We give a simple recursion labeled by binary sequences which computes rational $q,t$-Catalan power series, both in relatively prime and non relatively prime cases. It is inspired by, but not identical to recursions due to B. Elias, M. Hogancamp, and A. Mellit, obtained in their study of link homology. We also compare our recursion with the Hogancamp-Mellit's recursion and verify a connection between the Khovanov-Rozansky homology of $N,M$-torus links and the rational $q,t$-Catalan power series for general positive $N,M$.

中文翻译：

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有理q，t-加泰罗尼亚数的递归

我们给出了一个用二进制序列标记的简单递归，它计算有理数 $q，Ť$-加泰罗尼亚幂级数，在相对原始的和非相对原始的情况下均如此。它的灵感来自但不等同于B. Elias，M。Hogancamp和A. Mellit在进行链接同源性研究中获得的递归。我们还将递归与Hogancamp-Mellit的递归进行了比较，并验证了Khovanov-Rozansky同源性之间的联系。$ñ，\mathrm{中号}$-torus链接与理性 $q，Ť$-加泰罗尼亚语幂级数对于一般正 $ñ，\mathrm{中号}$。