The q, t-Catalan numbers and the space of diagonal harmonics : with an appendix on the combinatorics of Macdonald polynomials

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q,t-Catalan numbers and knot homology (2010, methodology): “nes. For a D-type step x ∈ π let nump(x) denote the number of peak lines to the east from x. Now let b(π) = bounce(T(π))+ X x nump(x), where summation is done over all D-steps x. Deﬁnition 2.15. ([5],[11]) The (q,t)-Schr¨oder polynomials are deﬁned as (2.6) Sn,k(q,t) = X π q( n 2)+ k 2−S(π)tb(π), wh”
Refined knot invariants and Hilbert schemes (2013, background): “ic β(P) Conjecture 6.7. The following equation holds: X D∈Y m/n qδm,n−| D|th+( ) Y P∈v(D) (1− ut−β(P)) = Pe n,m(u,q,t) For m = n + 1 a similar identity was conjectured in [15] and proved in [33] (see [34] and [47, Section A.3] for more details). 6.5. The Rational Shufﬂe Conjecture. The symmetric pol”
The representation of the symmetric group on m-Tamari intervals (2012, background): “This formula, which generalizes (1), is easily proved using the cycle lemma.”
Affine permutations and rational slope parking functions (2014, neutral): “[10, 11], related the combinatorics of Catalan numbers and parking functions to the space of diagonal harmonics.”