A389922 Standard composition numbers of integer compositions with minimum = length.

1, 10, 18, 20, 34, 40, 66, 80, 130, 160, 258, 292, 320, 514, 548, 580, 584, 640, 1026, 1060, 1092, 1096, 1156, 1160, 1168, 1280, 2050, 2084, 2116, 2120, 2180, 2192, 2308, 2312, 2320, 2336, 2560, 4098, 4132, 4164, 4168, 4228, 4240, 4356, 4384, 4612, 4616, 4624

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..48.

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

Formula

A000120(a(n)) = A333768(a(n)).

Example

The terms together with corresponding standard compositions begin:
     1: (1)
    10: (2,2)
    18: (3,2)
    20: (2,3)
    34: (4,2)
    40: (2,4)
    66: (5,2)
    80: (2,5)
   130: (6,2)
   160: (2,6)
   258: (7,2)
   292: (3,3,3)
   320: (2,7)
   514: (8,2)
   548: (4,3,3)
   580: (3,4,3)
   584: (3,3,4)
   640: (2,8)
  1026: (9,2)
  1060: (5,3,3)
  1092: (4,4,3)
  1096: (4,3,4)

Mathematica

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 1000], Min[stc[#]]==Length[stc[#]]&]

Crossrefs

Length of standard composition is A000120.

These compositions are counted by A098133, maximum version A098124.

For Heinz numbers of partitions the maximum version is A106529, counted by A047993.

For Heinz numbers of partitions we have A324522, counted by A006141, strict A096401.

Minimum of standard composition is A333768, maximum A333766.

For maximum instead of maximum we have A389532.

For last part instead of minimum we have A389923 = 2*A355489.

For first part instead of minimum we have A389924.

A005811 counts runs in binary expansion.

A011782 counts compositions.

A066099 lists standard compositions, sum A029837.

A124767 counts maximal runs in standard composition.

A171682 counts compositions with min = first, differences of A097939.

A212804 counts compositions with length = first, ranks A022340.

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

Cf. A001511, A029931, A048896, A070939, A133494, A333381, A351014, A358133, A389508.

Sequence in context: A043750 A043758 A043767 * A290002 A072992 A055481

Adjacent sequences: A389919 A389920 A389921 * A389923 A389924 A389925

Keyword

nonn

Author

Gus Wiseman, Oct 23 2025

Status

approved