05660514

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05660514 Claesson, Anders Kitaev, Sergey Classification of bijections between 321- and 132-avoiding permutations. S\'emin. Lothar. Comb. 60, B60d, 30 p., electronic only (2008). 2008 Universit\"at Wien, Fakult\"at f\"ur Mathematik, Wien EN bijection permutation statistics equidistribution pattern avoidance Catalan structures Zbl 0191.17903 Zbl 0711.05006 Zbl 0615.05002 Zbl 0994.05001

emis:journals/SLC/wpapers/s60claekit.html

Summary: It is well-known, and was first established by {\it D.E. Knuth} [The art of computer programming. Vol. 1: Fundamental algorithms, London: Addison-Wesley Publishing Company. (1968; Zbl 0191.17903)] that the number of 321-avoiding permutations is equal to that of 132-avoiding permutations. In the literature one can find many subsequent bijective proofs of this fact. It turns out that some of the published bijections can easily be obtained from others. In this paper we describe all bijections we were able to find in the literature and show how they are related to each other via "trivial" bijections. We classify the bijections according to statistics preserved (from a fixed, but large, set of statistics), obtaining substantial extensions of known results. Thus, we give a comprehensive survey and a systematic analysis of these bijections. We also give a recursive description of the algorithmic bijection given by {\it D. Richards} [``Ballot sequences and restricted permutations'', Ars Comb. 25, 83-86 (1988; Zbl 0711.05006)] in 1988 (combined with a bijection by Knuth from 1969). This bijection is equivalent to the celebrated bijection of {\it R. Simion} and {\it F.W. Schmidt} [``Restricted permutations'', Eur. J. Comb. 6, 383--406 (1985; Zbl 0615.05002)], as well as to the bijection given by {\it C. Krattenthaler} [``Permutations with restricted patterns and Dyck paths'', Adv. Appl. Math. 27, No.2-3, 510--530 (2001; Zbl 0994.05001)], and it respects 11 statistics --- the largest number of statistics any of the bijections respects.