cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Original entry on oeis.org

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

Views

Author

Wouter Meeussen, Mar 22 2003

Keywords

Comments

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)

Examples

			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;
  ...

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    << 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 *)

Formula

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)

A327884 Number T(n,k) of set partitions of [n] such that at least one of the block sizes is k or k=0; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 5, 4, 3, 1, 15, 11, 9, 4, 1, 52, 41, 35, 20, 5, 1, 203, 162, 150, 90, 30, 6, 1, 877, 715, 672, 455, 175, 42, 7, 1, 4140, 3425, 3269, 2352, 1015, 280, 56, 8, 1, 21147, 17722, 17271, 13132, 6237, 1890, 420, 72, 9, 1, 115975, 98253, 97155, 76540, 39480, 12978, 3150, 600, 90, 10, 1
Offset: 0

Views

Author

Alois P. Heinz, Sep 28 2019

Keywords

Examples

			T(4,1) = 11: 123|4, 124|3, 12|3|4, 134|2, 13|2|4, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
T(4,2) = 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.
T(4,3) = 4: 123|4, 124|3, 134|2, 1|234.
T(4,4) = 1: 1234.
T(5,1) = 41: 1234|5, 1235|4, 123|4|5, 1245|3, 124|3|5, 12|34|5, 125|3|4, 12|35|4, 12|3|45, 12|3|4|5, 1345|2, 134|2|5, 13|24|5, 135|2|4, 13|25|4, 13|2|45, 13|2|4|5, 14|23|5, 1|2345, 1|234|5, 15|23|4, 1|235|4, 1|23|45, 1|23|4|5, 145|2|3, 14|25|3, 14|2|35, 14|2|3|5, 15|24|3, 1|245|3, 1|24|35, 1|24|3|5, 15|2|34, 1|25|34, 1|2|345, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45, 1|2|3|4|5.
Triangle T(n,k) begins:
      1;
      1,     1;
      2,     1,     1;
      5,     4,     3,     1;
     15,    11,     9,     4,    1;
     52,    41,    35,    20,    5,    1;
    203,   162,   150,    90,   30,    6,   1;
    877,   715,   672,   455,  175,   42,   7,  1;
   4140,  3425,  3269,  2352, 1015,  280,  56,  8, 1;
  21147, 17722, 17271, 13132, 6237, 1890, 420, 72, 9, 1;
  ...

Links

Crossrefs

Columns k=0-3 give: A000110, A000296(n+1), A327885, A328153.
T(2n,n) gives A276961.
Cf. A080510, A327869.

Programs

  • Maple
    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:
T:= (n, k)-> b(n, 0)-`if`(k=0, 0, b(n, k)):
seq(seq(T(n, k), k=0..n), n=0..11);
  • Mathematica
    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}]];
T[n_, k_] := b[n, 0] - If[k == 0, 0, b[n, k]];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 30 2020, after Alois P. Heinz *)

Formula

E.g.f. of column k: exp(exp(x)-1) - [k>0] * exp(exp(x)-1-x^k/k!).
T(n,0) - T(n,1) = A000296(n).

A297924 Number of set partitions of [2n] in which the size of the last block is n.

Original entry on oeis.org

1, 1, 4, 20, 125, 952, 8494, 86025, 969862, 12020580, 162203607, 2363458396, 36930606254, 615302885459, 10878670826170, 203268056115256, 3999642836434361, 82617423216826640, 1786559190116778030, 40344863179696283037, 949348461372003462390
Offset: 0

Views

Author

Alois P. Heinz, Jan 08 2018

Keywords

Comments

The blocks are ordered with increasing least elements.
a(0) = 1 by convention.

Examples

			a(1) = 1: 1|2.
a(2) = 4: 12|34, 13|24, 14|23, 1|2|34.
a(3) = 20: 123|456, 124|356, 125|346, 126|345, 12|3|456, 134|256, 135|246, 136|245, 13|2|456, 145|236, 146|235, 156|234, 1|23|456, 14|2|356, 1|24|356, 15|2|346, 1|25|346, 16|2|345, 1|26|345, 1|2|3|456.

