Geodesic
A Shuffle Theorem for Paths Under Any Line
Forum of Mathematics, Pi, Tome 11 (2023)

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We generalize the shuffle theorem and its $(km,kn)$ version, as conjectured by Haglund et al. and Bergeron et al. and proven by Carlsson and Mellit, and Mellit, respectively. In our version the $(km,kn)$ Dyck paths on the combinatorial side are replaced by lattice paths lying under a line segment whose x and y intercepts need not be integers, and the algebraic side is given either by a Schiffmann algebra operator formula or an equivalent explicit raising operator formula. We derive our combinatorial identity as the polynomial truncation of an identity of infinite series of $\operatorname {\mathrm {GL}}_{l}$ characters, expressed in terms of infinite series versions of LLT polynomials. The series identity in question follows from a Cauchy identity for nonsymmetric Hall–Littlewood polynomials. 
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Jonah Blasiak; Mark Haiman; Jennifer Morse; Anna Pun; George H. Seelinger. A Shuffle Theorem for Paths Under Any Line. Forum of Mathematics, Pi, Tome 11 (2023). doi: 10.1017/fmp.2023.4

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