A111776 Triangle read by rows: number of idempotent order-preserving partial transformations (of an n-element chain) of waist k (waist(alpha) = max(Im(alpha))).
1, 1, 1, 1, 2, 3, 1, 4, 6, 10, 1, 8, 12, 20, 35, 1, 16, 24, 40, 70, 125, 1, 32, 48, 80, 140, 250, 450, 1, 64, 96, 160, 280, 500, 900, 1625
Offset: 0
Examples
G(3,2) = 6 because there are exactly 6 idempotent order-preserving partial transformations (on a 3-element chain) of waist 2, namely: (2)->(2), (1,2)->(1,2), (1,2)->(2,2),(1,3)->(3,3), (2,3)->(2,2), (2,3)->(3,3) - the mappings are coordinate-wise
References
- Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving partial transformations. Journal of Algebra 278, (2004), 342-359.
Crossrefs
Cf. A081567.
Formula
G(n,k)= (2^(n-k))*G(n,n)=(2^(n-k))*A081567(n-1), G(0,0) = 1
Comments