OFFSET
0,5
COMMENTS
F(n; n) is A001906(n), n >= 1.
LINKS
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
KEYWORD
nonn,tabl
AUTHOR
Abdullahi Umar, Aug 25 2008
EXTENSIONS
Minor edits by N. J. A. Sloane, Jan 01 2009
STATUS
approved