A371792 Number of non-biquanimous subsets of {1..n}. Sets with no subset having the same sum as the complement.

0, 1, 3, 6, 12, 24, 46, 90, 174, 337, 651, 1261, 2445, 4753, 9258, 18101

Offset

0,3

Comments

A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 and ranked by A357976.

Links

Table of n, a(n) for n=0..15.

Example

The subsets of S = {1,4,6,7} have distinct sums {0,1,4,5,6,7,8,10,11,12,13,14,17,18}. Since 9 is missing, S is counted under a(7).
The a(0) = 0 through a(4) = 12 subsets:
  .  {1}  {1}    {1}    {1}
          {2}    {2}    {2}
          {1,2}  {3}    {3}
                 {1,2}  {4}
                 {1,3}  {1,2}
                 {2,3}  {1,3}
                        {1,4}
                        {2,3}
                        {2,4}
                        {3,4}
                        {1,2,4}
                        {2,3,4}

Mathematica

biqQ[y_]:=MemberQ[Total/@Subsets[y], Total[y]/2];
Table[Length[Select[Subsets[Range[n]], Not@*biqQ]], {n, 0, 10}]

Crossrefs

This is the "bi-" version of A371789, differences A371790.

The complement is counted by A371791, differences A232466.

First differences are A371793.

The complement is the "bi-" version of A371796, differences A371797.

A002219 aerated counts biquanimous partitions, ranks A357976.

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

A108917 counts knapsack partitions, ranks A299702, strict A275972.

A237258 aerated counts biquanimous strict partitions, ranks A357854.

A321142 and A371794 count non-biquanimous strict partitions.

A321451 counts non-quanimous partitions, ranks A321453.

A321452 counts quanimous partitions, ranks A321454.

A366754 counts non-knapsack partitions, ranks A299729, strict A316402.

A371737 counts quanimous strict partitions, complement A371736.

A371781 lists numbers with biquanimous prime signature, complement A371782.

A371783 counts k-quanimous partitions.

Cf. A035470, A064914, A318434, A321455, A365543, A365661, A365663, A366320, A365381, A365925, A367094, A371788.

Sequence in context: A196787 A367217 A200662 * A264507 A137711 A068032

Adjacent sequences: A371789 A371790 A371791 * A371793 A371794 A371795

Keyword

nonn,more

Author

Gus Wiseman, Apr 07 2024

Status

approved