Conjectured statistics for the q,t-Catalan numbers |
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Authors: | J. Haglund |
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Affiliation: | Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA |
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Abstract: | We introduce the distribution function Fn(q,t) of a pair of statistics on Catalan words and conjecture Fn(q,t) equals Garsia and Haiman's q,t-Catalan sequence Cn(q,t), which they defined as a sum of rational functions. We show that Fn,s(q,t), defined as the sum of these statistics restricted to Catalan words ending in s ones, satisfies a recurrence relation. As a corollary we are able to verify that Fn(q,t)=Cn(q,t) when t=1/q. We also show the partial symmetry relation Fn(q,1)=Fn(1,q). By modifying a proof of Haiman of a q-Lagrange inversion formula based on results of Garsia and Gessel, we obtain a q-analogue of the general Lagrange inversion formula which involves Catalan words grouped according to the number of ones at the end of the word. |
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Keywords: | 05A15 05A30 (primary) 05E05 (secondary) |
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