Loehr–Warrington rational q,tq,t-Catalan specialization conjecture

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Let Cata,b(q,t)\mathsf{Cat}_{a,b}(q,t) be the rational q,tq,t-Catalan number and let [m]q[m]_q and [a+ba,b]q\genfrac{[}{]}{0pt}{}{a+b}{a,b}_q denote the qq-integer and qq-binomial coefficient. Rational q,tq,t-Catalan specialization conjecture.

q(a−1)(b−1)/2Cata,b(q,1/q)=1[a+b]q[a+ba,b]q.q^{(a-1)(b-1)/2}\mathsf{Cat}_{a,b}(q,1/q)=\frac{1}{[a+b]_q}\genfrac{[}{]}{0pt}{}{a+b}{a,b}_q.

This predicts that the rational q,tq,t-Catalan polynomial specializes, after the stated rescaling, to the rational qq-Catalan polynomial; the supplied text does not state its resolution.

References

Primary source

Drew Armstrong, Nicholas A. Loehr and Gregory S. Warrington, “Rational parking functions and Catalan numbers”, arXiv:1403.1845 (2014).

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