A new dinv statistic on fall-decorated rectangular Dyck paths yields a proven q,t-generating function formula equal to the skewing operator applied to e_{m,n+km}, extending the rational shuffle theorem and the rise Delta theorem.
A combinatorial skewing formula for the Rise Delta Theorem
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abstract
We prove that the symmetric function $\Delta'_{e_{k-1}}e_n$ appearing in the Delta Conjecture can be obtained from the symmetric function in the Rational Shuffle Theorem by applying a Schur skewing operator. This generalizes a formula by the first and third authors for the Delta Conjecture at $t=0$, and follows from work of Blasiak, Haiman, Morse, Pun, and Seelinger. Our main result is that we also provide a purely combinatorial proof of this skewing identity, giving a new proof of the Rise Delta Theorem from the Rational Shuffle Theorem.
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Falling stars: a fall-decorated rational shuffle theorem
A new dinv statistic on fall-decorated rectangular Dyck paths yields a proven q,t-generating function formula equal to the skewing operator applied to e_{m,n+km}, extending the rational shuffle theorem and the rise Delta theorem.