A375123 Weakly increasing run-leader transformation for standard compositions.

0, 1, 2, 1, 4, 5, 1, 1, 8, 9, 2, 5, 1, 3, 1, 1, 16, 17, 18, 9, 2, 5, 5, 5, 1, 3, 1, 3, 1, 3, 1, 1, 32, 33, 34, 17, 4, 37, 9, 9, 2, 5, 2, 5, 5, 11, 5, 5, 1, 3, 6, 3, 1, 3, 3, 3, 1, 3, 1, 3, 1, 3, 1, 1, 64, 65, 66, 33, 68, 69, 17, 17, 4, 9, 18, 37, 9, 19, 9, 9

Offset: 0

Author

Gus Wiseman, Aug 02 2024

Keywords

• nonn

Comments

The a(n)-th composition in standard order lists the leaders of weakly increasing runs of the n-th composition in standard order.

The leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.

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

			The 813th composition in standard order is (1,3,2,1,2,1), with weakly increasing runs ((1,3),(2),(1,2),(1)), with leaders (1,2,1,1). This is the 27th composition in standard order, so a(813) = 27.
		

Links

• Gus Wiseman, Statistics, classes, and transformations of standard compositions.
• Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Crossrefs

Positions of elements of A233564 are A374768, counted by A374632.

Positions of elements of A272919 are A374633, counted by A374631.

Ranks of rows of A374629.

The opposite version is A375124.

The strict version is A375125.

The strict opposite version is A375126.

A011782 counts compositions.

A238130, A238279, A333755 count compositions by number of runs.

All of the following pertain to compositions in standard order:

- Length is A000120.

- Sum is A029837(n+1).

- Leader is A065120.

- Parts are listed by A066099.

- Number of adjacent equal pairs is A124762, unequal A333382.

- Run-length transform is A333627, sum A070939.

- Run-sum transformation is A353847.

- Run-compression transform is A373948, sum A373953, excess A373954.

- Ranks of contiguous compositions are A374249, counted by A274174.

Six types of runs:

- Count: A124766, A124765, A124768, A124769, A333381, A124767.

- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.

- Rank: A375123, A375124, A375125, A375126, A375127, A373948.

Cf. A189076, A238343, A333213, A373949, A374634, A374635, A374637, A374743, A374744, A375128.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[First/@Split[stc[n],LessEqual]],{n,0,100}]

Formula

A000120(a(n)) = A124766(n).

A070939(a(n)) = A374630(n) for n > 0.

A065120(a(n)) = A065120(n).