A256672 Number of idempotents in the Motzkin monoid of degree n.

1, 2, 7, 31, 153, 834, 4839, 29612, 188695, 1243746, 8428597, 58476481, 413893789, 2980489256, 21787216989, 161374041945, 1209258743839, 9155914963702, 69969663242487, 539189056700627

Offset: 0

Author

Nick Loughlin, Apr 07 2015

Keywords

• nonn
• hard
• more

Comments

a(n) is the number of ways of drawing any number of nonintersecting chords joining n (labeled) points on a circle, such that when gluing the second half of one copy to the first half of the other so that each point k along the intersection is glued to n+1-k, the result is homotopic to the original.

a(n+1) > a(n) for every n.

The structure of the Motzkin monoid (and particularly its idempotents and some associated orderings) is governed intimately by the combinatorics of so-called Motzkin paths and Motzkin words, which are related to Dyck paths and words respectively by insertion of punctuation into the words, or marking/coloring subpaths.

Bounded above by A026945, strictly for n > 1. Bounded below by the square of A001006, strictly for n > 1.

Examples

			There is one empty graph, which is idempotent under the composition, hence a(0)=1.
There are two on 1 pair of points, the clique and the discrete graph; both are idempotents under the composition, hence a(1)=2.
		

Links

• I. Dolinka, J. East et al, Idempotent Statistics of the Motzkin and Jones Monoids, arXiv: 1507.04838 [math.CO], 2015, Table 2.
• Tom Halverson, Gelfand Models for Diagram Algebras, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013).
• Tom Halverson and Mike Reeks, Gelfand Models for Diagram Algebras, Journal of Algebraic Combinatorics (2014)41, 229-255.
• J. D. Mitchell et al., Semigroups - GAP package, Version 2.7.4, March, 2016.
• J. D. Mitchell, Counting idempotents in a monoid of partitions, C++ program, October, 2016
• Eliezer Posner, Kris Hatch, and Megan Ly, Presentation of the Motzkin Monoid, arXiv:1301.4518 [math.RT], 2013.

Crossrefs

Cf. A001006, A026945.

Extensions

a(9)-a(13) corrected and a(14)-a(16) computed using the Semigroups package for GAP added by James Mitchell, Apr 12 2016

a(17)-a(19) added by James Mitchell, Apr 01 2017