A102356 -id:A102356 - cp's OEIS frontend

A102456 a(n) = n!/A102356(n).

1, 1, 2, 2, 4, 8, 12, 24, 48, 96, 288, 576, 1152, 2304, 6912, 13824, 27648, 82944, 165888, 497664, 1327104, 2985984, 7962624, 19906560, 59719680, 143327232, 358318080, 955514880, 2866544640, 7644119040, 17199267840, 51597803520, 137594142720, 412782428160

Offset: 0

Vladeta Jovovic, Feb 23 2005

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..500

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if` (i<1, 0,
       max(seq(b(n-i*j, i-1) *n!/i!^j/(n-i*j)!/j!, j=0..n/i))))
    end:
a:= n-> n!/b(n, n):
seq(a(n), n=0..50);  # Alois P. Heinz, Jun 01 2012

b[0, ] = 1; b[, ?NonPositive] = 0; b[n, i_] := b[n, i] = Max[Table[b[n - i*j, i - 1]*n!/i!^j/(n - i*j)!/j!, {j, 0, n/i}]]; a[n_] := n!/b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)

A130760 Noncrossing set partition version of A102356.

Original entry on oeis.org

1, 1, 1, 3, 6, 10, 30, 105, 280, 756, 2520, 6930, 18480, 60060, 180180, 675675, 2162160, 6806800, 24504480, 77597520, 232792560, 888844320, 3259095840, 10708457760, 37479602160, 133855722000, 435031096500, 1445641797600, 5059746291600, 17468171721000, 58227239070000

Offset: 0

Dan Drake, Jul 13 2007

nonn

			a(7) = 105 because there are 105 noncrossing set partitions of {1,2,3,4,5,6,7} of type {3,2,1,1} and all other integer partitions of 7 produce fewer noncrossing set partitions.

D. E. Knuth, The Art of Computer Programming, vol. 4, section 7.2.1.5, problem 65.

Vaclav Kotesovec, Table of n, a(n) for n = 0..75

Cf. A102356.

Mathematica
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<Vaclav Kotesovec, Oct 23 2014 *)
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More terms from Vaclav Kotesovec, Oct 23 2014

A080575 Triangle of multinomial coefficients, read by rows (version 2).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 3, 6, 1, 1, 5, 10, 10, 15, 10, 1, 1, 6, 15, 15, 10, 60, 20, 15, 45, 15, 1, 1, 7, 21, 21, 35, 105, 35, 70, 105, 210, 35, 105, 105, 21, 1, 1, 8, 28, 28, 56, 168, 56, 35, 280, 210, 420, 70, 280, 280, 840, 560, 56, 105, 420, 210, 28, 1, 1, 9, 36, 36, 84, 252, 84, 126, 504, 378, 756, 126, 315, 1260, 1260, 1890, 1260, 126, 280, 2520, 840, 1260, 3780, 1260, 84, 945, 1260, 378, 36, 1, 1, 10, 45, 45, 120, 360, 120, 210, 840, 630, 1260, 210

Offset: 1

Wouter Meeussen, Mar 23 2003

This is different from A036040 and A178867.
T[n,m] = count of set partitions of n with block lengths given by the m-th partition of n in the canonical ordering.
From Tilman Neumann, Oct 05 2008: (Start)
These are also the coefficients occurring in complete Bell polynomials, Faa di Bruno's formula (in its simplest form) and computation of moments from cumulants.
Though the Bell polynomials seem quite unwieldy, they can be computed easily as the determinant of an n-dimensional square matrix. (see e.g. [Coffey] and program below)
The complete Bell polynomial of the first n primes gives A007446. (End)
The difference with A036040 and A178867 lies in the ordering of the monomials. This sequence uses lexicographic ordering, while in A036040 the total order (power) of the monomials prevails (Abramowitz-Stegun style): e.g., in row 6 we have ...+ 15*x[3]*x[5] + 15*x[3]*x[6]^2 + 10*x[4]^2 +...; in A036040 the coefficient of x[3]*x[6]^2 would come after that of x[4]^2 because the total order is higher, here it comes before in view of the lexicographic order. - M. F. Hasler, Jul 12 2015

