T, S), where T is a Standard Young Tableau of sha...

Many results involving Schur functions have analogues involving k-Schur functions. Standard strong marked tableaux play a role for k-Schur functions similar to the role standard Young tableaux play for Schur functions. We discuss results and conjectures toward an analogue of the hook-length formula.