A148092 -id:A148092 - cp's OEIS frontend

A080510 Triangle read by rows: T(n,k) gives the number of set partitions of {1,...,n} with maximum block length k.

1, 1, 1, 1, 3, 1, 1, 9, 4, 1, 1, 25, 20, 5, 1, 1, 75, 90, 30, 6, 1, 1, 231, 420, 175, 42, 7, 1, 1, 763, 2016, 1015, 280, 56, 8, 1, 1, 2619, 10024, 6111, 1890, 420, 72, 9, 1, 1, 9495, 51640, 38010, 12978, 3150, 600, 90, 10, 1, 1, 35695, 276980, 244035, 91938, 24024, 4950, 825, 110, 11, 1

Offset: 1

Wouter Meeussen, Mar 22 2003

Row sums are A000110 (Bell numbers). Second column is A001189 (Degree n permutations of order exactly 2).
From Peter Luschny, Mar 09 2009: (Start)
Partition product of Product_{j=0..n-1} ((k + 1)*j - 1) and n! at k = -1, summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A036040.
Same partition product with length statistic is A008277.
Diagonal a(A000217) = A000012.
Row sum is A000110. (End)
From Gary W. Adamson, Feb 24 2011: (Start)
Construct an array in which the n-th row is the partition function G(n,k), where G(n,1),...,G(n,6) = A000012, A000085, A001680, A001681, A110038, A148092, with the first few rows
  1,   1,   1,   1,   1,   1,    1, ... = A000012
  1,   2,   4,  10,  26,  76,  232, ... = A000085
  1,   2,   5,  14,  46, 166,  652, ... = A001680
  1,   2,   5,  15,  51, 196,  827, ... = A001681
  1,   2    5   15   52  202   869, ... = A110038
  1,   2,   5   15   52  203   876, ... = A148092
  ...
Rows tend to A000110, the Bell numbers. Taking finite differences from the top, then reorienting, we obtain triangle A080510.
The n-th row of the array is the eigensequence of an infinite lower triangular matrix with n diagonals of Pascal's triangle starting from the right and the rest zeros. (End)

			T(4,3) = 4 since there are 4 set partitions with longest block of length 3: {{1},{2,3,4}}, {{1,3,4},{2}}, {{1,2,3},{4}} and {{1,2,4},{3}}.
Triangle begins:
  1;
  1,    1;
  1,    3,     1;
  1,    9,     4,    1;
  1,   25,    20,    5,    1;
  1,   75,    90,   30,    6,   1;
  1,  231,   420,  175,   42,   7,  1;
  1,  763,  2016, 1015,  280,  56,  8,  1;
  1, 2619, 10024, 6111, 1890, 420, 72,  9,  1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_2 Triangles.
J. Riordan, Letter, 11/23/1970. See second page of letter.

Cf. A080107, A080337, A008277, A178979, A276922, A327884.
Cf. A157396, A157397, A157398, A157399, A157400, A157401, A157402, A157403, A157404, A157405. - Peter Luschny, Mar 09 2009
Cf. A000012, A000085, A001680, A001681, A110038, A148092. - Gary W. Adamson, Feb 24 2011
Columns k=1..10 give: A000012 (for n>0), A001189, A229245, A229246, A229247, A229248, A229249, A229250, A229251, A229252. - Alois P. Heinz, Sep 17 2013
T(2n,n) gives A276961.
Take differences along rows of A229223. - N. J. A. Sloane, Jan 10 2018

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       add(b(n-i*j, i-1) *n!/i!^j/(n-i*j)!/j!, j=0..n/i)))
    end:
T:= (n, k)-> b(n, k) -b(n, k-1):
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Apr 20 2012

<< DiscreteMath`NewCombinatorica`; Table[Length/@Split[Sort[Max[Length/@# ]&/@SetPartitions[n]]], {n, 12}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n-i*j, i-1]*n!/i!^j/(n-i*j)!/j!, {j, 0, n/i}]]]; T[n_, k_] := b[n, k]-b[n, k-1]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Feb 25 2014, after Alois P. Heinz *)

