A101819 -id:A101819 - cp's OEIS frontend

A101820 Triangle read by rows: T(n,h)/(n-1), where T is the array in A101819.

1, 1, 3, 1, 14, 12, 1, 45, 150, 60, 1, 124, 1080, 1560, 360, 1, 315, 6020, 21000, 16800, 2520, 1, 762, 28980, 204120, 378000, 191520, 20160, 1, 1785, 127050, 1631700, 5838840, 6667920, 2328480, 181440, 1, 4088, 522480, 11459280, 71442000

Offset: 0

Clark Kimberling, Dec 17 2004

			First rows:
1
1 3
1 14 12
4 45 150 60

Paul Barry, The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018.

Cf. A019538, A101817, A101819, A101821.

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

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.

Original entry on oeis.org

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

Clark Kimberling, Dec 17 2004

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

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

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

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

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

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

A101821 Triangle read by rows: T(n,h) = number of functions f:{1,2,...,n}->{1,2,...,n-2} such that |Image(f)|=h, h=1,2,...,n-2; n=3,4,....

Original entry on oeis.org

2, 3, 42, 4, 180, 600, 5, 620, 5400, 7800, 6, 1890, 36120, 126000, 100800, 7, 5534, 202860, 1428840, 2646000, 1340640, 8, 14280, 1016400, 13053600, 46710720, 53343360, 18627840, 9, 36792, 4702320, 103133520, 642978000, 1380576960

Offset: 0

Clark Kimberling, Dec 17 2004

			First rows:
2
3 42
4 180 600
5 620 5400 7800

Cf. A019538, A101817, A101819.

T(n, h) = C(n-2, h)*U(n, h), where U(n, h) is the array in A019538.

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

Original entry on oeis.org

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

Clark Kimberling, Dec 17 2004

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

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

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

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

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

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

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

cp's OEIS Frontend

A101820 Triangle read by rows: T(n,h)/(n-1), where T is the array in A101819.

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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.

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A101821 Triangle read by rows: T(n,h) = number of functions f:{1,2,...,n}->{1,2,...,n-2} such that |Image(f)|=h, h=1,2,...,n-2; n=3,4,....

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

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