A355321 Numbers k such that the k-th composition in standard order has the same number of even parts as odd.

0, 5, 6, 17, 18, 20, 24, 43, 45, 46, 53, 54, 58, 65, 66, 68, 72, 80, 96, 139, 141, 142, 149, 150, 154, 163, 165, 166, 169, 172, 177, 178, 180, 184, 197, 198, 202, 209, 210, 212, 216, 226, 232, 257, 258, 260, 264, 272, 288, 320, 343, 347, 349, 350, 363, 365

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.

Links

Table of n, a(n) for n=1..56.

Example

The terms together with their corresponding compositions begin:
   0: ()
   5: (2,1)
   6: (1,2)
  17: (4,1)
  18: (3,2)
  20: (2,3)
  24: (1,4)
  43: (2,2,1,1)
  45: (2,1,2,1)
  46: (2,1,1,2)
  53: (1,2,2,1)
  54: (1,2,1,2)
  58: (1,1,2,2)
  65: (6,1)
  66: (5,2)
  68: (4,3)
  72: (3,4)
  80: (2,5)
  96: (1,6)

Mathematica

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], Count[stc[#], _?EvenQ]==Count[stc[#], _?OddQ]&]

Crossrefs

A subset of A001969 (evil numbers), complement A000069.

These compositions are counted by A098123, without multiplicity A242821.

The version for partitions is A325698, counted by A045931.

For partitions without multiplicity we have A325700, counted by A241638.

A047993 counts balanced partitions, ranked by A106529.

A108950/A108949 count partitions with more odd/even parts.

A130780/A171966 count partitions with more or as many odd/even parts.

Cf. A000712, A001405, A026424, A028260, A239241 (conjugate A352129), A240009, A242498, A277579 (ranked by A349157).

Sequence in context: A358091 A257338 A059013 * A327102 A191144 A327097

Adjacent sequences: A355318 A355319 A355320 * A355322 A355323 A355324

Keyword

nonn

Author

Gus Wiseman, Jun 28 2022

Status

approved