Sweep maps: A continuous family of sorting algorithms

TL;DR

Sweep maps are a family of bijective sorting algorithms on lattice paths that unify and generalize many known combinatorial algorithms, providing concise formulas for important q,t-polynomials and connecting to rational Catalan combinatorics.

Contribution

The paper introduces sweep maps as a unifying framework for various lattice path algorithms, offering new insights and conjectures on their bijectivity and inversion.

Findings

Sweep maps are bijective in many cases and unify existing algorithms.

They provide concise formulas for q,t-Catalan and related numbers.

Inversion can be achieved using bounce paths in special cases.

Abstract

We define a family of maps on lattice paths, called sweep maps, that assign levels to each step in the path and sort steps according to their level. Surprisingly, although sweep maps act by sorting, they appear to be bijective in general. The sweep maps give concise combinatorial formulas for the q,t-Catalan numbers, the higher q,t-Catalan numbers, the q,t-square numbers, and many more general polynomials connected to the nabla operator and rational Catalan combinatorics. We prove that many algorithms that have appeared in the literature (including maps studied by Andrews, Egge, Gorsky, Haglund, Hanusa, Jones, Killpatrick, Krattenthaler, Kremer, Orsina, Mazin, Papi, Vaille, and the present authors) are all special cases of the sweep maps or their inverses. The sweep maps provide a very simple unifying framework for understanding all of these algorithms. We explain how inversion of the sweep map (which is an open problem in general) can be solved in known special cases by finding a "bounce path" for the lattice paths under consideration. We also define a generalized sweep map acting on words over arbitrary alphabets with arbitrary weights, which is also conjectured to be bijective.