A281442 Triangle read by rows: T(n,r), 0 <= r <= n, is the number of idempotents of rank r in the Kauffman monoid K_n.

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 4, 0, 1, 0, 8, 0, 6, 0, 1, 0, 0, 22, 0, 8, 0, 1, 0, 42, 0, 40, 0, 10, 0, 1, 0, 0, 140, 0, 62, 0, 12, 0, 1, 0, 262, 0, 288, 0, 88, 0, 14, 0, 1, 0, 0, 992, 0, 492, 0, 118, 0, 16, 0, 1

Offset: 0

Author

James East, Oct 05 2017

Keywords

• nonn
• tabl

Comments

Values were computed using the Semigroups package for GAP.

T(n,r) is also the number of idempotent basis elements of rank r in the Temperley-Lieb algebra of degree n in the generic case (when the twisting parameter is not an m-th root of unity for any m <= n).

Links

• Igor Dolinka, James East, Athanasios Evangelou, Desmond FitzGerald, Nicholas Ham, James Hyde, Nicholas Loughlin, Idempotent Statistics of the Motzkin and Jones Monoids, arXiv:1507.04838 [math.CO], 2015-2016.

Crossrefs

Cf. A281438 (row sums), A281441, A289620.

Formula

T(2n-1,1) = A005315(n). Empirical: T(2n,2) = A077056(n); T(n+2,n-2) = 2*A028875(n) for n>2. - Andrey Zabolotskiy, Oct 19 2017