On Noncrossing and Nonnesting Partitions of Type D

Abstract

Abstract. We present an explicit bijection between noncrossing and nonnesting partitions of Coxeter systems of type D which preserves openers, closers and transients. OverviewThe lattice of set partitions of a set of n elements can be interpreted as the intersection lattice for the hyperplane arragement corresponding to a root system of type A n−1 , i.e. the symmetric group of n objects, S n . In particular, two of its subposets are very wellbehaved and widely studied, i.e. the lattice of, respectively, noncrossing and nonnesting partitions, which have a lot of interesting combinatorial properties, see e.g. [4,7,8,9,12,14,15,18,21,22,23,24,25] and the references therein.Recently, Victor Reiner [19] and Alexander Postnikov, see Christos Athanasiadis' paper [2, §6, Remark 2] generalized the notion of, respectively, noncrossing and nonnesting partitions to all classical reflection groups, and a new very active research area sprung up, namely generalizing previous known results held for noncrossing and nonnesting partitions of type A to their type B and D analogue, see e.g. [1,3,13,26,27] and the references therein.In particular, it is well known that in type A the number of noncrossing partition equals the number of nonnesting partition, and several bijections have been established. Recently one of us in [16,17] solved the corresponding problem for the type B (and type C) presenting an explicit bijection which falls back to one already known when restricted to the type A.The goal of this paper is to present an explicit bijection for the type D, therefore solving the problem for all classical Weyl groups, and to show that this bijection has interesting additional combinatorial properties. For instance, for any classical reflection group and any partition it is possible to define three different maps, called openers, closers, and transients, going from the partition itself to subsets of the involved reflection group, and we show that our bijection preserves all these functions. Furthermore, our map remains a bijection if restricted to the two sets of respectively, noncrossing and nonnesting partitions which are both of type B and type D, and therefore it is also a bijection between the partitions which are noncrossing in type D but which are actually crossing in type B, and the partition which are nonnesting in type D but which are actually nesting in type B.Very recently, two other bijections have been presented for solving the type D problem, see [10,20], but both of them have different designs and settings. Moreover in [10] openers, closers and transients are not preserved.2000 Mathematics Subject Classification. 05A18; 05E15.

Gray codes and lexicographical combinatorial generation for nonnesting and sparse nonnesting set partitions

“…[2,12,13,43], so it is quite natural to ask for a generalization of our results to Weyl groups of type B and D.…”

“…There are several bijections between noncrossing and nonnesting set partitions (see, for example [2,13,17,29,45]), and since in [19] a Gray code for noncrossing partitions is presented, it is tempting to try employing these bijections in order to obtain a Gray code for nonnesting partitions. But, as referred in [48], a Gray code for a combinatorial class is intrinsically bound to the representation of objects in the class, and in the present case, the Gray code is not preserved under bijection.…”

“…In particular, there exist several bijections between noncrossing and nonnesting partitions, see e.g. [12,13,17,23,29,45], so it is very natural to wonder if all results which hold for noncrossing partitions hold for nonnesting partitions as well. Unfortunately, nonnesting partitions are much more mysterious and intricate, and plenty of results valid for noncrossing partitions do not translate to nonnesting partitions.…”

“…[1,2,[11][12][13]17,23,29,30,34,43,45,49], and among other things they have applications to the theory of free probability, see [37,51,52]. In particular, there exist several bijections between noncrossing and nonnesting partitions, see e.g.…”

New interpretations for noncrossing partitions of classical types

“…In fact, they showed that their bijections are the unique ones preserving those statistics. There are other bijections between noncrossing and nonnesting partitions of classical types due to Rubey and Stump [18] for type B and Conflitti and Mamede [8] for type D. However their bijections preserve not the types but 'openers' and 'closers'.…”