A101817 -id:A101817 - cp's OEIS frontend

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

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

A090657 Triangle read by rows: T(n,k) = number of functions from [1,2,...,n] to [1,2,...,n] such that the image contains exactly k elements (0<=k<=n).

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 3, 18, 6, 0, 4, 84, 144, 24, 0, 5, 300, 1500, 1200, 120, 0, 6, 930, 10800, 23400, 10800, 720, 0, 7, 2646, 63210, 294000, 352800, 105840, 5040, 0, 8, 7112, 324576, 2857680, 7056000, 5362560, 1128960, 40320

Offset: 0

Philippe Deléham, Dec 14 2003

Another version is in A101817. - Philippe Deléham, Feb 16 2013

			Triangle begins:
  1;
  0,  1;
  0,  2,   2;
  0,  3,  18,   6;
  0,  4,  84, 144, 24;
  ...

Alois P. Heinz, Rows n = 0..62, flattened
C. M. Ringel, The Catalan combinatorics of the hereditary artin algebras, arXiv preprint arXiv:1502.06553, 2015

Row sums give: A000312. Columns k=0-2 give: A000007, A001477, A068605. Diagonal, lower diagonal give: A000142, A001804. Cf. A007318, A048993, A019538, A008279.

Cf. A101817, A288312.

T:= proc(n,k) option remember;
      if k=n then n!
    elif k=0 or k>n then 0
    else n * (T(n-1,k-1) + k/(n-k) * T(n-1,k))
      fi
    end:
seq(seq(T(n,k), k=0..n), n=0..10);

Table[Table[StirlingS2[n, k] Binomial[n, k] k!, {k, 0, n}], {n, 0,10}] // Flatten  (* Geoffrey Critzer, Sep 09 2011 *)

T(n,k) = C(n,k) * k! * A048993(n,k).
T(n,k) = A008279(n,k) * A048993(n,k).
T(n,k) = C(n,k) * A019538(n, k).
T(n,k) = C(n,k) * Sum_{j=0..k} (-1)^(k-j) * C(k,j) * j^n.
T(n,k) = n * (T(n-1,k-1) + k/(n-k) * T(n-1,k)) with T(n,n) = n! and T(n,0) = 0 for n>0.
T(2n,n) = A288312(n). - Alois P. Heinz, Jun 07 2017

Revised description from Jan Maciak, Apr 25 2004

Edited by Alois P. Heinz, Jan 17 2011

A101819 Triangle read by rows: T(n,h) = number of functions f:{1,2,...,n}->{1,2,...,n-1} such that |Image(f)|=h, h=1,2,...,n-1, n=2,3,...

Original entry on oeis.org

1, 2, 6, 3, 42, 36, 4, 180, 600, 240, 5, 620, 5400, 7800, 1800, 6, 1890, 36120, 12600, 100800, 15120, 7, 5334, 202860, 1428840, 2646000, 1340640, 141120, 8, 14280, 1016400, 13053600, 46710720, 53343360, 18627840, 1451520, 9, 36792, 4702320

Offset: 0

Clark Kimberling, Dec 17 2004

			First rows:
1
2 6
3 42 36
4 180 600 240
To see that T(4,2)=42, first count 7 functions from {1,2,3,4}
onto {1,2} with f(1)=1 and 7 with f(1)=2. Count 14 onto {1,3}
and 14 onto {2,3}, for a total of 42.

Cf. A019538, A101817, A101819, A101821.

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

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.

A288312 Number of endofunctions on [2n] such that the image size equals n.

Original entry on oeis.org

1, 2, 84, 10800, 2857680, 1285956000, 880599202560, 853262368358400, 1111400775560275200, 1873276460474747328000, 3967400888465895264384000, 10313998054713896966296473600, 32291970618091110826769565696000, 119851615755915509174015455948800000

Offset: 0

Alois P. Heinz, Jun 07 2017

nonn

			a(1) = 2: (1,1), (2,2).

Alois P. Heinz, Table of n, a(n) for n = 0..195

Cf. A000142, A048993, A090657, A101817, A219859.

b:= proc(n, k) option remember; `if`(k=n, n!,
      `if`(k=0, 0, n*(b(n-1, k-1)+b(n-1, k)*k/(n-k))))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..15);

Table[StirlingS2[2*n, n]*(2*n)!/n!, {n, 0, 20}] (* Vaclav Kotesovec, Jun 10 2017 *)

a(n)=stirling(2*n, n, 2)*n!*binomial(2*n, n); \\ Indranil Ghosh, Jul 04 2017

from mpmath import *
mp.dps=100
def a(n): return int(stirling2(2*n, n)*fac(n)*binomial(2*n, n))
print([a(n) for n in range(21)]) # Indranil Ghosh, Jul 04 2017

a(n) = Stirling2(2*n,n) * n! * binomial(2*n,n).
a(n) = A090657(2n,n) = A101817(2n,n) = A219859(2n,n).
a(n) ~ n^(2*n - 1/2) * 2^(4*n) / (sqrt(Pi*(1-c)) * c^n * (2-c)^n * exp(2*n)), where c = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599... - Vaclav Kotesovec, Jun 10 2017

A219859 Triangular array read by rows: T(n,k) is the number of endofunctions, functions f:{1,2,...,n}->{1,2,...,n}, that have exactly k elements with no preimage; n>=0, 0<=k<=n.

