A327884 -id:A327884 - 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)

A327869 Sum T(n,k) of multinomials M(n; lambda), where lambda ranges over all partitions of n into distinct parts incorporating k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 4, 3, 3, 1, 5, 4, 0, 4, 1, 16, 5, 10, 10, 5, 1, 82, 66, 75, 60, 15, 6, 1, 169, 112, 126, 35, 140, 21, 7, 1, 541, 456, 196, 336, 280, 224, 28, 8, 1, 2272, 765, 1548, 1848, 1386, 630, 336, 36, 9, 1, 17966, 15070, 15525, 16080, 14070, 3780, 1050, 480, 45, 10, 1

Offset: 0

Alois P. Heinz, Sep 28 2019

Here we assume that every list of parts has at least one 0 because its addition does not change the value of the multinomial.

Number T(n,k) of set partitions of [n] with distinct block sizes and one of the block sizes is k. T(5,3) = 10: 123|45, 124|35, 125|34, 12|345, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234.

			Triangle T(n,k) begins:
      1;
      1,     1;
      1,     0,     1;
      4,     3,     3,     1;
      5,     4,     0,     4,     1;
     16,     5,    10,    10,     5,    1;
     82,    66,    75,    60,    15,    6,    1;
    169,   112,   126,    35,   140,   21,    7,   1;
    541,   456,   196,   336,   280,  224,   28,   8,  1;
   2272,   765,  1548,  1848,  1386,  630,  336,  36,  9,  1;
  17966, 15070, 15525, 16080, 14070, 3780, 1050, 480, 45, 10, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)
Wikipedia, Partition of a set

Columns k=0-3 give: A007837, A327876, A327881, A328155.
Row sums give A327870.
T(2n,n) gives A328156.
Cf. A327801, A327884.

with(combinat):
T:= (n, k)-> add(multinomial(add(i, i=l), l[], 0),
             l=select(x-> nops(x)=nops({x[]}) and
             (k=0 or k in x), partition(n))):
seq(seq(T(n, k), k=0..n), n=0..11);
# second Maple program:
b:= proc(n, i, k) option remember; `if`(i*(i+1)/2 n!*(b(n$2, 0)-`if`(k=0, 0, b(n$2, k))):
seq(seq(T(n, k), k=0..n), n=0..11);

b[n_, i_, k_] := b[n, i, k] = If[i(i+1)/2 < n, 0, If[n==0, 1, If[i<2, 0, b[n, i-1, If[i==k, 0, k]]] + If[i==k, 0, b[n-i, Min[n-i, i-1], k]/i!]]];
T[n_, k_] := n! (b[n, n, 0] - If[k == 0, 0, b[n, n, k]]);
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 28 2020, from 2nd Maple program *)

A276961 Number of set partitions of [2n] with largest set of size n.

Original entry on oeis.org

1, 1, 9, 90, 1015, 12978, 187110, 3008148, 53275365, 1028142830, 21426984722, 478684639524, 11394222257054, 287518726261900, 7658231720886900, 214521099685649640, 6299407928673657135, 193373975592937777770, 6189939300880260745050, 206159811915115686404700

Offset: 0

Alois P. Heinz, Sep 22 2016

nonn

The blocks are ordered with increasing least elements.

a(0) = 1 by convention.

			a(1) = 1: 1|2.
a(2) = 9: 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|24|3, 1|2|34.

Alois P. Heinz, Table of n, a(n) for n = 0..445
Wikipedia, Partition of a set

Cf. A000110, A080510, A276923, A297924, A297926, A327884, A328156.

b:= proc(n, k) option remember; `if`(n=0, 1, add(
      b(n-i, k)*binomial(n-1, i-1), i=1..min(n, k)))
    end:
a:= n-> `if`(n=0, 1, b(2*n, n)-b(2*n, n-1)):
seq(a(n), n=0..20);

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n - i, k]*Binomial[n - 1, i - 1], {i, 1, Min[n, k]}]];
a[n_] := If[n == 0, 1, b[2*n, n] - b[2*n, n - 1]];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 20 2018, translated from Maple *)

a(n) = A080510(2n,n).
a(n) = A327884(2n,n).
a(n) = ceiling(C(2n,n)*(A000110(n)-1/2)). - Ludovic Schwob, Jan 15 2022

A328153 Number of set partitions of [n] such that at least one of the block sizes is 3.

