A q,t-Schroder polynomial

Darla Kremer

Gettysburg College

May 1,
refreshments at 3:45pm


In 1994 Garsia and Haiman conjectured that the Hilbert series for the space of diagonal harmonic alternates can be described by a certain rational function $C_n(q,t)$. They showed that $ C_n(1,1) =c_n$, the $n$th Catalan number; that $C_n(1,q)$ satisfies the recurrence of the Carlitz-Riordan $q$-Catalan polynomial $C_n(q)$; and that $ q^{\binom{n}{2}}C_n(q,1/q)$ evaluates to the $q$-Catalan polynomial. Haiman used techniques from algebraic geometry to show that $C_n(q,t)$ is a polynomial. In 2000, Garsia and Haglund gave a combinatorial proof that $C_n(q,t)$ has nonnegative integer coefficients. The 1994 conjecture was confirmed by Haiman in 2001. Garsia and Haglund's proof of nonnegativity gave a combinatorial interpretation of $C_n(q,t)$ as the generating series for two statistics $area$ and $dmaj$ defined on the set of Catalan paths of length $2n$. (A Catalan path is a lattice path from $(0,0)$ to $(n,n)$ which remains weakly above the line $y=x$, and which takes only horizontal and vertical steps.) After giving some background on the $q,t$-Catalan polynomial, I will extend the statistics $area$ and $dmaj$ to Schr\"oder paths: lattice paths from $(0,0)$ to $(n,n)$ which remain weakly above the line $y=x$, and which take horizontal, vertical, and diagonal steps. The generating series for $area$ and $dmaj$ over Schr\"oder paths having $d$ diagonal steps $S_{n,d}(q,t)$ defines a polynomial which is believed to be symmetric in $q$ and $t$. That is, $S_{n,d}(q,t)=S_{n,d}(t,q)$. I will discuss properties of $S_{n,d}(q,t)$ which support this conjecture.

Speaker's Contact Info: dkremer(at-sign)math.mit.edu

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Combinatorics Seminar, Mathematics Department, MIT, sara(at-sign)math.mit.edu

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