			For n=4 the 5 integer partitions in canonical ordering with corresponding set partitions and counts are:
   [4]       -> #{1234} = 1
   [3,1]     -> #{123/4, 124/3, 134/2, 1/234} = 4
   [2,2]     -> #{12/34, 13/24, 14/23} = 3
   [2,1,1]   -> #{12/3/4, 13/2/4, 1/23/4, 14/2/3, 1/24/3, 1/2/34} = 6
   [1,1,1,1] -> #{1/2/3/4} = 1
Thus row 4 is [1, 4, 3, 6, 1].
Triangle begins:
1;
1, 1;
1, 3,  1;
1, 4,  3,  6,  1;
1, 5, 10, 10, 15,  10,  1;
1, 6, 15, 15, 10,  60, 20, 15,  45,  15,  1;
1, 7, 21, 21, 35, 105, 35, 70, 105, 210, 35, 105, 105, 21, 1;
...
Row 4 represents 1*k(4)+4*k(3)*k(1)+3*k(2)^2+6*k(2)*k(1)^2+1*k(1)^4 and T(4,4)=6 since there are six ways of partitioning four labeled items into one part with two items and two parts each with one item.

See A036040 for the column labeled "M_3" in Abramowitz and Stegun, Handbook, p. 831.

Alois P. Heinz, Rows n = 1..26, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Berg, Kimmo Complexity of solution structures in nonlinear pricing, Ann. Oper. Res. 206, 23-37 (2013).
R. J. Cano, Sequencer program.
Mark W. Coffey, A Set of Identities for a Class of Alternating Binomial Sums Arising in Computing Applications, arXiv:math-ph/0608049, 2006.
Wikipedia, Cumulant.
Wikipedia, Bell polynomials

See A036040 for another version. Cf. A036036-A036039.
Row sums are A000110.
Row lengths are A000041.
Cf. A007446. - Tilman Neumann, Oct 05 2008
Cf. A178866 and A178867 (version 3). - Johannes W. Meijer, Jun 21 2010
Maximum value in row n gives A102356(n).

runs[li:{__Integer}] := ((Length/@ Split[ # ]))&[Sort@ li]; Table[Apply[Multinomial, IntegerPartitions[w], {1}]/Apply[Times, (runs/@ IntegerPartitions[w])!, {1}], {w, 6}]
(* Second program: *)
completeBellMatrix[x_, n_] := Module[{M, i, j}, M[, ] = 0; For[i=1, i <= n-1 , i++, M[i, i+1] = -1]; For[i=1, i <= n , i++, For[j=1, j <= i, j++, M[i, j] = Binomial[i-1, j-1]*x[i-j+1]]]; Array[M, {n, n}]]; completeBellPoly[x_, n_] := Det[completeBellMatrix[x, n]]; row[n_] := List @@ completeBellPoly[x, n] /. x[] -> 1 // Reverse; Table[row[n], {n, 1, 10}] // Flatten (* _Jean-François Alcover, Aug 31 2016, after Tilman Neumann *)
B[0] = 1;
B[n_] := B[n] = Sum[Binomial[n-1, k] B[n-k-1] x[k+1], {k, 0, n-1}]//Expand;
row[n_] := Reverse[List @@ B[n] /. x[_] -> 1];
Table[row[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Aug 10 2018, after Wolfdieter Lang *)

completeBellMatrix := proc(x,n) // x - vector x[1]...x[m], m>=n
local i,j,M; begin M:=matrix(n,n): // zero-initialized
for i from 1 to n-1 do M[i,i+1]:=-1: end_for:
for i from 1 to n do for j from 1 to i do
    M[i,j] := binomial(i-1,j-1)*x[i-j+1]:
end_for: end_for:
return (M): end_proc:
completeBellPoly := proc(x, n) begin
return (linalg::det(completeBellMatrix(x,n))): end_proc:
for i from 1 to 10 do print(i,completeBellPoly(x,i)): end_for:
// Tilman Neumann, Oct 05 2008

