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

Abstract

We present combinatorial Gray codes and explicit designs of efficient algorithms for lexicographical combinatorial generation of the sets of nonnesting and sparse nonnesting set partitions of length n.

Gray codes for noncrossing and nonnesting partitions of classical types

“…In [7], Huemer et al defined a graph structure on the set of classical noncrossing partitions by declaring two partitions adjacent if they differ by the move of a single element from one block to another, and showed that this set has a Hamilton cycle. Recently, this result was also obtained for the set of all classical nonnesting partitions [6]. Classical noncrossing and nonnesting partitions are members of a broader class of objects, known as Coxeter-Catalan objects, associated with the symmetric group S n .…”

“…In this paper we generalize the type A results of [6,7] for Weyl groups of type B and D, constructing Hamilton cycles for the sets of noncrossing partitions of types B and D, and nonnesting partitions of type B, where now we declare two type B (or D) partitions adjacent if they differ by the move of at most two elements from one block to another. Although computational examples suggest that the set of type D nonnesting partitions is hamiltonian as well, we were only able to construct a Hamilton cycle on the subset formed by all those type D nonnesting partitions without zero-block.…”

“…Although computational examples suggest that the set of type D nonnesting partitions is hamiltonian as well, we were only able to construct a Hamilton cycle on the subset formed by all those type D nonnesting partitions without zero-block. In [6] we designed an efficient algorithm for a lexicographic combinatorial generation of nonnesting set partitions of type A, using a characterization of such partitions in terms of arcs. This characterization is used in this paper to generate all nonnesting partitions of type B and all nonnesting partitions of type D with zero-block in lexicographic order.…”

“…Thus it forms a Hamilton cycle with distance 2 for the set N N B (n). In [6], an algorithm GenT ot(n) was presented to generate all type A nonnesting partitions of [n] in lexicographic order of their arcs, i.e. first according to the number of arcs and then, for partitions with the same number of arcs, according to their openers.…”

“…Apply the algorithm GenT ot(n) given in [6] to generate in lexicographic order the list of all type A nonnesting partitions of [n], and consider the sublist π 1 , . .…”