The art of computer programming, volume 3: (2nd ed.) sorting and searching
The art of computer programming, volume 3: (2nd ed.) sorting and searching
Permutations with Restricted Patterns and Dyck Paths
Advances in Applied Mathematics
Bijections for refined restricted permutations
Journal of Combinatorial Theory Series A
Decreasing Subsequences in Permutations and Wilf Equivalence for Involutions
Journal of Algebraic Combinatorics: An International Journal
Counting permutations with no long monotone subsequence via generating trees and the kernel method
Journal of Algebraic Combinatorics: An International Journal
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By considering bijections from the set of Dyck paths of length 2n onto each of S"n(321) and S"n(132), Elizalde and Pak in [S. Elizalde, I. Pak, Bijections for refined restricted permutations, J. Combin. Theory Ser. A 105 (2004) 207-219] gave a bijection @Q:S"n(321)-S"n(132) that preserves the number of fixed points and the number of excedances in each @s@?S"n(321). We show that a direct bijection @C:S"n(321)-S"n(132) introduced by Robertson in [A. Robertson, Restricted permutations from Catalan to Fine and back, Sem. Lothar. Combin. 50 (2004) B50g] also preserves the number of fixed points and the number of excedances in each @s. We also show that a bijection @f^*:S"n(213)-S"n(321) studied in [J. Backelin, J. West, G. Xin, Wilf-equivalence for singleton classes, Adv. in Appl. Math. 38 (2007) 133-148] and [M. Bousquet-Melou, E. Steingrimsson, Decreasing subsequences in permutations and Wilf equivalence for involutions, J. Algebraic Combin. 22 (2005) 383-409] preserves these same statistics, and we show that an analogous bijection from S"n(132) onto S"n(213) does the same.