TY - JOUR

T1 - Sweep maps

T2 - A continuous family of sorting algorithms

AU - Armstrong, Drew

AU - Loehr, Nicholas A.

AU - Warrington, Gregory S.

N1 - Publisher Copyright:
© 2015 Elsevier Inc.

PY - 2015/10/2

Y1 - 2015/10/2

N2 - 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, Vaillé, 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.

AB - 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, Vaillé, 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.

KW - Diagonal harmonics

KW - Dyck paths

KW - Lattice paths

KW - Q, t-Catalan numbers

KW - Sorting algorithms

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U2 - 10.1016/j.aim.2015.07.012

DO - 10.1016/j.aim.2015.07.012

M3 - Article

AN - SCOPUS:84939238602

VL - 284

SP - 159

EP - 185

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -