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A Combinatorial Approach to the Symmetry of <tex> $q,t$</tex>-Catalan Numbers
Lee, Kyungyong,Li, Li,Loehr, Nicholas A. Society for Industrial and Applied Mathematics 2018 SIAM Journal on Discrete Mathematics Vol.32 No.1
<P>The <italic toggle='yes'><TEX>$q,t$</TEX>-Catalan numbers</I> <TEX>$C_n(q,t)$</TEX> are polynomials in <TEX>$q$</TEX> and <TEX>$t$</TEX> that reduce to the ordinary Catalan numbers when <TEX>$q=t=1$</TEX>. These polynomials have important connections to representation theory, algebraic geometry, and symmetric functions. Haglund and Haiman discovered combinatorial formulas for <TEX>$C_n(q,t)$</TEX> 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 <TEX>$C_n(q,t)=C_n(t,q)$</TEX>. 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 <TEX>$n$</TEX> but induce the joint symmetry for all <TEX>$n$</TEX> simultaneously. Using these methods, we prove combinatorially that for <TEX>$0\leq k\leq 9$</TEX> and all <TEX>$n$</TEX>, the terms in <TEX>$C_n(q,t)$</TEX> of total degree <TEX>$\binom{n}{2}-k$</TEX> have the required symmetry property.</P>