A258579 Triangle read by rows: T(n,k) = number of partial idempotent mappings (of an n-chain) with (right) waist exactly k.

1, 1, 1, 1, 2, 3, 1, 4, 7, 11, 1, 8, 17, 30, 48, 1, 16, 43, 86, 150, 241, 1, 32, 113, 258, 492, 846, 1358, 1, 64, 307, 806, 1686, 3108, 5276, 8445, 1, 128, 857, 2610, 6012, 11904, 21392, 35904, 57256, 1, 256, 2443, 8726, 22230, 47376, 90224, 158880, 263976, 419233

Offset: 0

Author

Wafa AlNadabi, Jul 04 2015

Keywords

• nonn
• tabl

Examples

			T(3,2) = 7 because there are exactly 7 partial idempotent mappings (of a 3-chain) with right waist exactly 2, namely: (123-->222), (123-->122), (123-->121), (12-->22), (12-->12), (23-->22), (2-->2).
Triangle starts:
1;
1,1;
1,2,3;
1,4,7,11;
1,8,17,30,48;
...
		

References

• F. AlKharosi, W. AlNadabi and A. Umar, "Combinatorial results for idempotents in full and partial transformation semigroups", (submitted).

Programs

PARI

mybinom(x,y) = if ((x==-1) && (y==-1), 1, binomial(x,y));
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(sum(m=0, k, mybinom(k-1, m-1) * (m+1)^(n-m)), ", "); ); print(); ); } \\ Michel Marcus, Jul 15 2015

Formula

T(n,k) = Sum_{m=0..k} binomial(k-1,m-1) * (m+1)^(n-m).