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The emph{$q,t$-Catalan numbers} $C_n(q,t)$ are polynomials in $q$ and $t$ that reduce to the ordinary Catalan numbers when $q=t=1$. These polynomials have important connections to representation theory, algebraic geometry, and symmetric functions. Haglund and Haiman discovered combinatorial formulas for $C_n(q,t)$ as weighted sums of Dyck paths (or equivalently, integer partitions contained in a staircase shape). This paper undertakes a combinatorial investigation of the joint symmetry property $C_n(q,t)=C_n(t,q)$. We conjecture some structural decompositions of Dyck objects into mutually opposite subcollections that lead to a bijective explanation of joint symmetry in certain cases. A key new idea is the construction of infinite chains of partitions that are independent of $n$ but induce the joint symmetry for all $n$ simultaneously. Using these methods, we prove combinatorially that for $0leq kleq 9$ and all $n$, the terms in $C_n(q,t)$ of total degree $binom{n}{2}-k$ have the required symmetry property.
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
The $q,t$-Catalan numbers can be defined using rational functions, geometry related to Hilbert schemes, symmetric functions, representation theory, Dyck paths, partition statistics, or Dyck words. After decades of intensive study, it was eventually p
The $q,t$-Catalan number $mathrm{Cat}_n(q,t)$ enumerates integer partitions contained in an $ntimes n$ triangle by their dinv and external area statistics. The paper [LLL18 (Lee, Li, Loehr, SIAM J. Discrete Math. 32(2018))] proposed a new approach to
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.
The aim of this paper is two-fold. We first prove several new interpretations of a kind of $(q,t)$-Catalan numbers along with their corresponding $gamma$-expansions using pattern avoiding permutations. Secondly, we give a complete characterization of