Title: Chain decompositions for q,t-Catalan numbers
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The q,t-Catalan numbers Cat_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. Work of Garsia, Haglund, and Haiman has given us combinatorial formulas for Cat_n(q,t) as sums of Dyck vectors weighted by area and dinv. This talk narrates our ongoing quest for a bijective proof of the notorious symmetry property Cat_n(q,t)=Cat_n(t,q).
We describe some structural decompositions of integer partitions into infinite chains that can be paired to prove the symmetry of certain coefficients in Cat_n(q,t). The chains are built from initial objects by applying an operator NU that increases dinv by 1 and reduces area by 1. A remarkable feature of these chains is that they are independent of n and explain symmetry for all n simultaneously. Our chain construction leads to a combinatorial proof that for all k<12 and all n, the terms in Cat_n(q,t) of total degree n(n-1)/2-k satisfy the required symmetry property.