Original entry on oeis.org

1, 1, 0, 2, 2, 0, 6, 18, 3, 0, 24, 144, 84, 4, 0, 120, 1200, 1500, 300, 5, 0, 720, 10800, 23400, 10800, 930, 6, 0, 5040, 105840, 352800, 294000, 63210, 2646, 7, 0, 40320, 1128960, 5362560, 7056000, 2857680, 324576, 7112, 8, 0, 362880, 13063680, 83825280, 160030080, 105099120, 23496480, 1524600, 18360, 9, 0

Offset: 0

Geoffrey Critzer, Dec 01 2012

Equivalently, T(n,k) is the number of endofunctions whose functional digraph has exactly k leaves.
Equivalently, T(n,k) is the number of doubly rooted trees with k leaves.  Here, a doubly rooted tree is a labeled tree in which two special vertices have been selected and the order of the selection matters. [Bona page 266]
Row sums are n^n.

			Triangle T(n,k) begins:
    1;
    1,     0;
    2,     2,     0;
    6,    18,     3,     0;
   24,   144,    84,     4,   0;
  120,  1200,  1500,   300,   5, 0;
  720, 10800, 23400, 10800, 930, 6, 0;
  ...

M. Bona, Introduction to Enumerative Combinatorics, McGraw Hill, 2007.

Column k=0-1 give: A000142, A001804.
Row sums give A000312.
T(2n,n) gives A288312.
Cf. A055302, A055314, A090657, A101817, A209290, A344053.

Table[Table[n!/k!StirlingS2[n,n-k],{k,0,n}],{n,0,8}]//Grid

row(n) = vector(n+1, k, k--; n!/k! * stirling(n,n-k,2)); \\ Michel Marcus, Jan 24 2022

T(n,k) = n!/k! * Stirling2(n,n-k).
T(n,0) = n!.
T(n,k) = A055302(n,k)*(n-k) + A055302(n,k+1)*(k+1).  The first term (on rhs of this equation) is the number of such functions in which the preimage of f(n) contains more than one element. The second term is the number of such functions in which the preimage of f(n) contains exactly one element.
T(n,k) = binomial(n,k) Sum_{j=0..n-k}(-1)^j*binomial(n-k,j)*(n-k-j)^n. - Geoffrey Critzer, Aug 20 2013
E.g.f.:  1/(1 - (A(x,y) - y*x + x)) where A(x,y) is the e.g.f. for A055302. - Geoffrey Critzer, Jan 24 2022
From Alois P. Heinz, Jan 24 2022: (Start)
Sum_{k=0..n} k * T(n,k) = A209290(n).
Sum_{k=0..n} (-1)^k * T(n,k) = A344053(n). (End)

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

Original entry on oeis.org

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.

A181415 Irregular triangle a(n,k) = A049009(n,k)/n, read by rows 1<=k<=A000041(n).

Original entry on oeis.org

1, 1, 1, 1, 6, 2, 1, 12, 9, 36, 6, 1, 20, 40, 120, 180, 240, 24, 1, 30, 75, 50, 300, 1200, 300, 1200, 2700, 1800, 120, 1, 42, 126, 210, 630, 3150, 2100, 3150, 4200, 25200, 12600, 12600, 37800, 15120, 720, 1, 56, 196, 392, 245, 1176, 7056, 11760, 8820, 11760, 11760, 88200

Offset: 1

Alford Arnold, Oct 20 2010

			Row three is calculated as follows:
( 3 18 6) divided by (3 3 3) yielding (1 6 2)
1;
1,1;
1,6,2;
1,12,9,36,6;
1,20,40,120,180,240,24;
1,30,75,50,300,1200,300,1200,2700,1800,120;
1,42,126,210,630,3150,2100,3150,4200,25200,12600,12600,37800,15120,720;

Cf. A000169 (row sums), A000081 (unlabeled rooted trees) A179438 (a similar refinement), A054589, A135278, A019538, A101817, A101818

Sum_{k=1.. A000041(n)} a(n,k) = A000169(n). (Row sums)

a(n,k) = A098546(n,k) *A049019(n,k) /n. - Compare with the formula in A101818.

Edited by R. J. Mathar, May 17 2016

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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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A090657 Triangle read by rows: T(n,k) = number of functions from [1,2,...,n] to [1,2,...,n] such that the image contains exactly k elements (0<=k<=n).

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A101819 Triangle read by rows: T(n,h) = number of functions f:{1,2,...,n}->{1,2,...,n-1} such that |Image(f)|=h, h=1,2,...,n-1, n=2,3,...

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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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A288312 Number of endofunctions on [2n] such that the image size equals n.

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A219859 Triangular array read by rows: T(n,k) is the number of endofunctions, functions f:{1,2,...,n}->{1,2,...,n}, that have exactly k elements with no preimage; n>=0, 0<=k<=n.

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

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A181415 Irregular triangle a(n,k) = A049009(n,k)/n, read by rows 1<=k<=A000041(n).

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