A372119 Numbers k such that the k-th composition in standard order is not biquanimous.

1, 2, 4, 5, 6, 7, 8, 9, 12, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 40, 42, 48, 49, 56, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96

Offset

1,2

Comments

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

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=1..67.

Example

The terms and corresponding compositions begin:
   1: (1)
   2: (2)
   4: (3)
   5: (2,1)
   6: (1,2)
   7: (1,1,1)
   8: (4)
   9: (3,1)
  12: (1,3)
  16: (5)
  17: (4,1)
  18: (3,2)
  19: (3,1,1)
  20: (2,3)
  21: (2,2,1)
  22: (2,1,2)
  23: (2,1,1,1)

Mathematica

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], !MemberQ[Total/@Subsets[stc[#]], Total[stc[#]]/2]&]

Crossrefs

The unordered complement is A357976, counted by A002219.

The unordered version is A371731, counted by A371795, even case A006827.

These compositions are counted by A371956.

The complement is A372120, counted by A064914.

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

A321142 and A371794 count non-biquanimous strict partitions.

A371791 counts biquanimous sets, differences A232466.

A371792 counts non-biquanimous sets, differences A371793.

Cf. A027187, A035470, A357879, A367094, A371781, A371782, A371783.

Sequence in context: A088384 A026446 A167493 * A328107 A368829 A039121

Adjacent sequences: A372116 A372117 A372118 * A372120 A372121 A372122

Keyword

nonn

Author

Gus Wiseman, Apr 20 2024

Status

approved