Links

Crossrefs

Cf. A121207, A124496, A276961, A297926.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(n=k, 1,
      add(b(n-j, k)*binomial(n-1, j-1), j=1..n-k))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..25);
  • Mathematica
    b[n_, k_] := b[n, k] = If[n == k, 1, Sum[b[n - j, k]*Binomial[n - 1, j - 1], {j, 1, n - k}]];
a[n_] := b[2*n, n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 20 2018, translated from Maple *)

Formula

a(n) = A121207(2n,n) = A124496(2n,n).

A297926 Number of set partitions of [2n] in which the size of the first block is n.

Original entry on oeis.org

1, 1, 6, 50, 525, 6552, 93786, 1504932, 26640900, 514083570, 10713538550, 239342496120, 5697111804566, 143759365731100, 3829115870472600, 107260549881604200, 3149703964487098665, 96686987797052290440, 3094969650442399156350, 103079905957566679518300
Offset: 0

Views

Author

Alois P. Heinz, Jan 08 2018

Keywords

Comments

The blocks are ordered with increasing least elements.
a(0) = 1 by convention.

Examples

			a(1) = 1: 1|2.
a(2) = 6: 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 14|2|3.

Links

Crossrefs

Cf. A000110, A056857, A056860, A276961, A297924.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> binomial(2*n-1, n-1)*b(n):
seq(a(n), n=0..25);
  • Mathematica
    b[n_] := b[n] = If[n == 0, 1, Sum[b[n-j]*Binomial[n-1, j-1], {j, 1, n}]];
a[n_] := Binomial[2*n-1, n-1] * b[n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 20 2018, translated from Maple *)

Formula

a(n) = binomial(2*n-1,n-1) * Bell(n).
a(n) = A056857(2n,n) = A056860(2n,n).

A328156 Number of set partitions of [2n] with distinct block sizes and one of the block sizes is n.

Original entry on oeis.org

1, 0, 0, 60, 280, 3780, 74844, 576576, 6949800, 110416020, 3319141540, 31333878576, 545777101324, 8349081650000, 196469122903200, 8108831645948160, 99934219113287400, 1961077012271694900, 39215221761564594900, 860948656518718429200, 25274389422461123124180
Offset: 0

Views

Author

Alois P. Heinz, Oct 05 2019

Keywords

Links

Crossrefs

Cf. A266518, A276961, A327869.

Programs

  • Maple
    b:= proc(n, i, k) option remember; `if`(i*(i+1)/2 (2*n)!*(b(2*n$2, 0)-`if`(n=0, 0, b(2*n$2, n))):
seq(a(n), n=0..22);
  • Mathematica
    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!]]];
a[n_] := (2n)! (b[2n, 2n, 0] - If[n == 0, 0, b[2n, 2n, n]]);
a /@ Range[0, 22] (* Jean-François Alcover, May 02 2020, after Maple *)

Formula

a(n) = A327869(2n,n).

A276923 Number of ordered set partitions of [2n] where the maximal block size equals n.

Original entry on oeis.org

1, 2, 42, 860, 21490, 657972, 24011988, 1017804216, 49118959890, 2657929522820, 159340977018652, 10480673825750856, 750335572490293972, 58077997318270046600, 4832536579295065540200, 430136064463753547944560, 40779223639911413185024530
Offset: 0

Views

Author

Alois P. Heinz, Sep 22 2016

Keywords

Links

Crossrefs

Cf. A276922, A276961.

Programs

  • Maple
    A:= proc(n, k) option remember; `if`(n=0, 1, add(
       A(n-i, k)*binomial(n, i), i=1..min(n, k)))
    end:
a:= n-> A(2*n, n) -`if`(n=0, 0, A(2*n, n-1)):
seq(a(n), n=0..20);
  • Mathematica
    A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[A[n - i, k]*Binomial[n, i], {i, 1, Min[n, k]}]];
a[n_] := A[2*n, n] - If[n == 0, 0, A[2*n, n - 1]];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 13 2018, translated from Maple *)

Formula

a(n) = A276922(2n,n).
a(n) ~ 2^(2*n-3/2) * n^(n+1) / (exp(n) * log(2)^(n+2)). - Vaclav Kotesovec, Sep 24 2016
Showing 1-6 of 6 results.

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