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We study Okadas conjecture on $(q,t)$-hook formula of general $d$-complete posets. Proctor classified $d$-complete posets into 15 irreducible ones. We try to give a case-by-case proof of Okadas $(q,t)$-hook formula conjecture using the symmetric functions. Here we give a proof of the conjecture for birds and banners, in which we use Gaspers identity for VWP-series ${}_{12}W_{11}$.
The higher $q,t$-Catalan polynomial $C^{(m)}_n(q,t)$ can be defined combinatorially as a weighted sum of lattice paths contained in certain triangles, or algebraically as a complicated sum of rational functions indexed by partitions of $n$. This pape
We start with a (q,t)-generalization of a binomial coefficient. It can be viewed as a polynomial in t that depends upon an integer q, with combinatorial interpretations when q is a positive integer, and algebraic interpretations when q is the order o
We give two proofs of the $q,t$-symmetry of the generalized $q,t$-Catalan number $C_{vec{k}}(q,t)$ for $vec{k}=(k_1,k_2,k_3)$. One is by MacMahons partition analysis as we proposed; the other is by a direct bijection.
Inspired by the recent work on $q$-congruences and the quadratic summation formula of Rahman, we provide some new $q$-supercongruences. By taking $qto 1$ in one of our results, we obtain a new Ramanujan-type supercongruence, which has the same right-
We relate the mixed Hodge structure on the cohomology of open positroid varieties (in particular, their Betti numbers over $mathbb{C}$ and point counts over $mathbb{F}_q$) to Khovanov--Rozansky homology of associated links. We deduce that the mixed H