Susanna Fishel

Statistics for special q, t-kostka polynomials

Kirillov and Reshetikhin introduced rigged configurations as a new way to calculate the entries KA,U(t) of the Kostka matrix. Macdonald defined the two-parameter Kostka matrix whose entries KA, (q, t) generalize KA,,(t) . We use rigged configurations and a formula of Stembridge to provide a combinatorial interpretation of KA,u(q, t) in the case where p is a partition with no more than two columns. In particular, we show that in this case, KA,u(q, t) has nonnegative coefficients.

Chains of maximum length in the Tamari lattice

The Tamari lattice T[subscript n] was originally defined on bracketings of a set of n + 1 objects, with a cover relation based on the associativity rule in one direction. Although in several related lattices, the number of maximal chains is known, quoting Knuth, “The enumeration of such paths in Tamari lattices remains mysterious.” The lengths of maximal chains vary over a great range. In this paper, we focus on the chains with maximum length in these lattices. We establish a bijection between the maximum length chains in the Tamari lattice and the set of standard shifted tableaux of staircase shape. We thus derive an explicit formula for the number of maximum length chains, using the Thrall formula for the number of shifted tableaux. We describe the relationship between chains of maximum length in the Tamari lattice and certain maximal chains in weak Bruhat order on the symmetric group, using standard Young tableaux. Additionally, recently, Bergeron and Pr eville-Ratelle introduced a generalized Tamari lattice. Some of the results mentioned above carry over to their generalized Tamari lattice.

Results and conjectures on the number of standard strong marked tableaux

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.New tools, residue and quotient tables, are presented, which allow for efficient computation of strong covers and have the potential to describe many other phenomena in k-function theory.

Enumerations relating braid and commutation classes

Abstract We obtain an upper and lower bound for the number of reduced words for a permutation in terms of the number of braid classes and the number of commutation classes of the permutation. We classify the permutations that achieve each of these bounds, and enumerate both cases.