International audience We introduce $k$-crossings and $k$-nestings of permutations. We show that the crossing number and the nesting number of permutations have a symmetric joint distribution. As a corollary, the number of $k$-noncrossing permutations is equal to the number of $k$-nonnesting permutations. We also provide some enumerative results for $k$-noncrossing permutations for some values of $k$. Nous introduisons les $k$-chevauchement d'arcs et les $k$-empilements d'arcs de permutations. Nous montrons que l'index de chevauchement et l'index de empilement ont une distribution conjointe symétrique pour les permutations de taille $n$. Comme corollaire, nous obtenons que le nombre de permutations n'ayant pas un $k$-chevauchement est égal au nombre de permutations n'ayant un $k$-empilement. Nous fournissons également quelques résultats énumératifs.
Abstract. Walks on Young's lattice of integer partitions encode many objects of algebraic and combinatorial interest. Chen et al. established connections between such walks and arc diagrams. We show that walks that start at ∅, end at a row shape, and only visit partitions of bounded height are in bijection with a new type of arc diagram -open diagrams. Remarkably two subclasses of open diagrams are equinumerous with well known objects: standard Young tableaux of bounded height, and Baxter permutations. We give an explicit combinatorial bijection in the former case.
We describe a generating tree approach to the enumeration and exhaustive generation of knonnesting set partitions and permutations. Unlike previous work in the literature using the connections of these objects to Young tableaux and restricted lattice walks, our approach deals directly with partition and permutation diagrams. We provide explicit functional equations for the generating functions, with k as a parameter.The heart of the argument maps the objects to a sequence of Young tableaux, and the bijection is achieved by taking the transpose of the tableaux.From the enumerative point of view, results have been far less forthcoming. The enumeration of k-nonnesting matchings was completed by Chen et al. in their master work by using a bijection to collections of noncrossing Dyck paths whose enumeration was known. Consequently, the results are elegant. Also, the class of 2-noncrossing (or simply noncrossing) partitions is counted by Catalan numbers, and so one could hope for a similarly beautiful enumeration scheme. So far, only exact formulas exist for 3-noncrossing set partitions, and beyond that, a far more complicated structure is conjectured. Specifically, Bousquet-Mélou and Xin [4] did a complete functional equation analysis, and determined explicit exact and asymptotic enumeration formulas.Although their functional equations can be adapted to enumerate set partitions with higher noncrossing numbers, they conjecture that the structure of the generating function becomes more complicated. Conjecture 1.1 (Bousquet-Mélou,Xin 2005 [4]). For every k > 3, the generating function of k-noncrossing set partitions is not D-finite. Mishna and Yen [26] determined functional equations for k-nonnesting set partitions, and described a process for isolating coefficients, giving additional evidence for Conjecture 1.1. A more recent development considers set partitions where the maximum nesting size and the maximum crossing size are controlled simultaneously. The resulting generating functions are rational [25], and (in theory) can be determined explicitly. They can be summed together for a different picture of generating functions for k-nonnesting partitions. 1.2. Permutations. The arc diagram is a convenient way to view permutations, as it simultaneously highlights both the cycle structure, and the line notation structure. Corteel [14] used arc diagrams to directly connect various permutation statistics, such as exceedences and permutation patterns to occurrences of nestings and crossings. Burill, Mishna and Post [8] extended this definition, and directly adapted the results of Chen et al. to prove that k-nonnesting permutations are in bijection with k-noncrossing permutations. They computed enumerative data by brute force, by finding the nesting and crossing numbers of each permutation. Consequently, the data is restricted to small sizes only. The generating function for permutations in which both the maximum nesting and maximum crossing numbers are controlled is rational [31], similarly to the case of set partitions. Exten...
International audience Tableau sequences of bounded height have been central to the analysis of $k$-noncrossing set partitions and matchings. We show here that families of sequences that end with a row shape are particularly compelling and lead to some interesting connections. First, we prove that hesitating tableaux of height at most two ending with a row shape are counted by Baxter numbers. This permits us to define three new Baxter classes which, remarkably, do not obviously possess the antipodal symmetry of other known Baxter classes. Oscillating tableau of height bounded by $k$ ending in a row are in bijection with Young tableaux of bounded height 2$k$. We discuss this recent result, and somegenerating function implications. Many of our proofs are analytic in nature, so there are intriguing combinatorial bijections to be found. Les séquences de tableau de hauteur bornée sont au centre de l’analyse des partages et couplages. Nous montrons que les familles de séquences qui se terminent par une seule ligne sont particulièrement fascinantes. Tout d’abord, nous prouvons que les tableaux hésitants de hauteur au plus deux se terminant par une seule ligne sont dénombrés par les nombres de Baxter. Cela nous permet de définir trois nouvelles classes Baxter qui, remarquablement, ne possèdent évidemment pas la symétrie antipode des autres classes Baxter connus. Nous discutons le résultat récent qui dit que les tableaux oscillants de hauteur au plus $k$ se terminant dans une ligne sont en bijection avec les tableaux de Young de hauteur au plus 2$k$. Nos preuves sont analytiques, il y a donc des bijections combinatoiresintrigantes à trouver.
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