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A256672 Number of idempotents in the Motzkin monoid of degree n. 1
1, 2, 7, 31, 153, 834, 4839, 29612, 188695, 1243746, 8428597, 58476481, 413893789, 2980489256, 21787216989, 161374041945, 1209258743839, 9155914963702, 69969663242487, 539189056700627 (list; graph; refs; listen; history; text; internal format)



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.


Table of n, a(n) for n=0..19.

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, Journal of Algebraic Combinatorics (2014).

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, Megan Ly, Presentation of the Motzkin Monoid, arXiv:1301.4518 [math.RT], 2013.


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.


Cf. A001006, A026945.

Sequence in context: A335868 A126033 A323632 * A007863 A302061 A030823

Adjacent sequences:  A256669 A256670 A256671 * A256673 A256674 A256675




Nick Loughlin, Apr 07 2015


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



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Last modified June 16 21:55 EDT 2021. Contains 345080 sequences. (Running on oeis4.)