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

0,2

References

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

Links

Table of n, a(n) for n=0..54.

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.

Example

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;
...

Prog

(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

Crossrefs

Sequence in context: A181481 A335872 A239489 * A364817 A119607 A279968

Adjacent sequences: A259756 A259757 A259758 * A259760 A259761 A259762

Keyword

nonn,tabl

Author

Wafa AlNadabi, Jul 04 2015

Extensions

More terms from Michel Marcus, Jul 15 2015

Status

approved