A232466 Number of dependent sets with largest element n.

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

Offset

1,4

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

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

Table of n, a(n) for n=1..26.

Example

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)

Maple

b:= proc(n, i) option remember; `if`(i<1, `if`(n=0, {0}, {}),
      `if`(i*(i+1)/2<n, {}, b(n, i-1) union map(p-> 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 *)

Prog

(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

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.

Sequence in context: A307768 A370583 A297183 * A003407 A151523 A317708

Adjacent sequences: A232463 A232464 A232465 * A232467 A232468 A232469

Keyword

nonn,more

Author

David S. Newman, Nov 24 2013

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

Status

approved