A354582 Number of distinct contiguous constant subsequences (or partial runs) in the k-th composition in standard order.

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

Offset

0,4

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=0..86.

Example

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), so a(981) = 6.
As a triangle:
  1
  1 2
  1 2 2 3
  1 2 2 3 2 2 3 4
  1 2 2 3 2 3 2 4 2 2 3 3 3 3 4 5
  1 2 2 3 2 3 3 4 2 3 3 4 3 2 3 5 2 2 3 3 3 3 2 4 3 3 4 3 4 4 5 6

Mathematica

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

Crossrefs

The version for partitions is A001222, full A001221.

If we allow any constant subsequence we get A063787.

If we allow any contiguous subsequence we get A124771.

Positions of first appearances are A126646.

The version for binary indices is A330036, full A005811.

If we allow any subsequence we get A334299.

The full version is A351014, firsts A351015.

The version for run-sums of partitions is A353861, full A353835.

Counting distinct sums of partial runs gives A354907, full A353849.

A066099 lists all compositions in standard order.

A124767 counts runs in standard compositions.

A238279 and A333755 count compositions by number of runs.

A353852 ranks compositions with all distinct run-sums, counted by A353850.

A353853-A353859 pertain to composition run-sum trajectory.

A353932 lists run-sums of standard compositions, rows ranked by A353847.

Cf. A000120, A003242, A029837, A175413, A274174, A333381, A333489, A353832, A353860, A353864.

Sequence in context: A261923 A124736 A144016 * A359312 A179868 A104232

Adjacent sequences: A354579 A354580 A354581 * A354583 A354584 A354585

Keyword

nonn,tabf

Author

Gus Wiseman, Jun 13 2022

Status

approved