On bijections for pattern-avoiding permutations
On bijections for pattern-avoiding permutations
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DOI：
10.1016/j.jcta.2009.03.006
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发表时间：
2009-11
期刊：
影响因子：
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通讯作者：
Jonathan Bloom;D. Saracino
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文献类型：
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作者：
Jonathan Bloom;D. Saracino
通过考虑从长度为2n的Dyck路集合到Sn（321）和Sn（132）的双射，Elizalde和Pak在[S.埃利萨尔德岛Pak，Bijections for refined restricted permutations，J. Combin. Theory Ser. A 105（2004）207 - 219]给出了双射θ：Sn（321）→ Sn（132），其保持每个σ ∈ Sn（321）中的不动点的数目和超越的数目。本文证明了Robertson在[A.罗伯逊，限制排列从加泰罗尼亚语罚款和背部，Sém。洛萨组合50（2004）B50g]也保留了每个σ中不动点的个数和超越数。我们还证明了[J.Backelin，J.West，G. Xin，Wilf-等价于单例类，Adv. in Appl.Math.38（2007）133 - 148]和[M. Bousquet-Melou，E. Steingrimsson，置换中的递减连续性和对合的Wilf等价性，J. Algebras Combin。22（2005）383 - 409]保持了这些相同的统计量，并且我们证明了从Sn（132）到Sn（213）的类似双射也是如此。
By considering bijections from the set of Dyck paths of length 2n onto each of Sn(321) and Sn(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 Θ:Sn(321)→Sn(132) that preserves the number of fixed points and the number of excedances in each σ∈Sn(321). We show that a direct bijection Γ:Sn(321)→Sn(132) introduced by Robertson in [A. Robertson, Restricted permutations from Catalan to Fine and back, Sém. Lothar. Combin. 50 (2004) B50g] also preserves the number of fixed points and the number of excedances in each σ. We also show that a bijection ϕ∗:Sn(213)→Sn(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 Sn(132) onto Sn(213) does the same.