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 1-4 of 4 results.

A035470 Number of ways to break {1,2,3,...,n} into sets with equal sums.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 6, 12, 11, 2, 80, 166, 2, 665, 2918, 3309, 9296, 23730, 31875, 301030, 422897, 2, 13716867, 71504980, 100664385, 54148591, 880696662, 498017759, 27450476787, 111911522819, 179459955554, 2144502175214, 59115423983, 45837019664552, 375743493787258, 816118711787493, 2, 9492169507922
Offset: 1

Views

Author

Erich Friedman

Keywords

Comments

a(n) = 2 <=> |{d|n*(n+1)/2 : d>=n}| = 2. - Alois P. Heinz, Sep 03 2009

Examples

			a(7) = 6 since we have 1234567, 16/25/34/7, 167/2345, 257/1346, 347/1256, 356/1247.
From _Gus Wiseman_, Jul 13 2019: (Start)
The a(6) = 2 through a(9) = 11 set partitions with equal block-sums:
  {123456}      {1234567}        {12345678}        {123456789}
  {16}{25}{34}  {1247}{356}      {12348}{567}      {12345}{69}{78}
                {1256}{347}      {12357}{468}      {1239}{456}{78}
                {1346}{257}      {12456}{378}      {1248}{357}{69}
                {167}{2345}      {1278}{3456}      {1257}{348}{69}
                {16}{25}{34}{7}  {1368}{2457}      {1347}{258}{69}
                                 {1458}{2367}      {1356}{249}{78}
                                 {1467}{2358}      {159}{2346}{78}
                                 {1236}{48}{57}    {159}{267}{348}
                                 {138}{246}{57}    {168}{249}{357}
                                 {156}{237}{48}    {18}{27}{36}{45}{9}
                                 {18}{27}{36}{45}
(End)

Links

Crossrefs

Cf. A164977, A164978. - Alois P. Heinz, Sep 03 2009
Row sums of A275714.
Row sums of A320438.
Cf. A000110, A007837, A038041, A112956, A275780, A275781, A321455, A326512, A326513, A326518, A326534.

Programs

  • Maple
    with(numtheory): b:= proc() option remember; local i, j, t; `if`(args[1]=0, `if`(nargs=2, 1, b(args[t] $t=2..nargs)), add(`if`(args[j] -args[nargs] <0, 0, b(sort([seq(args[i] -`if`(i=j, args[nargs], 0), i=1..nargs-1)])[], args[nargs]-1)), j=1..nargs-1)) end: a:= proc(n) local i, m, x; m:= n*(n+1)/2; 1+ add(b(i$(m/i), n)/(m/i)!, i=[select(x-> x>=n, divisors(m) minus {m})[]]) end: seq(a(n), n=1..25);  # Alois P. Heinz, Sep 03 2009
  • Mathematica
    b[args_List] := b[args] = If[args[[1]] == 0, If[Length[args] == 2, 1, b[Rest[args]]], Sum[If[args[[j]] - args[[-1]] < 0, 0, b[Sort[Join[Table[ args[[i]] - If[i == j, args[[-1]], 0], {i, 1, Length[args]-1}]]], {args[[-1]]-1}]], {j, 1, Length[args]-1}]]; b[a1_List, a2_List] := b[Join[a1, a2]];
a[n_] := a[n] = With[{m = n*(n+1)/2}, 1+Sum[b[Append[Array[i&, m/i], n]] / (m/i)!, {i, Select[Divisors[m] ~Complement~ {m}, # >= n &]}]];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 25}] (* Jean-François Alcover, Mar 22 2017, after Alois P. Heinz *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],SameQ@@Total/@#&]],{n,0,10}] (* Gus Wiseman, Jul 13 2019 *)

Extensions

More terms from John W. Layman, Mar 18 2002
a(19)-a(33) from Alois P. Heinz, Sep 03 2009
a(34) from Alois P. Heinz, May 24 2015
a(35)-a(38) from Max Alekseyev, Feb 15 2024

A275714 Number T(n,k) of set partitions of [n] into k blocks with equal element sum; triangle T(n,k), n>=0, 0<=k<=ceiling(n/2), read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 4, 0, 1, 0, 1, 7, 3, 1, 0, 1, 0, 9, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 35, 43, 0, 0, 1, 0, 1, 62, 102, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 595, 0, 68, 0, 1, 0, 1, 361, 1480, 871, 187, 17, 0, 1
Offset: 0

Views

Author

Alois P. Heinz, Aug 06 2016

Keywords

Examples

			T(8,1) = 1: 12345678.
T(8,2) = 7: 12348|567, 12357|468, 12456|378, 1278|3456, 1368|2457, 1458|2367, 1467|2358.
T(8,3) = 3: 1236|48|57, 138|246|57, 156|237|48.
T(8,4) = 1: 18|27|36|45.
T(9,3) = 9: 12345|69|78, 1239|456|78, 1248|357|69, 1257|348|69, 1347|258|69, 1356|249|78, 159|2346|78, 168|249|357, 159|267|348.
Triangle T(n,k) begins:
00 :  1;
01 :  0,  1;
02 :  0,  1;
03 :  0,  1,   1;
04 :  0,  1,   1;
05 :  0,  1,   0,    1;
06 :  0,  1,   0,    1;
07 :  0,  1,   4,    0,   1;
08 :  0,  1,   7,    3,   1;
09 :  0,  1,   0,    9,   0,   1;
10 :  0,  1,   0,    0,   0,   1;
11 :  0,  1,  35,   43,   0,   0,  1;
12 :  0,  1,  62,  102,   0,   0,  1;
13 :  0,  1,   0,    0,   0,   0,  0, 1;
14 :  0,  1,   0,  595,   0,  68,  0, 1;
15 :  0,  1, 361, 1480, 871, 187, 17, 0, 1;

Links

Crossrefs

Columns k=0-5 give: A000007, A000012 (for n>0), A058377, A112972, A317806, A317807.
Row sums give A035470 = 1 + A112956.
T(n^2,n) gives A321282.
Cf. A248112.

