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arXiv · 2508.20935

Falling stars: a fall-decorated rational shuffle theorem

Alessandro Iraci·Roberto Pagaria·Giovanni Paolini
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Abstract

In this paper, we formulate a rational analog of the fall Delta theorem and the Delta square conjecture. We find a new dinv statistic on fall-decorated paths on a $(m+k) \times (n+k)$ rectangle that simultaneously extends the previously known dinv statistics on decorated square objects and non-decorated rectangular objects. We prove a symmetric function formula for the $q,t$-generating function of fall-decorated rectangular Dyck paths as a skewing operator applied to $e_{m,n+km}$ and, conditionally on the rectangular paths conjecture, an analog formula for fall-decorated rectangular paths.

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Alessandro Iraci, Roberto Pagaria, Giovanni Paolini. 2025-08-28. Falling stars: a fall-decorated rational shuffle theorem. https://doi.org/10.1017/fms.2026.10232

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