# Falling stars: a fall-decorated rational shuffle theorem

## TL;DR

This paper introduces a rational analog of the fall Delta theorem and Delta square conjecture, defining a new dinv statistic on fall-decorated paths and proving related symmetric function formulas.

## Contribution

It extends dinv statistics to fall-decorated paths and formulates a rational analog of key conjectures in algebraic combinatorics.

## Key findings

- Defined a new dinv statistic on fall-decorated paths.
- Proved a symmetric function formula for the q,t-generating function.
- Established an analog formula for fall-decorated rectangular paths, conditional on a conjecture.

## 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.