E.g.f. for k-th column: exp(exp(x)*GAMMA(k, x)/(k-1)!-1)*(exp(x^k/k!)-1). - Vladeta Jovovic, Feb 04 2005
From Peter Luschny, Mar 09 2009: (Start)
T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n.
T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,...,a_n such that
1*a_1 + 2*a_2 + ... + n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*...*a_n!),
f^a = (f_1/1!)^a_1*...*(f_n/n!)^a_n and f_n = Product_{j=0..n-1} (-1) = (-1)^n. (End)
From Ludovic Schwob, Jan 15 2022: (Start)
T(2n,n) = C(2n,n)*(A000110(n)-1/2) for n>0.
T(n,m) = C(n,m)*A000110(n-m) for 2m > n > 0. (End)

A229223 Number G(n,k) of set partitions of {1,...,n} into sets of size at most k; triangle G(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 10, 14, 15, 0, 1, 26, 46, 51, 52, 0, 1, 76, 166, 196, 202, 203, 0, 1, 232, 652, 827, 869, 876, 877, 0, 1, 764, 2780, 3795, 4075, 4131, 4139, 4140, 0, 1, 2620, 12644, 18755, 20645, 21065, 21137, 21146, 21147

Offset: 0

Alois P. Heinz, Sep 16 2013

John Riordan calls these Allied Bell Numbers. - N. J. A. Sloane, Jan 10 2018
G(n,k) is defined for n,k >= 0.  The triangle contains only the terms with k<=n. G(n,k) = G(n,n) = A000110(n) for k>n.
G(n,k) - G(n,k-1) = A080510(n,k).
A column G(n>=0,k) can be generated by a linear recurrence with polynomial coefficients, where the initial terms correspond with A000110, and the coefficients contain constant factors derived from A008279 (cf. recg(k) in the fourth Maple program below). - Georg Fischer, May 19 2021

			G(4,2) = 10: 1/2/3/4, 12/3/4, 13/2/4, 14/2/3, 1/23/4, 1/24/3, 1/2/34, 12/34, 13/24, 14/23.
Triangle G(n,k) begins:
  1;
  0,  1;
  0,  1,   2;
  0,  1,   4,    5;
  0,  1,  10,   14,   15,
  0,  1,  26,   46,   51,   52;
  0,  1,  76,  166,  196,  202,  203;
  0,  1, 232,  652,  827,  869,  876,  877;
  0,  1, 764, 2780, 3795, 4075, 4131, 4139, 4140;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
J. Riordan, Letter, 11/23/1970

Columns k=0-10 give: A000007, A000012, A000085, A001680, A001681, A110038, A148092, A229224, A229225, A229226, A229227.
Main diagonal gives: A000110. Lower diagonal gives: A058692.
Cf. A066223 (G(2n,2)), A229228 (G(2n,n)), A229229 (G(n^2,n)), A227223 (G(2^n,n)).
Cf. A008279, A080510.

G:= proc(n, k) option remember; `if`(n=0, 1, `if`(k<1, 0,
       add(G(n-k*j, k-1) *n!/k!^j/(n-k*j)!/j!, j=0..n/k)))
    end:
seq(seq(G(n, k), k=0..n), n=0..10);
# second Maple program:
G:= proc(n, k) option remember; local j; if k>n then G(n, n)
      elif n=0 then 1 elif k<1 then 0 else G(n-k, k);
      for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j,k) od; % fi
    end:
seq(seq(G(n, k), k=0..n), n=0..10);
# third Maple program:
G:= proc(n, k) option remember; `if`(n=0, 1, add(
      G(n-i, k)*binomial(n-1, i-1), i=1..min(n, k)))
    end:
seq(seq(G(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jun 26 2017
# fourth Maple program (for columns G(n>=0,k)):
init := n -> seq(a(j) = combinat:-bell(j), j=0..n): # A000110
b := (n, k) -> mul((n - j)/(j + 1), j = 0..k-1):
recg := k -> {(k-1)!*(add(j*b(n, j)*a(n-j), j = 1..k) - n*a(n)), init(k-1)}:
column := proc(k, len) local f; f := gfun:-rectoproc(recg(k), a(n), remember):
map(f, [$0..len-1]) end:
seq(print(column(k, 12)), k=1..9); # Georg Fischer, May 19 2021

g[n_, k_] := g[n, k] = If[n == 0, 1, If[k < 1, 0, Sum[g[n - k*j, k - 1] *n!/k!^j/(n - k*j)!/j!, { j, 0, n/k}]]]; Table[Table[g[n, k], { k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 09 2013, translated from Maple *)

G(0,k) = 1, G(n,k) = 0 for n>0 and k<1, otherwise G(n,k) = Sum_{j=0..floor(n/k)} G(n-k*j,k-1) * n!/(k!^j*(n-k*j)!*j!).
G(n,k) = G(n-1,k) +(n-1)/1 *(G(n-2,k) +(n-2)/2 *(G(n-3,k) +(n-3)/3 *(G(n-4,k) + ... +(n-(k-1))/(k-1) *G(n-k,k)...))).
E.g.f. of column k: exp(Sum_{j=1..k} x^j/j!).

