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 1-10 of 10 results.

A232466 Number of dependent sets with largest element n.

Original entry on oeis.org

0, 0, 1, 2, 4, 10, 20, 44, 93, 198, 414, 864, 1788, 3687, 7541, 15382, 31200, 63191, 127482, 256857, 516404, 1037104, 2080357, 4170283, 8354078, 16728270, 33485553, 67012082, 134083661, 268249350, 536617010, 1073391040, 2147014212, 4294321453, 8589084469, 17178702571, 34358228044, 68717407217, 137436320023, 274874294012, 549751307200, 1099505394507, 2199015662477, 4398035921221, 8796080392378, 17592168222674
Offset: 1

Views

Author

David S. Newman, Nov 24 2013

Keywords

Comments

Let S be a set of positive integers. If S can be divided into two subsets which have equal sums, then S is said to be a dependent set.
Dependent sets are also called biquanimous sets. Biquanimous partitions are counted by A002219 and ranked by A357976. - Gus Wiseman, Apr 18 2024

Examples

			From _Gus Wiseman_, Apr 18 2024: (Start)
The a(1) = 0 through a(6) = 10 sets:
  .  .  {1,2,3}  {1,3,4}    {1,4,5}    {1,5,6}
                 {1,2,3,4}  {2,3,5}    {2,4,6}
                            {1,2,4,5}  {1,2,3,6}
                            {2,3,4,5}  {1,2,5,6}
                                       {1,3,4,6}
                                       {2,3,5,6}
                                       {3,4,5,6}
                                       {1,2,3,4,6}
                                       {1,2,4,5,6}
                                       {2,3,4,5,6}
(End)

References

  • J. Bourgain, Λ_p-sets in analysis: results, problems and related aspects. Handbook of the geometry of Banach spaces, Vol. I,195-232, North-Holland, Amsterdam, 2001.

Links

Crossrefs

Cf. A161943, A232534.
Column k=2 of A248112.
First differences of A371791.
The complement is counted by A371793, differences of A371792.
This is the "bi-" case of A371797, differences of A371796.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A006827 and A371795 count non-biquanimous partitions, ranks A371731.
A237258 (aerated) counts biquanimous strict partitions, ranks A357854.
A321142 and A371794 count non-biquanimous strict partitions.
Cf. A035470, A064914, A321451, A321452, A366320, A367094, A371783, A371789.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(i<1, `if`(n=0, {0}, {}),
      `if`(i*(i+1)/2 p+x^i,
       b(n+i, i-1) union b(abs(n-i), i-1))))
    end:
a:= n-> nops(b(n, n-1)):
seq(a(n), n=1..15);  # Alois P. Heinz, Nov 24 2013
  • Mathematica
    b[n_, i_] := b[n, i] = If[i<1, If[n == 0, {0}, {}], If[i*(i+1)/2 < n, {}, b[n, i-1] ~Union~ Map[Function[p, p+x^i], b[n+i, i-1] ~Union~ b[Abs[n-i], i-1]]]]; a[n_] := Length[b[n, n-1]]; Table[Print[a[n]]; a[n], {n, 1, 24}] (* Jean-François Alcover, Mar 04 2014, after Alois P. Heinz *)
biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
Table[Length[Select[Subsets[Range[n]], MemberQ[#,n]&&biqQ[#]&]],{n,10}] (* Gus Wiseman, Apr 18 2024 *)
  • PARI
    dep(S,k=0)=if(#S<2,return(if(#S,S[1],0)==k)); my(T=S[1..#S-1]);dep(T,abs(k-S[#S]))||dep(T,k+S[#S])
a(n)=my(S=[1..n-1]);sum(i=1,2^(n-1)-1,dep(vecextract(S,i),n)) \\ Charles R Greathouse IV, Nov 25 2013
  • PARI
    a(n)=my(r=0);forsubset(n-1,s,my(t=sum(i=1,#s,s[i])+n);if(t%2==0,my(b=1);for(i=1,#s,b=bitor(b,b<Martin Fuller, Mar 21 2025

Formula

a(n) < 2^(n-2) because there are 2^(n-1) sets of which half have an even sum. - Martin Fuller, Mar 21 2025

Extensions

a(9)-a(24) from Alois P. Heinz, Nov 24 2013
a(25) from Alois P. Heinz, Sep 30 2014
a(26) from Alois P. Heinz, Sep 17 2022
a(27) onwards from Martin Fuller, Mar 21 2025

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 *)

A232534 Number of subsets of {1,...,n} containing n and having at least one set partition into 3 blocks with equal element sum.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 5, 12, 29, 63, 146, 329, 722, 1613, 3505, 7567, 16119, 34194, 71455, 148917, 307432, 631816, 1290905, 2628736, 5330368
Offset: 1

Views

Author

Alois P. Heinz, Nov 25 2013

Keywords

Comments

Subsets with more than one set partition into 3 blocks with equal element sum are counted only once: {1,2,3,4,5,6,7,8}-> 1236/48/57, 138/246/57, 156/237/48.

