A354907 Number of distinct sums of contiguous constant subsequences (partial runs) of the n-th composition in standard order.

0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 4, 1, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 5, 1, 2, 2, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 2, 2, 4, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 2, 3, 3, 4, 6, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 3, 2, 3, 3, 4, 2, 3, 2, 3, 3, 4, 3

Offset: 0

Author

Gus Wiseman, Jun 23 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.

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

Examples

			Composition number 981 in standard order is (1,1,1,2,2,2,1), with partial runs (1), (2), (1,1), (2,2), (1,1,1), (2,2,2), with distinct sums {1,2,3,4,6}, so a(981) = 5.
		

Crossrefs

Positions of 1's are A000051.

Positions of first appearances are A000079.

The standard compositions used here are A066099, run-sums A353847/A353932.

If we allow any subsequence we get A334968.

The case of full runs is A353849, firsts A246534.

A version for nonempty partitions is A353861, full A353835.

Counting all distinct runs (instead of their distinct sums) gives A354582.

A124767 counts runs in standard compositions.

A238279 and A333755 count compositions by number of runs.

A330036 counts distinct partial runs of prime indices, full A005811.

A351014 counts distinct runs of standard compositions, firsts A351015.

A353853-A353859 pertain to composition run-sum trajectory.

A353860 counts collapsible compositions.

A354584 lists run-sums of prime indices, rows ranked by A353832.

Cf. A029837, A124771, A274174, A333381, A334299, A353864.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
pre[y_]:=NestWhileList[Most,y,Length[#]>1&];
Table[Length[Union[Total/@Join@@pre/@Split[stc[n]]]],{n,0,100}]