A110038 The partition function G(n,5).

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 202, 869, 4075, 20645, 112124, 648649, 3976633, 25719630, 174839120, 1245131903, 9263053753, 71806323461, 578719497070, 4839515883625, 41916097982471, 375401824277096, 3471395994487422, 33099042344383885, 325005134436155395

Offset: 0

N. J. A. Sloane, May 13 2009

nonn

Set partitions into sets of size at most 5. The e.g.f. for partitions into sets of size at most s is exp( sum(j=1..s, x^j/j!) ). [Joerg Arndt, Dec 07 2012]

Seiichi Manyama, Table of n, a(n) for n = 0..591 (terms 0..200 from Alois P. Heinz)
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
F. L. Miksa, L. Moser and M. Wyman, Restricted partitions of finite sets, Canad. Math. Bull., 1 (1958), 87-96.

The sequences G(n,1), G(n,2), G(n,3), G(n,4), G(n,5), G(n,6) are given by A000012, A000085, A001680, A001681, A110038, A148092 respectively.
Column k=5 of A229223.
Cf. A276925.

G:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       add(G(n-i*j, i-1) *n!/i!^j/(n-i*j)!/j!, j=0..n/i)))
    end:
a:= n-> G(n, 5):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 20 2012
# second Maple program:
a:= proc(n) option remember; `if`(n<5, [1, 1, 2, 5, 15][n+1],
      a(n-1)+(n-1)*(a(n-2)+(n-2)/2*(a(n-3)+(n-3)/3*(a(n-4)
      +(n-4)/4*a(n-5)))))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 15 2013

G[n_, i_] := G[n, i] = If[n == 0, 1, If[i<1, 0, Sum[G[n-i*j, i-1] *n!/i!^j/(n-i*j)!/j!, {j, 0, n/i}]]]; a[n_] := G[n, 5]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)

E.g.f.: exp( x + x^2/2 + x^3/6 + x^4/24 + x^5/120 ).
a(n) = n! * sum(k=1..n, 1/k! * sum(r=0..k, C(k,r) * sum(m=0..r, 2^(m-r) * C(r,m) * sum(j=0..m, C(m,j) * C(j,n-m-k-j-r) * 6^(j-m) * 24^(n-r-m-k-2*j) * 120^(m+k+j+r-n))))). - Vladimir Kruchinin, Jan 25 2011
a(n) = G(n,5) with G(0,i) = 1, G(n,i) = 0 for n>0 and i<1, otherwise G(n,i) = Sum_{j=0..floor(n/i)} G(n-i*j,i-1) * n!/(i!^j*(n-i*j)!*j!). - Alois P. Heinz, Apr 20 2012

A151511 The triangle in A151359 read by rows downwards.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 7, 6, 1, 0, 1, 15, 25, 10, 1, 0, 1, 31, 90, 65, 15, 1, 0, 0, 63, 301, 350, 140, 21, 1, 0, 0, 119, 966, 1701, 1050, 266, 28, 1, 0, 0, 210, 2989, 7770, 6951, 2646, 462, 36, 1, 0, 0, 336, 8925, 33985, 42525, 22827, 5880, 750, 45, 1, 0, 0, 462, 25641

Offset: 0

N. J. A. Sloane, May 14 2009

The Bell transform of g(n) = 1 if n<6 else 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

			Triangle begins:
1
0 1
0 1 1
0 1 3 1
0 1 7 6 1
0 1 15 25 10 1
0 1 31 90 65 15 1
0 0 63 301 350 140 21 1
0 0 119 966 1701 1050 266 28 1

Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394, 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009 (see Table 7 E5(n,k) page 16).

Begins in same way as triangle of Stirling numbers of second kind, A048993, but is strictly different. N. J. A. Sloane, Aug 09 2017

Cf. A148092 (row sums), A122848, A111246, A144644, A151509.

