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%I #24 Oct 21 2017 22:08:15
%S 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,
%T 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,
%U 0,118,0,16,0,1
%N Triangle read by rows: T(n,r), 0 <= r <= n, is the number of idempotents of rank r in the Kauffman monoid K_n.
%C Values were computed using the Semigroups package for GAP.
%C 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).
%H Igor Dolinka, James East, Athanasios Evangelou, Desmond FitzGerald, Nicholas Ham, James Hyde, Nicholas Loughlin, <a href="https://arxiv.org/abs/1507.04838">Idempotent Statistics of the Motzkin and Jones Monoids</a>, arXiv:1507.04838 [math.CO], 2015-2016.
%F 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
%Y Cf. A281438 (row sums), A281441, A289620.
%K nonn,tabl
%O 0,8
%A _James East_, Oct 05 2017
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