Decreasing Subsequences in Permutations and Wilf Equivalence for Involutions Article

Received: 04 June 2004 Revised: 04 May 2005 Accepted: 05 May 2005 DOI :
10.1007/s10801-005-4625-1

Cite this article as: Bousquet-Mélou, M. & Steingrímsson, E. J Algebr Comb (2005) 22: 383. doi:10.1007/s10801-005-4625-1
Abstract In a recent paper, Backelin, West and Xin describe a map φ^{*} that recursively replaces all occurrences of the pattern k ... 21 in a permutation σ by occurrences of the pattern (k −1)... 21 k . The resulting permutation φ^{*} (σ) contains no decreasing subsequence of length k . We prove that, rather unexpectedly, the map φ^{*} commutes with taking the inverse of a permutation.

In the BWX paper, the definition of φ^{*} is actually extended to full rook placements on a Ferrers board (the permutations correspond to square boards), and the construction of the map φ^{*} is the key step in proving the following result. Let T be a set of patterns starting with the prefix 12... k . Let T ′ be the set of patterns obtained by replacing this prefix by k ... 21 in every pattern of T . Then for all n , the number of permutations of the symmetric group \({\cal S}\) _{n} that avoid T equals the number of permutations of \({\cal S}\) _{n} that avoid T ′.

Our commutation result, generalized to Ferrers boards, implies that the number of involutions of \({\cal S}\) _{n} that avoid T is equal to the number of involutions of \({\cal S}\) _{n} avoiding T ′, as recently conjectured by Jaggard.

Keywords pattern avoiding permutations Wilf equivalence involutions decreasing subsequences prefix exchange Both authors were partially supported by the European Commission's IHRP Programme, grant HPRN-CT-2001-00272, “Algebraic Combinatorics in Europe”

References 1.

E. Babson and E. Steingrímsson, “Generalized permutation patterns and a classification of the Mahonian statistics,” Sém. Lothar. Combin ., 44:Art. B44b, (2000), 18 pp. (electronic).

2.

E. Babson and J. West, “The permutations 123

p _{4} ...

p _{m} and 321

p _{4} ...

p _{m} are Wilf-equivalent,”

Graphs Combin .

16 (4) (2000), 373–380.

MathSciNet 3.

J. Backelin, J. West, and G. Xin, “Wilf-equivalence for singleton classes,” in Proceeedings of the 13th Conference on Formal Power Series and Algebraic Combinatorics , H. Barcelo and V. Welker (Eds.), Arizona State University, 2001, pp. 29–38 (To appear in Adv. in Appl. Math .).

4.

E. Barcucci, A. Del Lungo, E. Pergola, and R. Pinzani, “Some permutations with forbidden subsequences and their inversion number,”

Discrete Math .

234 (1–3) (2001), 1–15.

MathSciNet 5.

M. Bóna, “Exact enumeration of 1342-avoiding permutations: A close link with labeled trees and planar maps,”

J. Combin. Theory Ser. A 80 (2) (1997), 257–272.

CrossRef MATH MathSciNet 6.

M. Bousquet-Mélou, “Four classes of pattern-avoiding permutations under one roof: Generating trees with two labels,” Electron. J. Combin . 9 (2), (2003).

7.

A. Claesson and T. Mansour, “Enumerating permutations avoiding a pair of Babson-Steingrí msson patterns,” Ars Combinatoria , to appear.

8.

I. Gessel, “Symmetric functions and P-recursiveness,”

J. Combin. Theory Ser. A 53 (2) (1990), 257–285.

CrossRef MATH MathSciNet 9.

I. Gessel, J. Weinstein, and H.S. Wilf, “Lattice walks in

Z ^{d} and permutations with no long ascending subsequences,”

Electron. J. Combin .

5 (1) (1998), 11.

MathSciNet 10.

D. Gouyou-Beauchamps, “Standard Young tableaux of height 4 and 5,”

European J. Combin .

10 (1) (1989), 69–82.

MATH MathSciNet 11.

O. Guibert, “Combinatoire des permutations à motifs exclus, en liaison avec mots, cartes planaires et tableaux de Young,” Ph.D. thesis, LaBRI, Université Bordeaux 1, 1995.

12.

O. Guibert, E. Pergola, and R. Pinzani, “Vexillary involutions are enumerated by Motzkin numbers,”

Ann. Comb .

5 (2) (2001), 153–174.

MathSciNet 13.

G. Huet, “Confluent reductions: Abstract properties and applications to term rewriting systems,”

J. Assoc. Comput. Mach .

27 (4) (1980) 797–821.

MATH MathSciNet 14.

A.D. Jaggard, “Prefix exchanging and pattern avoidance for involutions,”

Electron. J. Combin .

9 (2)(2003), 16.

MathSciNet 15.

S. Kitaev and T. Mansour, “A survey of certain pattern problems,” Preprint, 2004.

16.

C. Krattenthaler, “Permutations with restricted patterns and Dyck paths,”

Adv. in Appl. Math .

27 (2/3) (2001), 510–530.

MATH MathSciNet 17.

A. Regev, “Asymptotic values for degrees associated with strips of Young diagrams,”

Adv. in Math .

41 (2) (1981) 115–136.

CrossRef MATH MathSciNet 18.

C. Schensted, “Longest increasing and decreasing subsequences,”

Canad. J. Math .

13 (1961), 179–191.

MATH MathSciNet 19.

R. Simion and F.W. Schmidt, “Restricted permutations,”

European J. Combin .

6 (4) (1985) 383–406.

MathSciNet 20.

Z.E. Stankova, “Forbidden subsequences,”

Discrete Math .

132 (1/3) (1994), 291–316.

MATH MathSciNet 21.

Z.E. Stankova, “Classification of forbidden subsequences of length 4,”

European J. Combin .

17 (5) (1996), 501–517.

CrossRef MATH MathSciNet 22.

Z.E. Stankova and J. West, “A new class of Wilf-equivalent permutations,”

J. Algebraic Combin .

15 (3) (2002), 271–290.

CrossRef MathSciNet 23.

G. Viennot, “Une forme géométrique de la correspondance de Robinson-Schensted,” in Combinatoire et représentation du groupe symétrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur, Strasbourg, 1976) . Lecture Notes in Math., vol. 579. Springer, Berlin, 1977, pp. 29–58.

24.

J. West, “Permutations with Forbidden Subsequences, and Stack-Sortable Permutations,” Ph.D. thesis, MIT, 1990.

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations 1. CNRS LaBRI, Université Bordeaux 1 Talence Cedex France 2. Matematik Chalmers tekniska Högskola och Göteborgs Universitet Göteborg Sweden