login
Number of dependent sets with largest element n.
24

%I #47 Apr 20 2024 16:36:25

%S 0,0,1,2,4,10,20,44,93,198,414,864,1788,3687,7541,15382,31200,63191,

%T 127482,256857,516404,1037104,2080357,4170283,8354078,16728270

%N Number of dependent sets with largest element n.

%C 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.

%C Dependent sets are also called biquanimous sets. Biquanimous partitions are counted by A002219 and ranked by A357976. - _Gus Wiseman_, Apr 18 2024

%D 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.

%e From _Gus Wiseman_, Apr 18 2024: (Start)

%e The a(1) = 0 through a(6) = 10 sets:

%e . . {1,2,3} {1,3,4} {1,4,5} {1,5,6}

%e {1,2,3,4} {2,3,5} {2,4,6}

%e {1,2,4,5} {1,2,3,6}

%e {2,3,4,5} {1,2,5,6}

%e {1,3,4,6}

%e {2,3,5,6}

%e {3,4,5,6}

%e {1,2,3,4,6}

%e {1,2,4,5,6}

%e {2,3,4,5,6}

%e (End)

%p b:= proc(n, i) option remember; `if`(i<1, `if`(n=0, {0}, {}),

%p `if`(i*(i+1)/2<n, {}, b(n, i-1) union map(p-> p+x^i,

%p b(n+i, i-1) union b(abs(n-i), i-1))))

%p end:

%p a:= n-> nops(b(n, n-1)):

%p seq(a(n), n=1..15); # _Alois P. Heinz_, Nov 24 2013

%t 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_ *)

%t biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];

%t Table[Length[Select[Subsets[Range[n]], MemberQ[#,n]&&biqQ[#]&]],{n,10}] (* _Gus Wiseman_, Apr 18 2024 *)

%o (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])

%o 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

%Y Cf. A161943, A232534.

%Y Column k=2 of A248112.

%Y First differences of A371791.

%Y The complement is counted by A371793, differences of A371792.

%Y This is the "bi-" case of A371797, differences of A371796.

%Y A002219 (aerated) counts biquanimous partitions, ranks A357976.

%Y A006827 and A371795 count non-biquanimous partitions, ranks A371731.

%Y A237258 (aerated) counts biquanimous strict partitions, ranks A357854.

%Y A321142 and A371794 count non-biquanimous strict partitions.

%Y Cf. A035470, A064914, A321451, A321452, A366320, A367094, A371783, A371789.

%K nonn,more

%O 1,4

%A _David S. Newman_, Nov 24 2013

%E a(9)-a(24) from _Alois P. Heinz_, Nov 24 2013

%E a(25) from _Alois P. Heinz_, Sep 30 2014

%E a(26) from _Alois P. Heinz_, Sep 17 2022