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

1,2

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.

Prog

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

Crossrefs

Cf. A000108, A227545, A225797.

Sequence in context: A323397 A292169 A283799 * A303204 A333411 A032203

Adjacent sequences: A225795 A225796 A225797 * A225799 A225800 A225801

Keyword

nonn

Author

James Mitchell, Jul 27 2013

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

Status

approved