Ž. P. M. Terwilliger 1992, J. Algebraic Combin. 1, 363᎐388 considered the ‫-ރ‬alge-bra generated by a given Bose Mesner algebra M and the associated dual Bose Mesner algebra M U . This algebra is now known as the Terwilliger algebra and is usually denoted by T. Terwilliger showed that each vanishing intersection number and Krein parameter of M gives rise to a relation on certain generators of T. These relations determine much of the structure of T, thought not all of it in general. To illuminate the role these relations play, we consider a certain generalization T T of T. To go from T to T T, we replace M and M U with a pair of dual character algebras C and C U . We define T T by generators and relations; intuitively T T is the associative ‫-ރ‬algebra with identity generated by C and C U subject to the analogues of Terwilliger's relations. T T is infinite dimensional and noncommutative in general. We construct an irreducible T T-module which we call the primary module; the dimension of this module is the same as that of C and C U . We find two bases of the primary module; one diagonalizes C and the other diagonalizes C U . We compute the action of the generators of T T on these bases. We show T T is a direct sum of two sided ideals T T and T T with T T isomorphic to a full matrix 0 1 0 algebra. We show that the irreducible module associated with T T is isomorphic to 0 the primary module. We compute the central primitive idempotent of T T associated with T T in terms of the generators of T T. ᮊ

Garsia and Haiman (J. Algebraic. Combin. $\bf5$ $(1996)$, $191-244$) conjectured that a certain sum $C_n(q,t)$ of rational functions in $q,t$ reduces to a polynomial in $q,t$ with nonnegative integral coefficients. Haglund later discovered (Adv. Math., in press), and with Garsia proved (Proc. Nat. Acad. Sci. ${\bf98}$ $(2001)$, $4313-4316$) the refined conjecture $C_n(q,t) = \sum q^{{\rm area}}t^{{\rm bounce}}$. Here the sum is over all Catalan lattice paths and ${\rm area}$ and ${\rm bounce}$ have simple descriptions in terms of the path. In this article we give an extension of $({\rm area},{\rm bounce})$ to Schröder lattice paths, and introduce polynomials defined by summing $q^{{\rm area}}t^{{\rm bounce}}$ over certain sets of Schröder paths. We derive recurrences and special values for these polynomials, and conjecture they are symmetric in $q,t$. We also describe a much stronger conjecture involving rational functions in $q,t$ and the $\nabla$ operator from the theory of Macdonald symmetric functions.

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We study generating functions for the number of involutions, even involutions, and odd involutions in S n subject to two restrictions. One restriction is that the involution avoid 3412 or contain 3412 exactly once. The other restriction is that the involution avoid another pattern τ or contain τ exactly once. In many cases we express these generating functions in terms of Chebyshev polynomials of the second kind.