A225798 The number of idempotents in the Jones (or Temperley-Lieb) monoid on the set [1..n].

1, 2, 5, 12, 36, 96, 311, 886, 3000, 8944, 31192, 96138, 342562, 1083028, 3923351, 12656024, 46455770, 152325850, 565212506, 1878551444, 7033866580, 23645970022, 89222991344, 302879546290, 1150480017950, 3938480377496, 15047312553918, 51892071842570, 199274492098480, 691680497233180

Offset: 1

Author

James Mitchell, Jul 27 2013

Keywords

• nonn

Comments

The Jones monoid is the set of partitions on [1..2n] with classes of size 2, which can be drawn as a planar graph, and multiplication inherited from the Brauer monoid, which contains the Jones monoid as a subsemigroup. The multiplication is defined in Halverson and Ram.

These numbers were produced using the Semigroups (2.0) package for GAP 4.7.

No general formula is known for the number of idempotents in the Jones monoid.

Links

• Attila Egri-Nagy, Nick Loughlin, and James Mitchell Table of n, a(n) for n = 1..30 (a(1) to a(21) from Attila Egri-Nagy, a(22)-a(24) from Nick Loughlin, a(25)-a(30) from James Mitchell)
• I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, et al., Enumeration of idempotents in diagram semigroups and algebras, arXiv preprint arXiv:1408.2021 [math.GR], 2014.
• I. Dolinka, J. East et al, Idempotent Statistics of the Motzkin and Jones Monoids, arXiv:1507.04838 [math.CO], 2015. Table 4 and 5.
• T. Halverson, A. Ram, Partition algebras, European J. Combin. 26 (6) (2005) 869-921.
• J. D. Mitchell et al., Semigroups package for GAP.

Crossrefs

Cf. A000108, A227545, A225797.

Programs

GAP

for i in [1..18] do
Print(NrIdempotents(JonesMonoid(i)), "\n");
od;

Extensions

a(20)-a(21) from Attila Egri-Nagy, Sep 12 2014

a(22)-a(24) from Nick Loughlin, Jan 23 2015

a(25)-a(30) from James Mitchell, May 21 2016