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Triangle read by rows: T(n,h)/(n-1), where T is the array in A101819.
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1
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
FORMULA
T(n, h) = C(n-1, h)*U(n, h)/(n-1), where U(n, h) is the array in A019538.
EXAMPLE
First rows:
1
1 3
1 14 12
4 45 150 60
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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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
COMMENTS
Row sums = n^n. T(n,1) = n, T(n,n) = n!.
REFERENCES
H. Picquet, Note #124, L'Intermédiaire des Mathématiciens, 1 (1894), pp. 125-127. - N. J. A. Sloane, Feb 28 2022
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).
EXAMPLE
First rows:
1;
2, 2;
3, 18, 6;
4, 84, 144, 24;
MATHEMATICA
Table[Table[StirlingS2[n, k] Binomial[n, k] k!, {k, 1, n}], {n, 1, 8}] // Grid
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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4
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
FORMULA
T(n, h) = C(n-2, h)*U(n, h), where U(n, h) is the array in A019538.
EXAMPLE
First rows:
2
3 42
4 180 600
5 620 5400 7800
Triangle read by rows: (1/n)*T(n,h), where T(n,h) is the array in A101817.
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3
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
COMMENTS
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.
FORMULA
T(n, h) = (1/n)*C(n, h)*U(n, h), where U(n, h) is the array in A019538.
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,
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