PARI
```
\\ See links.
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A080575_poly(n,V=vector(n,i,eval(Str('x,i))))={matdet(matrix(n,n,i,j,if(j<=i,binomial(i-1,j-1)*V[n-i+j],-(j==i+1))))}
A080575_row(n)={(f(s)=if(type(s)!="t_INT",concat(apply(f,select(t->t,Vec(s)))),s))(A080575_poly(n))} \\ M. F. Hasler, Jul 12 2015

A102462 Max{ k!/(a(1)!a(2)!..a(n)!) : a(1) + 2a(2) + 3a(3) + ... + na(n) = n, a(1) + a(2) + ... + a(n) = k }.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 12, 20, 30, 60, 105, 168, 280, 504, 840, 1512, 2520, 5040, 9240, 15840, 27720, 55440, 102960, 180180, 360360, 675675, 1201200, 2162160, 4084080, 7351344, 12697776, 24504480, 46558512, 84651840, 155195040, 296281440, 543182640, 961015440

Offset: 0

Vladeta Jovovic, Feb 23 2005

nonn

a(n) is the greatest number in row n of A048996 and in row n of A072811. Thus a(n) is the greatest number of compositions (permutations) obtainable from some partition of n. Example: a(7)=12 is the greatest number of compositions from some partition of 7, specifically, the partition {3,2,1,1}. - Clark Kimberling, Dec 24 2006
The partition(s) giving this optimum is always one where #{parts equal to i} >= #{parts equal to j} if i <= j. These partitions are counted in A007294. - Franklin T. Adams-Watters, Apr 08 2008
The number of partition(s) giving this optimum is given by A198254. - Olivier Gérard, Nov 17 2011

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A048992, A059171, A072811, A102356.

b:= proc(n,i,p) option remember; `if`(n=0 or i=1, (p+n)!/n!,
       max(seq(b(n-i*j, i-1, p+j)/j!, j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 15 2015

b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, (p + n)!/n!, Max[Table[ b[n-i*j, i-1, p+j]/j!, {j, 0, n/i}]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Dec 19 2015, after Alois P. Heinz *)

A211350 Refined triangle A124323: T(n,k) is the number of partitions of an n-set that are of type k (k-th integer partition, defined by A194602).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 4, 3, 1, 1, 10, 10, 15, 5, 10, 1, 1, 15, 20, 45, 15, 60, 6, 15, 15, 10, 1, 1, 21, 35, 105, 35, 210, 21, 105, 105, 70, 7, 105, 21, 35, 1, 1, 28, 56, 210, 70, 560, 56, 420, 420, 280, 28, 840, 168, 280, 8, 105, 210, 280, 28

Offset: 1

Tilman Piesk, Apr 09 2012

Name could also be "Triangle of multinomial coefficients, read by rows (version 4)", compare A036040, A080575, A178867. The latter and this one differ only in the order of columns.
The rows are counted from 1, the columns from 0.
Row lengths: 1,2,3,5,7,11... (partition numbers A000041)
Row sums: 1,2,5,15,52,203... (Bell numbers A000110)
Row maxima: 1,1,3,6,15,60,210,840,3780,12600,69300,415800... (A102356)
Distinct entries per row: 1,1,2,4,4,7,7,13,17,23,26,40... (A102465)
Rightmost columns are those from Pascal's triangle A007318 without the second one (i.e. triangle A184049). The other columns - (always?) without a 1 at the top - are multiples of these columns from Pascal's triangle; so actually only the top elements of each column are needed to calculate the other entries; these top elements are in A211360. (The top elements of the related triangle A178867 are in A178866.)

Tilman Piesk, Rows n=1..12 of triangle, flattened
Tilman Piesk, Partition related number triangles

Cf. A124323, A194602, A178867, A000041, A000110, A102356, A102465, A184049, A211360.

