# A q,t-Schroder polynomial

## ABSTRACT

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