Conjectured statistics for the q,t-Catalan numbers

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• 作者：Haglund ; J.
• 关键词：05A15 ; 05A30 (primary) ; 05E05 (secondary) ; Catalan words ; Lagrange inversion ; Major index statistic
• 刊名：Advances in Mathematics
• 出版年：2003
• 期刊代码：62_00018708
• 类别：mt
• 出版时间：May 10, 2003
• 卷：175
• 期：2
• 页码：319-334
• 文件大小：169 K

摘要

We introduce the distribution function F_n(q,t) of a pair of statistics on Catalan words and conjecture F_n(q,t) equals Garsia and Haiman's q,t-Catalan sequence C_n(q,t), which they defined as a sum of rational functions. We show that F_n,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 F_n(q,t)=C_n(q,t) when t=1/q. We also show the partial symmetry relation F_n(q,1)=F_n(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.