A353744 Numbers k such that the k-th composition in standard order has all equal run-lengths.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 17, 18, 20, 22, 24, 25, 31, 32, 33, 34, 36, 37, 38, 40, 41, 42, 43, 44, 45, 48, 49, 50, 52, 54, 58, 63, 64, 65, 66, 68, 69, 70, 72, 76, 77, 80, 81, 82, 88, 89, 96, 97, 98, 101, 102, 104, 105, 108, 109, 127, 128

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..66.

Example

Composition 2362 in standard order is (3,3,1,1,2,2), with run-lengths (2,2,2), so 2362 is in the sequence.

Mathematica

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

Crossrefs

Standard compositions are listed by A066099.

The version for partitions is A072774, counted by A047966.

These compositions are counted by A329738.

For distinct instead of equal run-lengths we have A351596.

For run-sums instead of lengths we have A353848, counted by A353851.

For distinct run-sums we have A353852, counted by A353850.

A003242 counts anti-run compositions, ranked by A333489.

A005811 counts runs in binary expansion.

A300273 ranks collapsible partitions, counted by A275870.

A353838 ranks partitions with all distinct run-sums, counted by A353837.

A353847 represents the composition run-sum transformation.

A353853-A353859 pertain to composition run-sum trajectory.

A353860 counts collapsible compositions.

A353833 ranks partitions with all equal run-sums, counted by A304442.

Cf. A029837, A071625, A124767, A175413, A238279, A333381, A333755, A353834, A353849.

Sequence in context: A301373 A354581 A193096 * A309129 A307345 A033110

Adjacent sequences: A353741 A353742 A353743 * A353745 A353746 A353747

Keyword

nonn

Author

Gus Wiseman, Jun 11 2022

Status

approved