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

Author

Gus Wiseman, Jun 11 2022

Keywords

• nonn

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.

Examples

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

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.

Programs

Mathematica

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