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Register a new userTwo proofs of the $q,t$-symmetry of the generalized $q,t$-Catalan number $C_{(k_1,k_2,k_3)}(q,t)$
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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.
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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 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.
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 proved that all these definitions are equivalent. In this paper, we study the similar situation for higher $q,t$-Catalan numbers, where the equivalence of the algebraic and combinatorial definitions is still conjectural. We compute the limits of sever
We define two symmetric $q,t$-Catalan polynomials in terms of the area and depth statistic and in terms of the dinv and dinv of depth statistics. We prove symmetry using an involution on plane trees. The same involution proves symmetry of the Tutte polynomials. We also answer an open question of Garsia et al. regarding parking functions and the number of connected graphs on a fixed number of vertices.
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 understanding the symmetry property $mathrm{Cat}_n(q,t)=mathrm{Cat}_n(t,q)$ based on decomposing the set of all integer partitions into infinite chains. Each such global chain $mathcal{C}_{mu}$ has an opposite chain $mathcal{C}_{mu^*}$; these combine to give a new small slice of $mathrm{Cat}_n(q,t)$ that is symmetric in $q$ and $t$. Here we advance the agenda of [LLL18] by developing a new general method for building the global chains $mathcal{C}_{mu}$ from smaller elements called local chains. We define a local opposite property for local chains that implies the needed opposite property of the global chains. This local property is much easier to verify in specific cases compared to the corresponding global property. We apply this machinery to construct all global chains for partitions with deficit at most $11$. This proves that for all $n$, the terms in $mathrm{Cat}_n(q,t)$ of degree at least $binom{n}{2}-11$ are symmetric in $q$ and $t$.
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