The Foata correspondence, cycle lengths and anomalies

Abstract : In their study of the densest jammed configurations for theater models, Krapivsky and Luck observe that two classes of permutations have the same cardinalities and ask for a bijection between them. In this note we show that the Foata correspondence provides the desired bijection. Krapivsky and Luck (2019) introduced the theater model as a variant of directed random sequential adsorption, where spectators sequentially select a seat in a row of L seats, with the constraint that they cannot go past a cluster of b or more consecutive occupied seats. Configurations where all the seats are eventually occupied are parametrized by permutations σ of {1,. .. , L} such that for any i between 1 and L, one cannot find b consecutive integers j + 1,. .. , j + b with j + b < i and σ(j + k) > σ(i) for all k between 1 and b. Krapivsky and Luck (2019) showed that the number D (b) L of such permutations satisfies a linear recurrence relation which implies that they have the same cardinality as the permutations of L elements with cycles of lengths at most b. The authors then asked for a bijective proof of this fact. The goal of this note is to show that the Foata correspondence provides such a bijection. In Section 1 we recall the Foata correspondence and in Section 2 we show that it provides the desired bijection. 1 The Foata correspondence Let S L be the group of permutations of {1,. .. , L}. We will represent permutations in S L by words with L distinct letters in {1,. .. , L}. For example 1423 denotes the permutation s ∈ S 4 such that s(1) = 1, s(2) = 4, s(3) = 2 and s(4) = 3. One can associate to every permutation in S L its cycle decomposition. Including the fixed points in that decomposition, the above s ∈ S 4 has cycle decomposition [1][243]. This way of writing is however not unique for two reasons: • each cycle of length d can be written in d different ways (one can freely choose what element to put first) ; • if a permutation has k cycles (including singletons corresponding to fixed points) one can have them appear in k! different orders.
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Sanjay Ramassamy. The Foata correspondence, cycle lengths and anomalies. 2019. ⟨hal-02165342⟩

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