A101818 Triangle read by rows: (1/n)*T(n,h), where T(n,h) is the array in A101817.

1, 1, 1, 1, 6, 2, 1, 21, 36, 6, 1, 60, 300, 240, 24, 1, 155, 1800, 3900, 1800, 120, 1, 378, 9030, 42000, 50400, 15120, 720, 1, 889, 40572, 357210, 882000, 670320, 141120, 5040, 1, 2040, 169400, 2610720, 11677680, 17781120, 9313920, 1451520, 40320

Offset: 1

Author

Clark Kimberling, Dec 17 2004

Keywords

• nonn
• tabl

Comments

Column 2 is A066524.

T(n,h) is the number of partial functions f:{1,2,...,n-1}->{1,2,...,n-1} such that |Image(f)| = h-1. Equivalently T(n,h) = |D_h(a)| where D_h(a) is Green's D-class containing a, with a in the semigroup of partial transformations on [n-1] and rank(a) = h-1. - Geoffrey Critzer, Jan 02 2022

Examples

			First rows:
1
1 1
1 6 2
1 21 36 6
		

References

• O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, 2009, page 61.

Crossrefs

Cf. A019538, A066524, A090657, A101817, A101819, A101820, A101821, A000169 (row sums).

Programs

Mathematica

Table[Table[StirlingS2[n, k] (n-1)!/(n - k)!, {k, 1, n}], {n, 1,
   6}] // Grid (* Geoffrey Critzer, Jan 02 2022 *)

Formula

T(n, h) = (1/n)*C(n, h)*U(n, h), where U(n, h) is the array in A019538.

T(n, h) = Stirling2(n,h)*(n-1)!/(n-h)!. - Geoffrey Critzer, Jan 02 2022

Extensions

Offset changed to 1 by Alois P. Heinz, Jan 03 2022