Triangle read by rows: T(n,r), 0 <= r <= n, is the number of idempotents of rank r in the Kauffman monoid K_n.
(history;
published version)
Discussion
Fri Oct 20
00:06
Andrey Zabolotskiy: Thanks for the correction. The more explicit formulas the better, I think, although the most valuable ones are non-trivial ones, of course.
Discussion
Thu Oct 19
23:26
James East: Changed T(n+1,n-2) to T(n+2,n-2); note that T(n,r) = 0 if n != r (mod 2). It is also easy to show that T(n,n-2) = 2(n-2), but I doubt this is worth adding.
Discussion
Thu Oct 19
23:08
James East: Andrey's first formula is indeed proved (I haven't checked the second one). The "interface graphs" discussed in the article generally involve multiple (nested) open and closed meanders. In the case of rank-1 idempotents from K_n (with n odd), there is exactly one such meander.
23:12
James East: I also added a connection with Temperley-Lieb algebras.
COMMENTS
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).
Discussion
Thu Oct 19
17:42
Andrey Zabolotskiy: James, I'm not sure about the connection to meandric numbers: is it proved or only observed?
Discussion
Thu Oct 19
10:21
Michel Marcus: Andrey , you entered 2 formulas ?
