0,3

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).

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.

Mathematics Stack Exchange,What is a sequence run? (answered 2011-12-01)

As a triangle:

0

1

2 2

4 5 6 4

8 9 8 10 12 13 10 8

16 17 18 18 20 17 22 20 24 25 24 26 20 21 18 16

These are the standard composition numbers of the following compositions (transposed):

() (1) (2) (3) (4) (5)

(2) (2,1) (3,1) (4,1)

(1,2) (4) (3,2)

(3) (2,2) (3,2)

(1,3) (2,3)

(1,2,1) (4,1)

(2,2) (2,1,2)

(4) (2,3)

(1,4)

(1,3,1)

(1,4)

(1,2,2)

(2,3)

(2,2,1)

(3,2)

(5)

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;

stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;

Table[stcinv[Total/@Split[stc[n]]], {n, 0, 100}]

Standard compositions are listed byA066099.

The version for partitions isA353832.

The run-sums themselves are listed byA353932,withA353849distinct terms.

A005811counts runs in binary expansion.

A300273ranks collapsible partitions, counted byA275870.

A353838ranks partitions with all distinct run-sums, counted byA353837.

A353851counts compositions with all equal run-sums, ranked byA353848.

A353840-A353846pertain to partition run-sum trajectory.

A353852ranks compositions with all distinct run-sums, counted byA353850.

A353853-A353859pertain to composition run-sum trajectory.

A353860counts collapsible compositions.

A353863counts run-sum-complete partitions.

Cf.A003242.A175413,A181819,A238279,A274174,A333381,A333489,A333755,A353835,A353839,A353864.

Sequence in context:A133937A181523A353855*A181537A130498A263433

Adjacent sequences:A353844A353845A353846*A353848A353849A353850

Gus Wiseman,May 30 2022

approved