A111589 Triangle read by rows: number of idempotent order-preserving partial transformations (of an n-element totally ordered set) of width k (width(alpha) = |Dom(alpha)|

1, 1, 1, 1, 2, 3, 1, 3, 9, 8, 1, 4, 18, 32, 21, 1, 5, 30, 80, 105, 55, 1, 6, 45, 160, 315, 330, 144, 1, 7, 63, 280, 735, 1155, 1008, 377

Offset

0,5

Comments

F(n; n) is A001906(n), n >= 1.

Links

Table of n, a(n) for n=0..35.

Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.

Laradji, A. and Umar, A. Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8

Formula

F(n,k)= C(n,k)*A001906(k-1), (n>=k>0),F(0,0)=1

Example

F(3,2) = 9 because there are exactly 9 idempotent order-preserving partial transformations (on a 3-element chain) of width 2, namely: (1,2)->(1,1), (1,2)->(1,2), (1,2)->(2,2), (1,2)->(3,3), (1,3)->(1,1), (1,3)->(1,3),(1,3)->(3,3), (2,3)->(2,2), (2,3)->(2,3),( 2,3)->(3,3) - the mappings are coordinate-wise
Triangle begins:
1,
1,1,
1,2,3,
1,3,9,8,
1,4,18,32,21,
1,5,30,80,105,55,
1,6,45,160,315,330,144, ...

Crossrefs

Cf. A001906.

Sequence in context: A134319 A135091 A171150 * A259760 A010027 A151880

Adjacent sequences: A111586 A111587 A111588 * A111590 A111591 A111592

Keyword

nonn,tabl

Author

Abdullahi Umar, Aug 25 2008

Extensions

Minor edits by N. J. A. Sloane, Jan 01 2009

Status

approved