Tree-like tableaux
TL;DR: An elementary insertion procedure is exhibited which gives a clear proof that tableaux of size n are counted by n!, and which moreover respects most of the well-known statistics studied originally on alternative and permutation tableaux.
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Abstract: In this work we introduce and study tree-like tableaux, which are certain fillings of Ferrers diagrams in simple bijection with permutation tableaux and alternative tableaux. We exhibit an elementary insertion procedure on our tableaux which gives a clear proof that tableaux of size n are counted by n!, and which moreover respects most of the well-known statistics studied originally on alternative and permutation tableaux. Our insertion procedure allows to define in particular two simple new bijections between tree-like tableaux and permutations: the first one is conceived specifically to respect the generalized pattern 2-31, while the second one respects the underlying tree of a tree-like tableau.
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Bijections for Permutation Tableaux
Sylvie Corteel,Philippe Nadeau +1 more
TL;DR: In this paper, a new bijection between permutation tableaux and permutations is proposed, which shows how natural statistics on tableaux are equidistributed to classical statistics on permutations: descents, RL-minima and pattern enumerations.
52
Asymptotic behavior of some statistics in Ewens random permutations
TL;DR: In this article, the authors present a general method to find limiting laws for some renormalized statistics on random permutations, including the number of occurrences of any given dashed pattern.
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Combinatorics of non-ambiguous trees
TL;DR: The enumeration of non-ambiguous trees satisfying some additional constraints allows us to give elegant combinatorial proofs of identities due to Carlitz, and to Ehrenborg and Steingrimsson.
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Weighted Random Staircase Tableaux
Pawel Hitczenko,Svante Janson +1 more
TL;DR: A general model of random staircase tableaux in which symbols (Greek letters) that appear in staircase tableau may have arbitrary positive weights and is derived, which leads to a two-parameter family of polynomials, generalizing the classical Eulerian poynomials.
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Combinatorics of the permutation tableaux of type B
TL;DR: The signed permutation statistics arising here are of several kinds. First, there are several variants of descents and excedances as discussed by the authors, and more precisely of flag descents or excedance.
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Total positivity, Grassmannians, and networks
TL;DR: In this article, the authors discuss the relationship between total positivity and planar directed networks and show that the inverse boundary problem for these networks is naturally linked with the study of the totally nonnegative Grassmannian.
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On Some Properties of Permutation Tableaux
TL;DR: In this paper, the relations between various permutation statistics and properties of permutation tableaux are investigated, in particular on the distribution of the bistatistic of numbers of rows and essential ones in permutation tables.
Total positivity for cominuscule Grassmannians
Thomas Lam,Lauren Williams +1 more
TL;DR: In this paper, the combinatorics of the nonneg-ative part (G/P ) of a cominuscule Grassmannian are explored in terms of pattern avoidance.
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