A256492 Number of idempotents in the partial Jones monoid.

1, 2, 7, 24, 103, 416, 1998, 8822, 45661, 213674, 1167797, 5694690, 32445914, 163151262, 960580840, 4945645808, 29899013071, 156834641076, 968947169139

Offset

0,2

Comments

The partial Jones monoid contains all the elements of the Motzkin monoid whose pictorial representatives are subgraphs of those in the Jones monoid. The number a(n) counts the idempotent elements in this monoid in each degree n, starting from zero. This monoid was discovered by the sequence's original author and a collaborator during work on a paper yet to appear at the time of posting.

References

V. F. R. Jones, The Potts model and the symmetric group, in: Subfactors: Proceedings of the Taniguchi Symposium on Operator Algebras (Kyuzeso, 1993), World Sci. Publishing, 1994, 259-267.

Links

Table of n, a(n) for n=0..18.

Egri-Nagy Attila, Organic semigroup theory: ferns growing in the Jones/Temperley-Lieb monoid, on Computational Semigroup Theory at Wordpress, September 1, 2014.

I. Dolinka, J. East et al, Idempotent Statistics of the Motzkin and Jones Monoids, arXiv: 1507.04838 [math.CO] (2015).

J. East, Egri-Nagy A., A. R. Francis, J. D. Mitchell, Finite Diagram Semigroups: Extending the Computational Horizon, arXiv:1502.07150 [math.GR], 2015.

K. Hatch, E. Ly, E. Posner, Presentation of the Motzkin Monoid, arXiv:1301.4518 [math.RT], 2013.

K. W. Lau & D. G. FitzGerald, Ideal Structure of the Kauffman and Related Monoids, Communications in Algebra, 30:7 (2006), 2617-2629. doi:10.1080/00927870600651414

J. D. Mitchell et al., Semigroups package for GAP.

Example

In degree at most 1, the idempotents are all partial identities, giving a(0)=1 and a(1)=2. In degree 2 ,there are 7; the four partial identities, the Temperly-Lieb cup-and-cap, and its 3 subpictures (one of which is the empty picture, which is also a partial identity, hence the overcount by 1).

Crossrefs

Sequence in context: A150446 A150447 A150448 * A150449 A150450 A150451

Adjacent sequences: A256489 A256490 A256491 * A256493 A256494 A256495

Keyword

hard,more,nonn,walk

Author

Nick Loughlin, Mar 30 2015

Extensions

a(11)-a(18) computed using the GAP package Semigroups and added by James Mitchell, May 21 2016

Status

approved