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Register a new userCombinatorics of certain higher $q,t$-Catalan polynomials: chains, joint symmetry, and the Garsia-Haiman formula
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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.
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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
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 $t$.
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 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.
A polynomial $A(q)=sum_{i=0}^n a_iq^i$ is said to be unimodal if $a_0le a_1le cdots le a_kge a_{k+1} ge cdots ge a_n$. We investigate the unimodality of rational $q$-Catalan polynomials, which is defined to be $C_{m,n}(q)= frac{1}{[n+m]} left[ m+n atop nright]$ for a coprime pair of positive integers $(m,n)$. We conjecture that they are unimodal with respect to parity, or equivalently, $(1+q)C_{m+n}(q)$ is unimodal. By using generating functions and the constant term method, we verify our conjecture for $mle 5$ in a straightforward way.
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