A362925 -id:A362925 - cp's OEIS frontend

A113547 Triangle read by rows: number of labeled partitions of n with maximin m.

1, 1, 1, 1, 2, 2, 1, 4, 5, 5, 1, 8, 13, 15, 15, 1, 16, 35, 47, 52, 52, 1, 32, 97, 153, 188, 203, 203, 1, 64, 275, 515, 706, 825, 877, 877, 1, 128, 793, 1785, 2744, 3479, 3937, 4140, 4140, 1, 256, 2315, 6347, 11002, 15177, 18313, 20270, 21147, 21147, 1, 512, 6817, 23073, 45368, 68303, 88033, 102678, 111835, 115975, 115975

Offset: 1

Franklin T. Adams-Watters, Jan 13 2006

The maximin of a partition is the maximum over all parts of the minimum label in each part. If the rows are reversed, the result is the number of partitions of n with minimax m.

The number of restricted growth functions of length n where the maximum appears first at position m. The RGF's are defined here as f(1)=1 and f(i) <=1+max_{1<=jR. J. Mathar, Mar 18 2016

			Maximin [123]=max(1)=1, maximin [12|3]=max(1,3)=3, maximin [13|2]=max(1,2)=2, maximin [1|23]=max(1,2)=2 and maximin [1|2|3]=max(1,2,3)=3, so for n=3 the multiset of maximins is {1,2,2,3,3}, making the 3rd line 1,2,2.
1;
1,  1;
1,  2,   2;
1,  4,   5,   5;
1,  8,  13,  15,  15;
1, 16,  35,  47,  52,  52;
1, 32,  97, 153, 188, 203, 203;
1, 64, 275, 515, 706, 825, 877, 877;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Todd Tichenor, A note on graph compositions and their connection to minimax of set partitions, arXiv:1709.00393 [math.CO], 2017.

Cf. A008277, A000110, A271466.

A362924 and A362925 are other versions of this triangle. - N. J. A. Sloane, Aug 10 2023

A113547 := proc(n,m)
    add(combinat[stirling2](m-1,k-1)*k^(n-m),k=1..m) ;
end proc:
seq(seq( A113547(n,m),m=1..n),n=1..10) ; # R. J. Mathar, Mar 13 2016

T[n_, n_] := BellB[n - 1]; T[n_, n_ - 1] := BellB[n - 1]; T[n_, n_ - 2] := BellB[n - 1] - BellB[n - 3]; T[n_, m_] := Sum[StirlingS2[m - 1, k - 1]*k^(n - m), {k, 1, m}]; Table[T[n, m], {n, 1, 5}, {m, 1, n}] (* G. C. Greubel, May 06 2017 *)

T(n, m) = Sum_{k=1..m} S2(m-1, k-1)*k^(n-m), where S2 is the Stirling numbers of the second kind (A008277). T(n, n)=T(n, n-1)=B(n-1), where B is the Bell numbers (A000110). T(n, n-2)=B(n-1)-B(n-3).

Conjectures: T(n,3) = A007689(n-3). T(n,4) = 2^(n-4)+3^(n-3)+4^(n-4).- R. J. Mathar, Mar 13 2016

A362924 Triangle read by rows: T(n,m), n >= 1, 1 <= m <= n, is number of partitions of the set {1,2,...,n} that have at most one block contained in {1,...,m}.

Original entry on oeis.org

1, 2, 1, 5, 4, 1, 15, 13, 8, 1, 52, 47, 35, 16, 1, 203, 188, 153, 97, 32, 1, 877, 825, 706, 515, 275, 64, 1, 4140, 3937, 3479, 2744, 1785, 793, 128, 1, 21147, 20270, 18313, 15177, 11002, 6347, 2315, 256, 1, 115975, 111835, 102678, 88033, 68303, 45368, 23073, 6817, 512, 1, 678570, 657423, 610989, 536882, 436297, 316305, 191866, 85475, 20195, 1024, 1

Offset: 1

N. J. A. Sloane, Aug 10 2023, based on an email from Don Knuth

Also, the maximum number of solutions to an exact cover problem with n items, of which m are secondary.

