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Cycles and sorting index for matchings and restricted permutations

Abstract : We prove that the Mahonian-Stirling pairs of permutation statistics $(sor, cyc)$ and $(∈v , \mathrm{rlmin})$ are equidistributed on the set of permutations that correspond to arrangements of $n$ non-atacking rooks on a fixed Ferrers board with $n$ rows and $n$ columns. The proofs are combinatorial and use bijections between matchings and Dyck paths and a new statistic, sorting index for matchings, that we define. We also prove a refinement of this equidistribution result which describes the minimal elements in the permutation cycles and the right-to-left minimum letters.
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Svetlana Poznanović. Cycles and sorting index for matchings and restricted permutations. 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. pp.731-742, ⟨10.46298/dmtcs.2338⟩. ⟨hal-01229666⟩

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