Examples

			a(5) = 1: {1,2,3,4,5}-> 14/23/5.
a(6) = 2: {1,2,4,5,6}-> 15/24/6, {1,2,3,4,5,6}-> 16/25/34.
a(7) = 5: {2,3,4,5,7}-> 25/34/7, {1,3,4,6,7}-> 16/34/7, {1,2,5,6,7}-> 16/25/7, {1,2,3,5,6,7}-> 17/26/35, {2,3,4,5,6,7}-> 27/36/45.
a(8) = 12: {2,3,5,6,8}, {1,3,5,7,8}, {1,2,6,7,8}, {2,3,4,6,7,8}, {1,2,3,4,5,7,8}, {1,3,4,5,6,8}, {1,2,4,5,6,7,8}, {1,2,3,6,7,8}, {3,4,5,6,7,8}, {1,2,4,5,7,8}, {1,2,3,4,5,6,7,8}, {1,2,3,4,6,8}.

Crossrefs

Cf. A164934, A232466 (2 blocks).
Column k=3 of A248112.

Programs

  • Maple
    b:= proc(n, k, i) option remember; local m; m:= i*(i+1)/2;
      `if`(k>n, b(k, n, i), `if`(i<1, `if`(n=0 and k=0, {0}, {}),
      `if`(k>=0 and n+k>m or k<0 and n-2*k>m, {}, b(n, k, i-1)
       union map(p-> p+x^i, b(n+i, k+i, i-1) union b(n-i, k, i-1)
       union b(n, k-i, i-1)))))
    end:
a:= n-> nops(b(n, n, n-1)):
seq(a(n), n=1..15);
  • Mathematica
    b[n_, k_, i_] := b[n, k, i] = Module[{m = i*(i + 1)/2}, If[k > n, b[k, n, i], If[i < 1, If[n == 0 && k == 0, {0}, {}], If[k >= 0 && n + k > m || k < 0 && n - 2*k > m, {}, b[n, k, i - 1] ~Union~ Map[# + x^i &, b[n + i, k + i, i - 1] ~Union~ b[n - i, k, i - 1] ~Union~ b[n, k - i, i - 1]]]]]];
a[n_] := Length[b[n, n, n - 1]];
Table[a[n], {n, 1, 20}] (* Jean-François Alcover, May 25 2018, translated from Maple *)

Extensions

a(25) from Alois P. Heinz, Mar 26 2016

A248113 Number of subsets of {1,...,n} containing n and having at least one set partition into 4 blocks with equal element sum.

Original entry on oeis.org

1, 2, 6, 14, 37, 88, 218, 515, 1226, 2848, 6566, 14997, 33483, 74406, 162422, 351621, 751171, 1594028
Offset: 7

Views

Author

Alois P. Heinz, Oct 01 2014

Keywords

Examples

			a(7) = 1: {1,2,3,4,5,6,7}-> 16/25/34/7.
a(8) = 2: {1,2,3,5,6,7,8}-> 17/26/35/8, {1,2,3,4,5,6,7,8}-> 18/27/36/45.

Crossrefs

Column k=4 of A248112.

A248114 Number of subsets of {1,...,n} containing n and having at least one set partition into 5 blocks with equal element sum.

Original entry on oeis.org

1, 2, 7, 16, 49, 118, 313, 764, 1937, 4752, 11541, 27610, 65186, 151426, 346412, 784064
Offset: 9

Views

Author

Alois P. Heinz, Oct 02 2014

Keywords

Crossrefs

Column k=5 of A248112.

A248115 Number of subsets of {1,...,n} containing n and having at least one set partition into 6 blocks with equal element sum.

Original entry on oeis.org

1, 2, 8, 19, 62, 149, 435, 1097, 2937, 7442, 19179, 47777, 118337, 288220
Offset: 11

Views

Author

Alois P. Heinz, Oct 02 2014

Keywords

Crossrefs

Column k=6 of A248112.

A248116 Number of subsets of {1,...,n} containing n and having at least one set partition into 7 blocks with equal element sum.

Original entry on oeis.org

1, 2, 9, 21, 76, 191, 581, 1498, 4289, 11178, 30228, 78312, 203322
Offset: 13

Views

Author

Alois P. Heinz, Oct 02 2014

Keywords

Crossrefs

Column k=7 of A248112.

A248117 Number of subsets of {1,...,n} containing n and having at least one set partition into 8 blocks with equal element sum.

Original entry on oeis.org

1, 2, 10, 24, 92, 230, 767, 2006, 6037, 16133, 45868, 122539
Offset: 15

Views

Author

Alois P. Heinz, Oct 02 2014

Keywords

Crossrefs

Column k=8 of A248112.

A248118 Number of subsets of {1,...,n} containing n and having at least one set partition into 9 blocks with equal element sum.

Original entry on oeis.org

1, 2, 11, 26, 111, 280, 980, 2599, 8292, 22644, 67300
Offset: 17

Views

Author

Alois P. Heinz, Oct 02 2014

Keywords

Crossrefs

Column k=9 of A248112.

A248119 Number of subsets of {1,...,n} containing n and having at least one set partition into 10 blocks with equal element sum.

Original entry on oeis.org

1, 2, 12, 29, 130, 329, 1239, 3329, 11128, 30766, 95946
Offset: 19

Views

Author

Alois P. Heinz, Oct 02 2014

Keywords

Crossrefs

Column k=10 of A248112.
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.