We outline here a proof that a certain rational function Cn(q, t), which has come to be known as the “q, t-Catalan,” is in fact a polynomial with positive integer coefficients. This has been an open problem since 1994. Because Cn(q, t) evaluates to the Catalan number at t = q = 1, it has also been an open problem to find a pair of statistics a, b on the collection 𝒟n of Dyck paths Π of length 2n yielding Cn(q, t) = ∑π ta(Π)qb(Π). Our proof is based on a recursion for Cn(q, t) suggested by a pair of statistics recently proposed by J. Haglund. One of the byproducts of our results is a proof of the validity of Haglund'sconjecture.
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机译：我们在这里概述一个证明，即某个已知的有理函数Cn（q，t）实际上被称为“ q，t-Catalan”，它是具有正整数系数的多项式。自1994年以来，这一直是一个未解决的问题。由于Cn（q，t）在t = q = 1时计算加泰罗尼亚数，因此在集合＆＃x1d49f上找到一对统计值a，b也一直是一个未解决的问题。 ; n长度为2n的Dyck路径Π产生Cn（ q ， t ）= ∑π t a （Π） q b （Π）。我们的证明基于J.Haglund最近提出的一对统计数据对 Cn （ q ， t ）的递归。我们结果的副产品之一是Haglund检验的有效性的证明。推测。
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