A389923 Standard composition numbers of integer compositions with last part = length.

1, 6, 10, 18, 28, 34, 44, 52, 66, 76, 84, 100, 120, 130, 140, 148, 164, 184, 196, 216, 232, 258, 268, 276, 292, 312, 324, 344, 360, 388, 408, 424, 456, 496, 514, 524, 532, 548, 568, 580, 600, 616, 644, 664, 680, 712, 752, 772, 792, 808, 840, 880, 904, 944, 976

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..55.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

Formula

a(n>1) = 2*A355489(n-1).

A000120(a(n)) = A001511(a(n)).

Example

The terms together with corresponding standard compositions begin:
    1: (1)
    6: (1,2)
   10: (2,2)
   18: (3,2)
   28: (1,1,3)
   34: (4,2)
   44: (2,1,3)
   52: (1,2,3)
   66: (5,2)
   76: (3,1,3)
   84: (2,2,3)
  100: (1,3,3)
  120: (1,1,1,4)
  130: (6,2)
  140: (4,1,3)
  148: (3,2,3)
  164: (2,3,3)
  184: (2,1,1,4)
  196: (1,4,3)
  216: (1,2,1,4)
  232: (1,1,2,4)

Mathematica

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

Crossrefs

Length of standard composition is A000120.

Last part of standard composition is A001511.

First part of standard composition is A065120.

For Heinz numbers of partitions we get A106529, counted by A047993.

These compositions are counted by A212804.

This is 1 followed by 2*A355489.

For maximum instead of last we get A389532, counted by A098124.

For minimum instead of last we get A389922, counted by A098133.

For first instead of last we get A389924, counted by A212804.

A005811 counts runs in binary expansion.

A011782 counts compositions.

A066099 lists standard compositions, sum A029837.

A124767 counts maximal runs in standard compositions.

A171682 counts compositions with minimum = first part, differences of A097939.

A333768 gives minimum of standard composition, maximum A333766.

A389530 ranks compositions whose runs all satisfy length = first, counted by A301503.

Cf. A006141, A022340, A029931, A070939, A096401, A133494, A324522, A333381.

Sequence in context: A315366 A248056 A331861 * A032641 A293555 A077626

Adjacent sequences: A389920 A389921 A389922 * A389924 A389925 A389926

Keyword

nonn

Author

Gus Wiseman, Oct 24 2025

Status

approved