Programs

  • Mathematica
    Needs["Combinatorica`"]; T[n_, k_] := Count[(Equal @@ (Total /@ #)&) /@ KSetPartitions[n, k], True]; Table[row = Table[T[n, k], {k, 0, Ceiling[n/2]}]; Print[n, " ", row]; row, {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 20 2017 *)

A112972 Number of ways the set {1,2,...,n} can be split into three subsets of equal sums.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 0, 3, 9, 0, 43, 102, 0, 595, 1480, 0, 9294, 23728, 0, 157991, 411474, 0, 2849968, 7562583, 0, 53987864, 145173095, 0, 1061533318, 2885383960, 0, 21515805520, 59003023409, 0, 447142442841, 1235311936936, 0, 9489835046489, 26382363207307
Offset: 1

Views

Author

Floor van Lamoen, Oct 07 2005

Keywords

Examples

			For n=8 we have 84/75/6321, 84/732/651 and 831/75/642 so a(8)=3.

Links

Crossrefs

Cf. A112956, A058377.
Column k=3 of A275714.
Similar sequences: A327448, A327449, A327450.

Programs

  • Maple
    A112972:= n-> coeff(coeff(mul((x^(-2*k)+x^k*(y^k+y^(-k)))
              , k=1..n), x, 0), y, 0)/6:
seq(A112972(n), n=1..20);
# second Maple program:
b:= proc() option remember; local i, j, t; `if`(args[1]=0,
      `if`(nargs=2, 1, b(args[t] $t=2..nargs)), add(
      `if`(args[j] -args[nargs]<0, 0, b(sort([seq(args[i]-
      `if`(i=j, args[nargs], 0), i=1..nargs-1)])[],
                args[nargs]-1)), j=1..nargs-1))
    end:
a:= n-> (m-> `if`(irem(m, 3)=0, b((m/3)$3, n)/6, 0))(n*(n+1)/2):
seq(a(n), n=1..42);  # Alois P. Heinz, Sep 03 2009
  • Mathematica
    b[args_List] := b[args] = Module[{nargs = Length[args]}, If[args[[1]] == 0, If[nargs == 2, 1, b[args // Rest]], Sum[If[args[[j]] - Last[args] < 0, 0, b[Append[Sort[Flatten[Table[args[[i]] - If[i == j, Last[args], 0], {i, 1, nargs-1}]]], Last[args]-1]]], {j, 1, nargs-1}]]];
a[n_] := If[Mod[#, 3] == 0, b[{#/3, #/3, #/3, n}]/6, 0]&[n(n+1)/2];
Array[a, 42] (* Jean-François Alcover, Oct 30 2020, after Alois P. Heinz *)

Formula

a(n) is 1/6 of the coefficient of x^0*y^0 in Product_{k=1..n} (x^(-2*k)+x^k*(y^k+y^(-k))).

Extensions

Extended beyond a(25) by Alois P. Heinz, Sep 03 2009

A320438 Irregular triangle read by rows where T(n,k) is the number of set partitions of {1,...,n} with all block-sums equal to d, where d is the k-th divisor of n*(n+1)/2 that is >= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 3, 7, 1, 1, 9, 1, 1, 1, 1, 43, 35, 1, 1, 102, 62, 1, 1, 1, 1, 68, 595, 1, 1, 17, 187, 871, 1480, 361, 1, 1, 2650, 657, 1, 1, 9294, 1, 1, 23728, 1, 1, 27763, 4110, 1, 1, 1850, 25035, 108516, 157991, 7636, 1, 1, 11421, 411474, 1
Offset: 1

Views

Author

Gus Wiseman, Jan 08 2019

Keywords

Examples

			Triangle begins:
    1
    1
    1    1
    1    1
    1    1
    1    1
    1    4    1
    1    3    7    1
    1    9    1
    1    1
    1   43   35    1
    1  102   62    1
    1    1
    1   68  595    1
    1   17  187  871 1480  361    1
    1 2650  657    1
Row 8 counts the following set partitions:
  {{18}{27}{36}{45}}  {{1236}{48}{57}}  {{12348}{567}}  {{12345678}}
                      {{138}{246}{57}}  {{12357}{468}}
                      {{156}{237}{48}}  {{12456}{378}}
                                        {{1278}{3456}}
                                        {{1368}{2457}}
                                        {{1458}{2367}}
                                        {{1467}{2358}}

Crossrefs

Row lengths are A164978. Row sums are A035470.
Cf. A000110, A000258, A008277, A112956, A164977, A275714, A279375, A300335, A320423, A320424, A321455, A321469.

Programs

  • Mathematica
    spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[Select[Subsets[Range[n]],Total[#]==d&],Range[n]]],{n,12},{d,Select[Divisors[n*(n+1)/2],#>=n&]}]

Extensions

More terms from Jinyuan Wang, Feb 27 2025
Name edited by Peter Munn, Mar 06 2025
Showing 1-4 of 4 results.

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