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

1,5

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

References

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

Links

Table of n, a(n) for n=1..45.

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

Example

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

Mathematica

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

Crossrefs

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

Sequence in context: A124913 A181415 A289711 * A138186 A110321 A377763

Adjacent sequences: A101815 A101816 A101817 * A101819 A101820 A101821

Keyword

nonn,tabl

Author

Clark Kimberling, Dec 17 2004

Extensions

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

Status

approved