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Register a new user(q,t)-hook formula for Birds and Banners
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Authors
Masao Ishikawa
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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 ${}_{12}W_{11}$.
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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 paper proves the equivalence of the two definitions for all $mgeq 1$ and all $nleq 4$. We also give a bijective proof of the joint symmetry property $C^{(m)}_n(q,t)=C^{(m)}_n(t,q)$ for all $mgeq 1$ and all $nleq 4$. The proof is based on a general approach for proving joint symmetry that dissects a collection of objects into chains, and then passes from a joint symmetry property of initial points and terminal points to joint symmetry of the full set of objects. Further consequences include unimodality results and specific formulas for the coefficients in $C^{(m)}_n(q,t)$ for all $mgeq 1$ and all $nleq 4$. We give analogous results for certain rational-slope $q,t$-Catalan polynomials.
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 of a finite field. These (q,t)-binomial coefficients and their interpretations generalize further in two directions, one relating to column-strict tableaux and Macdonalds ``seventh variation of Schur functions, the other relating to permutation statistics and Hilbert series from the invariant theory of GL_n(F_q).
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-hand side as Van Hammes (G.2) supercongruence for $pequiv 1 pmod 4$. We also formulate some related challenging conjectures on supercongruences and $q$-supercongruences.
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 Hodge polynomials of top-dimensional open positroid varieties are given by rational $q,t$-Catalan numbers. Via the curious Lefschetz property of cluster varieties, this implies the $q,t$-symmetry and unimodality properties of rational $q,t$-Catalan numbers. We show that the $q,t$-symmetry phenomenon is a manifestation of Koszul duality for category $mathcal{O}$, and discuss relations with open Richardson varieties and extension groups of Verma modules.
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