			Triangle begins:
  [1],
  [2, 1],
  [5, 4, 1],
  [15, 13, 8, 1],
  [52, 47, 35, 16, 1],
  [203, 188, 153, 97, 32, 1],
  [877, 825, 706, 515, 275, 64, 1],
  [4140, 3937, 3479, 2744, 1785, 793, 128, 1],
  [21147, 20270, 18313, 15177, 11002, 6347, 2315, 256, 1],
  [115975, 111835, 102678, 88033, 68303, 45368, 23073, 6817, 512, 1],
  [678570, 657423, 610989, 536882, 436297, 316305, 191866, 85475, 20195, 1024, 1],
...
For example, if n=4, m=3, then T(4,3) = 8, because out of the A000110(4) = 15 set partitions of {1,2,3,4}, those that have 2 or more blocks contained in {1,2,3} are
  {12,3,4},
  {13,2,4},
  {14,2,3},
  {23,1,4},
  {24,1,3},
  {34,1,2},
  {1,2,3,4},
  while
  {1234},
  {123,4},
  {124,3}
  {134,2}
  {234,1},
  {12,34}
  {13. 24}.
  {14, 23}
  do not.

D. E. Knuth, The Art of Computer Programming, Volume 4B, exercise 7.2.2.1--185, answer on page 468.

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of the triangle, flattened).

See A113547 and A362925 for other versions of this triangle.
Row sums give A005493.
Cf. A000110, A008277, A078468, A143494.

with(combinat);
T:=proc(n,m) local k;
add(stirling2(n-m,k)*(k+1)^m, k=0..n-m);
end;

A362924[n_,m_]:=Sum[StirlingS2[n-m,k](k+1)^m,{k,0,n-m}];
Table[A362924[n,m],{n,15},{m,n}] (* Paolo Xausa, Dec 02 2023 *)

T(n, 1) = Bell number (all set partitions) A000110(n);
T(n, n) = 1 when m=n (the only possibility is a single block);
T(n, n-1) = 2^{n-1} when m=n-1 (a single block or two blocks);
T(n, 2) =  A078468(2).
In general, T(n, m) = Sum_{k=0..n-m} Stirling_2(n-m,k)*(k+1)^m.

A383049 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) is the n-th term of the inverse Stirling transform of j-> (j+1)^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 4, 1, 0, 1, 8, 5, -1, 0, 1, 16, 19, -3, 2, 0, 1, 32, 65, -1, 4, -6, 0, 1, 64, 211, 45, -10, -8, 24, 0, 1, 128, 665, 359, -116, 48, 20, -120, 0, 1, 256, 2059, 2037, -538, 340, -234, -52, 720, 0, 1, 512, 6305, 10079, -1316, 984, -1240, 1302, 72, -5040, 0

Offset: 0

Seiichi Manyama, Apr 14 2025

			Square array begins:
  1,  1,  1,    1,     1,     1,     1, ...
  1,  2,  4,    8,    16,    32,    64, ...
  0,  1,  5,   19,    65,   211,   665, ...
  0, -1, -3,   -1,    45,   359,  2037, ...
  0,  2,  4,  -10,  -116,  -538, -1316, ...
  0, -6, -8,   48,   340,   984, -1148, ...
  0, 24, 20, -234, -1240, -1866, 16400, ...

Christian G. Bower, PARI programs for transforms, 2007.
N. J. A. Sloane, Maple programs for transforms, 2001-2020.

Columns k=0..6 give A019590(n+1), A302190 (for n > 0), A222627, A222636, A222748, A223023, A383050.
Main diagonal gives A383051.
Cf. A362924, A362925.

a(n, k) = sum(j=0, n, (j+1)^k*stirling(n, j, 1));

A(n,k) = Sum_{j=0..n} (j+1)^k * Stirling1(n,j).
E.g.f. of column k: Sum_{j>=0} (j+1)^k * log(1+x)^j / j!.
E.g.f. of column k: (1+x) * Sum_{j=0..k} Stirling2(k+1,j+1) * log(1+x)^j.

A367820 Number of partitions of [2n] that have at most one block contained in [n].

Original entry on oeis.org

1, 2, 13, 153, 2744, 68303, 2224417, 90995838, 4538437039, 269755223485, 18766884323562, 1506040068195721, 137740473851280141, 14212098473767962472, 1640078704487165930485, 210103319793655159244093, 29684467774817808296383256, 4598958815992575305097910699

Offset: 0

Alois P. Heinz, Dec 01 2023

nonn

			a(2) = 13: 1234, 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3.

