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%I #48 Jan 13 2024 04:55:53
%S 1,2,7,31,153,834,4839,29612,188695,1243746,8428597,58476481,
%T 413893789,2980489256,21787216989,161374041945,1209258743839,
%U 9155914963702,69969663242487,539189056700627
%N Number of idempotents in the Motzkin monoid of degree n.
%C 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.
%C a(n+1) > a(n) for every n.
%C 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.
%C Bounded above by A026945, strictly for n > 1. Bounded below by the square of A001006, strictly for n > 1.
%H I. Dolinka, J. East et al, <a href="http://arxiv.org/abs/1507.04838">Idempotent Statistics of the Motzkin and Jones Monoids</a>, arXiv: 1507.04838 [math.CO], 2015, Table 2.
%H Tom Halverson, <a href="https://doi.org/10.46298/dmtcs.2347">Gelfand Models for Diagram Algebras</a>, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013).
%H Tom Halverson and Mike Reeks, <a href="https://doi.org/10.1007/s10801-014-0534-5">Gelfand Models for Diagram Algebras</a>, Journal of Algebraic Combinatorics (2014)41, 229-255.
%H J. D. Mitchell et al., <a href="http://www.gap-system.org/Packages/semigroups.html">Semigroups - GAP package</a>, Version 2.7.4, March, 2016.
%H J. D. Mitchell, <a href="https://github.com/james-d-mitchell/Jones">Counting idempotents in a monoid of partitions</a>, C++ program, October, 2016
%H Eliezer Posner, Kris Hatch, and Megan Ly, <a href="http://arxiv.org/abs/1301.4518">Presentation of the Motzkin Monoid</a>, arXiv:1301.4518 [math.RT], 2013.
%e There is one empty graph, which is idempotent under the composition, hence a(0)=1.
%e There are two on 1 pair of points, the clique and the discrete graph; both are idempotents under the composition, hence a(1)=2.
%Y Cf. A001006, A026945.
%K nonn,hard,more
%O 0,2
%A _Nick Loughlin_, Apr 07 2015
%E a(9)-a(13) corrected and a(14)-a(16) computed using the Semigroups package for GAP added by _James Mitchell_, Apr 12 2016
%E a(17)-a(19) added by _James Mitchell_, Apr 01 2017