A356841 Numbers k such that the k-th composition in standard order covers an interval of positive integers (gapless).

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 18, 20, 21, 22, 23, 26, 27, 29, 30, 31, 32, 36, 37, 38, 41, 42, 43, 44, 45, 46, 47, 50, 52, 53, 54, 55, 58, 59, 61, 62, 63, 64, 68, 72, 74, 75, 77, 78, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 101

Offset

1,3

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..67.

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

Example

The terms and their corresponding standard compositions begin:
   0: ()
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   5: (2,1)
   6: (1,2)
   7: (1,1,1)
   8: (4)
  10: (2,2)
  11: (2,1,1)
  13: (1,2,1)
  14: (1,1,2)
  15: (1,1,1,1)
  16: (5)
  18: (3,2)
  20: (2,3)
  21: (2,2,1)

Mathematica

nogapQ[m_]:=m=={}||Union[m]==Range[Min[m], Max[m]];
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], nogapQ[stc[#]]&]

Crossrefs

See link for sequences related to standard compositions.

An unordered version is A073491, complement A073492.

These compositions are counted by A107428.

The complement is A356842.

The non-initial case is A356843, unordered A356845.

A356230 ranks gapless factorization lengths, firsts A356603.

A356233 counts factorizations into gapless numbers.

A356844 ranks compositions with at least one 1.

Cf. A053251, A055932, A073493, A132747, A137921, A286470, A356224/A356225.

Sequence in context: A039218 A076487 A317717 * A181709 A320058 A320057

Adjacent sequences: A356838 A356839 A356840 * A356842 A356843 A356844

Keyword

nonn

Author

Gus Wiseman, Aug 31 2022

Status

approved