Reduction of m-regular noncrossing partitions

Abstract

In this paper, we present a reduction algorithm which transforms m-regular partitions of [n] = {1, 2, . . . , n} to (m − 1)-regular partitions of [n − 1]. We show that this algorithm preserves the noncrossing property. This yields a simple explanation of an identity due to Simion-Ullman and Klazar in connection with enumeration problems on noncrossing partitions and RNA secondary structures. For ordinary noncrossing partitions, the reduction algorithm leads to a representation of noncrossing partitions in terms of independent arcs and loops, as well as an identity of Simion and Ullman which expresses the Narayana numbers in terms of the Catalan numbers.

Identities from weighted Motzkin paths

“…Note that identity (1.1) can be derived from the following identity due to Simion and Ullman [14], see also [3]:…”

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

“…[24,42,56], and combining the main result of [11] with one of the known bijections linking noncrossing and nonnesting partitions, we get that # NN (n, k) = # SNN (n + 1, k), but the resulting bijection which exhibits such equality is extremely complicated and "ugly".…”

“…We remark that in [11] an algorithm has been presented for the corresponding bijection when one replaces the "nonnesting" condition with the "noncrossing" one.…”

“…Denote by SP(n) the set of sparse set partitions of [n]; it is easy to check that the cardinality of SP(n) is equal to the number of set partitions of [n − 1]. A bijection between these two sets can be construct as follows (see [11,24,42]). Take a set partition π of [n − 1], and consider each factor of two or more consecutive letters in the canonical sequential form of π , where a factor is the maximal interval of consecutive occurrences of one letter (xaa .…”

“…[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.…”