A372120 Numbers k such that the k-th composition in standard order is biquanimous.

0, 3, 10, 11, 13, 14, 15, 36, 37, 38, 39, 41, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 136, 137, 138, 139, 140, 141, 142, 143, 145, 147, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 162, 163, 165, 166, 167, 168, 169

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..58.

Example

The terms and corresponding compositions begin:
   0: ()
   3: (1,1)
  10: (2,2)
  11: (2,1,1)
  13: (1,2,1)
  14: (1,1,2)
  15: (1,1,1,1)
  36: (3,3)
  37: (3,2,1)
  38: (3,1,2)
  39: (3,1,1,1)
  41: (2,3,1)
  43: (2,2,1,1)
  44: (2,1,3)
  45: (2,1,2,1)
  46: (2,1,1,2)
  47: (2,1,1,1,1)
  50: (1,3,2)
  51: (1,3,1,1)
  52: (1,2,3)
  53: (1,2,2,1)
  54: (1,2,1,2)

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

These compositions are counted by A064914.

The unordered version (integer partitions) is A357976, counted by A002219.

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

The complement is A372119, counted by A371956.

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: A024575 A114134 A167519 * A357708 A169939 A357627

Adjacent sequences: A372117 A372118 A372119 * A372121 A372122 A372123

Keyword

nonn

Author

Gus Wiseman, Apr 20 2024

Status

approved