A259759 Triangle read by rows: T(n,k) = number of partial idempotent mappings (of an n-chain) with collapse exactly k.

1, 2, 0, 4, 0, 2, 8, 0, 12, 3, 16, 0, 48, 24, 16, 32, 0, 160, 120, 160, 65, 64, 0, 480, 480, 960, 780, 336, 128, 0, 1344, 1680, 4480, 5460, 4704, 1897, 256, 0, 3584, 5376, 17920, 29120, 37632, 30352, 11824, 512, 0, 9216, 16128, 64512, 131040, 225792, 273168, 212832, 80145

Offset: 0

Author

Wafa AlNadabi, Jul 04 2015

Keywords

• nonn
• tabl

Examples

			T (3,2) = 12 because there are exactly 12 partial idempotent mappings (of a 3-chain) with collapse exactly 2, namely: (123-->113), (123-->121), (123-->122), (123-->223), (123-->133), (123--> 323), (12-->11), (12-->22), (23-->22), (23-->33), (13-->11), (13-->33).
Triangle starts:
1;
2,0;
4,0,2;
8,0,12,3;
16,0,48,24,16;
...
		

References

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

Programs

PARI

tabl(nn) = {for (n=0, nn, for (k=0, n, print1(binomial(n,k)*sum(r=k, n, binomial(n-k,r-k)*sum(j=0, k, binomial(k,j)*stirling(k-j,j,2)*j!)), ", ");); print(););} \\ Michel Marcus, Jul 15 2015

Formula

T(n,k) = binomial(n,k)Sum_{r=k}^n binomial(n-k,r-k)Sum_{j=0}^k binomial(k,j)S(k-j,j)j!, where S (x,y) is the Stirling numbers of the second kind, which gives the number of ways to partition x into y nonempty subsets.

Extensions

More terms from Michel Marcus, Jul 15 2015