Alois P. Heinz, Table of n, a(n) for n = 0..288

Cf. A008277, A048993, A011655, A108459, A113547, A362925.

b:= proc(n) option remember; expand(`if`(n=0, 1,
      x*add(b(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
a:= n-> add(coeff(b(n), x, j)*(j+1)^n, j=0..n):
seq(a(n), n=0..21);

A367820[n_]:=Sum[StirlingS2[n,j](j+1)^n,{j,0,n}];
Array[A367820,25,0] (* Paolo Xausa, Dec 04 2023 *)

a(n) = A113547(2n+1,n+1) = A362925(2n,n).
a(n) = Sum_{j=0..n} (j+1)^n * Stirling2(n,j).
a(n) mod 2 = A011655(n+2).

A383052 a(n) = Sum_{k=0..n} (k+1)^3 * Stirling2(n,k).

Original entry on oeis.org

1, 8, 35, 153, 706, 3479, 18313, 102678, 610989, 3844525, 25492752, 177579961, 1295811637, 9879799744, 78525094847, 649253421173, 5573667453498, 49595062947091, 456689512735421, 4345710521536150, 42675672248378721, 431963852263306569, 4501627598926298992

Offset: 0

Seiichi Manyama, Apr 14 2025

nonn

Stirling transform of (n+1)^3.

Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A000110, A222636, A362925.

a(n) = sum(k=0, n, (k+1)^3*stirling(n, k, 2));

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^3*(exp(x)-1)^k/k!)))

a(n) = A362925(n+3,3).
E.g.f.: Sum_{k>=0} (k+1)^3 * (exp(x) - 1)^k / k!.
E.g.f.: exp(exp(x) - 1) * Sum_{k=0..3} Stirling2(4,k+1) * (exp(x) - 1)^k.

A383053 a(n) = Sum_{k=0..n} (k+1)^4 * Stirling2(n,k).

Original entry on oeis.org

1, 16, 97, 515, 2744, 15177, 88033, 536882, 3441439, 23151411, 163135410, 1201594675, 9232595661, 73858810120, 614045917741, 5296398334735, 47321198203496, 437310785441381, 4174403973827181, 41107555809612466, 417122543915965091, 4356601173778017487

Offset: 0

Seiichi Manyama, Apr 14 2025

nonn

Stirling transform of (n+1)^4.

Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A000110, A222748, A362925.

a(n) = sum(k=0, n, (k+1)^4*stirling(n, k, 2));

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^4*(exp(x)-1)^k/k!)))

a(n) = A362925(n+4,4).
E.g.f.: Sum_{k>=0} (k+1)^4 * (exp(x) - 1)^k / k!.
E.g.f.: exp(exp(x) - 1) * Sum_{k=0..4} Stirling2(5,k+1) * (exp(x) - 1)^k.

A383054 a(n) = Sum_{k=0..n} (k+1)^5 * Stirling2(n,k).

Original entry on oeis.org

1, 32, 275, 1785, 11002, 68303, 436297, 2891670, 19947717, 143327725, 1072207680, 8342947657, 67440657877, 565603592392, 4914839764895, 44191989524117, 410644596021954, 3938713285932859, 38950532224469117, 396712750010963782, 4157217331880368521

Offset: 0

Seiichi Manyama, Apr 14 2025

nonn

Stirling transform of (n+1)^5.

Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A000110, A223023, A362925.

a(n) = sum(k=0, n, (k+1)^5*stirling(n, k, 2));

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^5*(exp(x)-1)^k/k!)))

a(n) = A362925(n+5,5).
E.g.f.: Sum_{k>=0} (k+1)^5 * (exp(x) - 1)^k / k!.
E.g.f.: exp(exp(x) - 1) * Sum_{k=0..5} Stirling2(6,k+1) * (exp(x) - 1)^k.

cp's OEIS Frontend

A113547 Triangle read by rows: number of labeled partitions of n with maximin m.

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A362924 Triangle read by rows: T(n,m), n >= 1, 1 <= m <= n, is number of partitions of the set {1,2,...,n} that have at most one block contained in {1,...,m}.

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A383049 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) is the n-th term of the inverse Stirling transform of j-> (j+1)^k.

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A367820 Number of partitions of [2n] that have at most one block contained in [n].

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A383052 a(n) = Sum_{k=0..n} (k+1)^3 * Stirling2(n,k).

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A383053 a(n) = Sum_{k=0..n} (k+1)^4 * Stirling2(n,k).

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A383054 a(n) = Sum_{k=0..n} (k+1)^5 * Stirling2(n,k).

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