A101817 Triangle read by rows: T(n,h) = number of functions f:{1,2,...,n}->{1,2,...,n} such that |Image(f)|=h; h=1,2,...,n, n=1,2,3,... . Essentially A090657, but without zeros.

1, 2, 2, 3, 18, 6, 4, 84, 144, 24, 5, 300, 1500, 1200, 120, 6, 930, 10800, 23400, 10800, 720, 7, 2646, 63210, 294000, 352800, 105840, 5040, 8, 7112, 324576, 2857680, 7056000, 5362560, 1128960, 40320, 9, 18360, 1524600, 23496480, 105099120

Offset: 1

Author

Clark Kimberling, Dec 17 2004

Keywords

• nonn
• tabl

Comments

Row sums = n^n. T(n,1) = n, T(n,n) = n!.

Examples

			First rows:
1;
2,   2;
3,  18,   6;
4,  84, 144,  24;
		

References

• H. Picquet, Note #124, L'Intermédiaire des Mathématiciens, 1 (1894), pp. 125-127. - N. J. A. Sloane, Feb 28 2022

Crossrefs

Cf. A090657, A048993, A101818, A101819, A101821, A288312.

Programs

Mathematica

Table[Table[StirlingS2[n, k] Binomial[n, k] k!, {k, 1, n}], {n, 1, 8}] // Grid

Formula

T(n, h) = C(n, h)*U(n, h), where U(n, h) is the array in A019538. Thus T(n, h) = C(n, h)*h!*S(n, h), where S(n, h) is a Stirling number of the second kind (given by A048993 with zeros removed).

T(2n,n) = A288312(n). - Alois P. Heinz, Jun 07 2017