Original entry on oeis.org

0, 0, 0, 1, 4, 20, 90, 455, 2352, 13132, 76540, 473660, 3069220, 20922330, 149021600, 1109629885, 8604815520, 69437698160, 581661169640, 5051885815603, 45411759404560, 421977921782270, 4047693372023070, 40034523497947132, 407818256494533984, 4274309903558446900

Offset: 0

Alois P. Heinz, Oct 05 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..576
Wikipedia, Partition of a set

Column k=3 of A327884.

Cf. A000110, A124504, A328155.

b:= proc(n, k) option remember; `if`(n=0, 1, add(
      `if`(j=k, 0, b(n-j, k)*binomial(n-1, j-1)), j=1..n))
    end:
a:= n-> b(n, 0)-b(n, 3):
seq(a(n), n=0..27);

b[n_, k_] := b[n, k] = If[n==0, 1, Sum[If[j==k, 0, b[n-j, k] Binomial[n-1, j-1]], {j, 1, n}]];
a[n_] := b[n, 0] - b[n, 3];
a /@ Range[0, 27] (* Jean-François Alcover, May 02 2020, after Maple *)

E.g.f.: exp(exp(x)-1) - exp(exp(x)-1-x^3/6).

a(n) = A000110(n) - A124504(n).

A327885 Number of set partitions of [n] such that at least one of the block sizes is 2.

Original entry on oeis.org

0, 0, 1, 3, 9, 35, 150, 672, 3269, 17271, 97155, 578985, 3654750, 24331320, 170074177, 1244911605, 9520843575, 75890001665, 629104453236, 5413637745144, 48277814341765, 445463898405225, 4246785220234557, 41775507558584283, 423516880995944532

Offset: 0

Alois P. Heinz, Sep 28 2019

nonn

			a(2) = 1: 12.
a(3) = 3: 12|3, 13|2, 1|23.
a(4) = 9: 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
a(5) = 35: 123|45, 124|35, 125|34, 12|345, 12|34|5, 12|35|4, 12|3|45, 12|3|4|5, 134|25, 135|24, 13|245, 13|24|5, 13|25|4, 13|2|45, 13|2|4|5, 145|23, 14|235, 14|23|5, 15|234, 15|23|4, 1|23|45, 1|23|4|5, 14|25|3, 14|2|35, 14|2|3|5, 15|24|3, 1|24|35, 1|24|3|5, 15|2|34, 1|25|34, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45.

Alois P. Heinz, Table of n, a(n) for n = 0..576
Wikipedia, Partition of a set

Column k=2 of A327884.

Cf. A000110, A097514.

b:= proc(n, k) option remember; `if`(n=0, 1, add(
      `if`(j=k, 0, b(n-j, k)*binomial(n-1, j-1)), j=1..n))
    end:
a:= n-> b(n, 0)-b(n, 2):
seq(a(n), n=0..27);

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[If[j == k, 0, b[n - j, k]* Binomial[n - 1, j - 1]], {j, n}]];
a[n_] := b[n, 0] - b[n, 2];
a /@ Range[0, 27] (* Jean-François Alcover, May 04 2020, after Maple *)

E.g.f.: exp(exp(x)-1) - exp(exp(x)-1-x^2/2).

a(n) = A000110(n) - A097514(n).

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.

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A327869 Sum T(n,k) of multinomials M(n; lambda), where lambda ranges over all partitions of n into distinct parts incorporating k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A276961 Number of set partitions of [2n] with largest set of size n.

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A328153 Number of set partitions of [n] such that at least one of the block sizes is 3.

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A327885 Number of set partitions of [n] such that at least one of the block sizes is 2.

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