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A Weight-Preserving Bijection Between Schröder Paths and Schröder Permutations
Jason Bandlow1, Eric S. Egge2, and Kendra Killpatrick1
1Mathematics Department, Colorado State University, Fort Collins, CO 80523, USA
{bandlow, killpatr}@math.colostate.edu
2Department of Mathematics, Gettysburg College,Gettysburg, PA 17325, USA
eggee@member.ams.org
Annals of Combinatorics 6 (3) p.235-248 September, 2002
AMS Subject Classification: 05A15, 05A19, 05E99
Abstract:
In 1993 Bonin, Shapiro, and Simion showed that the Schröder numbers count certain kinds of lattice paths; these paths are now called Schröder paths. In 1995 West showed that the Schröder numbers also count permutations which avoid the patterns 4231 and 4132. Using some technical machinery, Barcucci, Del Lungo, Pergola, and Pinzani showed in 1999 that a certain q-analog of the Schröder numbers, called the Schröder polynomial, is the generating function for a statistic called the area statistic on Schröder paths and is also the generating function for the inversion number on permutations which avoid 4231 and 4132. In this paper we give a constructive bijection from Schröder paths to permutations which avoid 4231 and 4132 that takes the area statistic on Schröder paths to the inversion number on permutations which avoid 4231 and 4132.
Keywords: Schröder paths, Schröder permutations, Schröder polynomials, Catalan polynomials, pattern-avoiding permutations, lattice paths, inversion number

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