Enumeration of idempotents in planar diagram monoids

We classify and enumerate the idempotents in several planar diagram monoids: namely, the Motzkin, Jones (a.k.a. Temperley-Lieb) and Kauffman monoids. The classification is in terms of certain vertex- and edge-coloured graphs associated to Motzkin diagrams. The enumeration is necessarily algorithmic in nature, and is based on parameters associated to cycle components of these graphs. We compare our algorithms to existing algorithms for enumerating idempotents in arbitrary (regular *-) semigroups, and give several tables of calculated values.Comment: Majorly revised (new title, new abstract, one additional author), 24 pages, 6 figures, 8 tables, 5 algorithm

Dyck paths and pattern-avoiding matchings

How many matchings on the vertex set V={1,2,...,2n} avoid a given configuration of three edges? Chen, Deng and Du have shown that the number of matchings that avoid three nesting edges is equal to the number of matchings avoiding three pairwise crossing edges. In this paper, we consider other forbidden configurations of size three. We present a bijection between matchings avoiding three crossing edges and matchings avoiding an edge nested below two crossing edges. This bijection uses non-crossing pairs of Dyck paths of length 2n as an intermediate step. Apart from that, we give a bijection that maps matchings avoiding two nested edges crossed by a third edge onto the matchings avoiding all configurations from an infinite family, which contains the configuration consisting of three crossing edges. We use this bijection to show that for matchings of size n>3, it is easier to avoid three crossing edges than to avoid two nested edges crossed by a third edge. In this updated version of this paper, we add new references to papers that have obtained analogous results in a different context.Comment: 18 pages, 4 figures, important references adde