A353932 Irregular triangle read by rows where row k lists the run-sums of the k-th composition in standard order.

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

Offset

1,2

Comments

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.

Links

Table of n, a(n) for n=1..84.

Example

Triangle begins:
  1
  2
  2
  3
  2 1
  1 2
  3
  4
  3 1
  4
  2 2
  1 3
  1 2 1
For example, composition 350 in standard order is (2,2,1,1,1,2), so row 350 is (4,3,2).

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Total/@Split[stc[n]], {n, 0, 30}]

Crossrefs

Row-sums are A029837.

Standard compositions are listed by A066099.

Row-lengths are A124767.

These compositions are ranked by A353847.

Row k has A353849(k) distinct parts.

The version for partitions is A354584, ranked by A353832.

A005811 counts runs in binary expansion.

A300273 ranks collapsible partitions, counted by A275870.

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

A353851 counts compositions with all equal run-sums, ranked by A353848.

A353840-A353846 pertain to partition run-sum trajectory.

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

A353853-A353859 pertain to composition run-sum trajectory.

A353860 counts collapsible compositions.

Cf. A003242, A175413, A181819, A238279, A274174, A333381, A333489, A333755, A353835, A353839, A353863, A353864.

Sequence in context: A259632 A304041 A238509 * A368798 A253141 A100890

Adjacent sequences: A353929 A353930 A353931 * A353933 A353934 A353935

Keyword

nonn,tabf

Author

Gus Wiseman, Jun 10 2022

Status

approved