A255404 Number of different integer partitions of n that produce the maximum number of set partitions for a set of cardinality n.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 3, 2, 1, 4, 2, 1, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 4, 6, 4, 1, 2, 1, 5, 5, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 5, 2, 2, 1, 1, 4, 1, 1, 2, 3, 1, 8, 2, 1, 1, 3, 1, 1, 1, 3, 1, 1, 3, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 3, 2, 1, 1, 1, 1

Offset: 0

Andrei Cretu, Feb 22 2015

nonn

If n=Sum_i[n_i], the number of set partitions can be written as sp=n!/Prod_i,j(n_i!m_j!) where m_j is the multiplicity of the integer j in the n_i's. For certain integers, this number is maximized by more than one partition.

			For n=9, {1,1,2,2,3} maximizes the number of set partitions, while for n=10, this number is maximized by {1,2,3,4}, {1,1,2,3,3}, {1,2,2,2,3} and {1,1,1,2,2,3}.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A102356, A102456.

Prod[l_] := Apply[Times, Map[#! &, l]]*
    Apply[Times, Map[Count[l, #]! &, Range[Max[Length[l]]]]]
b[n_] := (Min[Map[Prod, IntegerPartitions[n]]])
a[n_] := Count[Map[Prod, IntegerPartitions[n]], b[n]]
Table[a[n], {n, 0, 20}] (* after A102356 *)

More terms from Alois P. Heinz, Feb 25 2015

A333305 Irregular array read by rows, a refinement of A256894.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 3, 3, 1, 4, 3, 5, 1, 6, 1, 1, 1, 1, 4, 6, 4, 1, 5, 10, 9, 8, 7, 1, 10, 15, 9, 1, 10, 1, 1, 1, 1, 5, 10, 10, 5, 1, 6, 15, 14, 10, 35, 16, 15, 9, 1, 15, 60, 19, 15, 33, 12, 1, 20, 45, 14, 1, 15, 1, 1

Offset: 0

Peter Luschny, May 19 2020

			Irregular table (the refinement is indicated by round brackets) starts:
[0] [1]
[1] [1, 1]
[2] [1, (1, 1), 1]
[3] [1, (1, 2, 1), (3, 1), 1]
[4] [1, (1, 3, 3, 1), (4, 3, 5, 1), (6, 1), 1]
[5] [1, (1, 4, 6, 4, 1), (5, 10, 9, 8, 7, 1), (10, 15, 9, 1), (10, 1), 1]
[6] [1, (1, 5, 10, 10, 5, 1), (6, 15, 14, 10, 35, 16, 15, 9, 1), (15, 60, 19, 15,
     33, 12, 1), (20, 45, 14, 1), (15, 1), 1]

Peter Luschny, The Bell transform

Cf. A000070 (length of rows), A102356 (max in rows), A186021 (sum of rows).

Cf. A256894.

def BellBlocks(n):
    R = InfinitePolynomialRing(ZZ, 'v') # Thanks to F. Chapoton.
    V = R.gen()
    @cached_function
    def T(n, k):
        if k == 0: return V[0]^n
        return sum(binomial(n-1, j-1)*V[j]*T(n-j, k-1) for j in (0..n-k+1))
    P = (T(n, k) for k in (0..n))
    return flatten([p.coefficients() for p in P])
for n in (0..8): print(BellBlocks(n))

cp's OEIS Frontend

A102456 a(n) = n!/A102356(n).

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A130760 Noncrossing set partition version of A102356.

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A080575 Triangle of multinomial coefficients, read by rows (version 2).

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A102462 Max{ k!/(a(1)!*a(2)!*..*a(n)!) : a(1) + 2*a(2) + 3*a(3) + ... + n*a(n) = n, a(1) + a(2) + ... + a(n) = k }.

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Maple

Mathematica

A211350 Refined triangle A124323: T(n,k) is the number of partitions of an n-set that are of type k (k-th integer partition, defined by A194602).

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Comments

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Crossrefs

A255404 Number of different integer partitions of n that produce the maximum number of set partitions for a set of cardinality n.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A333305 Irregular array read by rows, a refinement of A256894.

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Author

Keywords

Examples

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SageMath

A102462 Max{ k!/(a(1)!a(2)!..a(n)!) : a(1) + 2a(2) + 3a(3) + ... + na(n) = n, a(1) + a(2) + ... + a(n) = k }.