Unprotect[Power]; 0^0 = 1; a[n_ /; 1 <= n <= 6] = 1; a[] = 0; T[n, k_] := T[n, k] = If[k == 0, a[0]^n, Sum[Binomial[n - 1, j - 1] a[j] T[n - j, k - 1], {j, 0, n - k + 1}]]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 20 2016, after Peter Luschny *)

# uses[bell_matrix from A264428]
bell_matrix(lambda n: 1 if n<6 else 0, 12) # Peter Luschny, Jan 19 2016

Bivariate e.g.f. A151511(x,t) = Sum_{n>=0, k>=0} T(n,k)*x^n*t^k/n! = exp(t*G6(x)), where G6(x) = Sum_{i=1..6} x^i/i! is the e.g.f. of column 1. - R. J. Mathar, May 28 2019

Row 9 added by Michel Marcus, Feb 13 2014

More rows from R. J. Mathar, May 28 2019

A276926 Number of ordered set partitions of [n] with at most six elements per block.

Original entry on oeis.org

1, 1, 3, 13, 75, 541, 4683, 47292, 545818, 7086954, 102241902, 1622523210, 28089336198, 526810157874, 10640241569958, 230256584914356, 5314976561846952, 130352566702702344, 3385021286975255928, 92786398312428612792, 2677217312112863784264

Offset: 0

Alois P. Heinz, Sep 22 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..424

Column k=6 of A276921.

Cf. A148092.

a:= proc(n) option remember; `if`(n=0, 1, add(
       a(n-i)*binomial(n, i), i=1..min(n, 6)))
    end:
seq(a(n), n=0..25);

max = 25; CoefficientList[1/(1-Sum[x^i/i!, {i, 1, 6}]) + O[x]^(max+1), x]* Range[0, max]! (* Jean-François Alcover, May 24 2018 *)

E.g.f.: 1/(1-Sum_{i=1..6} x^i/i!).

A151512 The triangle in A151359 read by rows upwards.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 7, 1, 0, 1, 10, 25, 15, 1, 0, 1, 15, 65, 90, 31, 1, 0, 1, 21, 140, 350, 301, 63, 0, 0, 1, 28, 266, 1050, 1701, 966, 119, 0, 0, 1, 36, 462, 2646, 6951, 7770, 2989, 210, 0, 0, 1, 45, 750, 5880, 22827, 42525, 33985, 8925, 336, 0, 0

Offset: 0

N. J. A. Sloane, May 14 2009

Conjectured: The i-th element of row j is the number of different equivalence relationships, within a set of (j-1) element, having (j-i) equivalence classes. For example: row 5 = [1, 6, 7, 1, 0] means that, in a set of 4 elements, there exists 7 equivalence relationships having 3 different equivalence classes. - Philippe Beaudoin, Nov 09 2013

			Triangle begins:
  1
  1  0
  1  1   0
  1  3   1    0
  1  6   7    1    0
  1 10  25   15    1   0
  1 15  65   90   31   1   0
  1 21 140  350  301  63   0 0
  1 28 266 1050 1701 966 119 0 0

Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394, 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 2009) (see Table 7 E5(n,k) page 16).

Cf. A148092 (row sums), A151511 (row-reversed).

Unprotect[Power]; 0^0 = 1; a[n_ /; 1 <= n <= 6] = 1; a[] = 0; t[n, k_] := t[n, k] = If[k == 0, a[0]^n, Sum[Binomial[n - 1, j - 1] a[j] t[n - j, k - 1], {j, 0, n - k + 1}]]; Table[Table[t[n - 1, k], {k, n - 1, 0, -1}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 20 2016, using Peter Luschny's Bell transform *)

Row 9 added by Michel Marcus, Feb 13 2014

Row 10 from R. J. Mathar, May 28 2019

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A080510 Triangle read by rows: T(n,k) gives the number of set partitions of {1,...,n} with maximum block length k.

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A229223 Number G(n,k) of set partitions of {1,...,n} into sets of size at most k; triangle G(n,k), n>=0, 0<=k<=n, read by rows.

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A110038 The partition function G(n,5).

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A151511 The triangle in A151359 read by rows downwards.

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A276926 Number of ordered set partitions of [n] with at most six elements per block.

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A151512 The triangle in A151359 read by rows upwards.

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