Refined enumeration of pattern-avoiding permutations.

    Sergi Elizalde, (MIT)

Abstract: The talk will begin with a survey of some of the main enumerative results in the subject of restricted (or pattern-avoiding) permutations. Next, recent developments and new directions will be discussed, including simultaneous avoidance of several patterns, enumeration of occurrences of a particular pattern in permutations, and generalized patterns (i.e., with the requirement that some elements occur in adjacent positions). The second part of the talk will focus on the study of statistics in restricted permutations, in which bijections to Dyck paths play an important role. We give a new bijection between 321-avoiding permutations and 132-avoiding permutations that preserves the number of fixed points and the number of excedances. We will discuss recent work with Emeric Deutsch, Toufik Mansour, Marc Noy and Igor Pak.

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