# {A q,t-Schr\"oder polynomial} {"Darla Kremer} {"Gettysburg College} {"dkremer(at-sign)math.mit.edu} {" 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 s\ howed that $C_n(1,1) = \frac{1}{n+1}\binom{2n}{n}$, the $n$th Catalan number; that $C_ n(1,q)$ satisfies the recurrence of the Carlitz-Riordan $q$-Catalan polynomial $C_n(q)$ \ $C_n(q)=\sum_{k=1}^nq^{k-1}C_{k-1}(q,1)C_{n-k}(q,1)$; and that $q^{\binom{n}{2}}C_n(q,1/q)=\frac{1}{[n+1]_q}\qbinom{2n}{n}.$ Haiman, in 1995, used al\ gebraic techniques 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 1\ 994 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 ver\ tical steps.) After giving some background on the $q,t$-Catalan polynomial, I will exte\ nd 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 h\ aving $d$ diagonal steps $S_{n,d}(q,t)=\sum q^{area}t^{dmaj}$ 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 w\ ill discuss properties of $S_{n,d}(q,t)$ which support this conjecture. }

## ABSTRACT

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 Combinatorics Seminar, Mathematics Department, MIT, 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 ver\ tical steps.) After giving some background on the $q,t$-Catalan polynomial, I will exte\ nd 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 h\ aving $d$ diagonal steps $S_{n,d}(q,t)=\sum q^{area}t^{dmaj}$ 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 w\ ill discuss properties of $S_{n,d}(q,t)$ which support this conjecture. } ">sara(at-sign)math.mit.edu