A375124 Weakly decreasing run-leader transformation for standard compositions.

0, 1, 2, 1, 4, 2, 6, 1, 8, 4, 2, 2, 12, 6, 6, 1, 16, 8, 4, 4, 20, 2, 10, 2, 24, 12, 6, 6, 12, 6, 6, 1, 32, 16, 8, 8, 4, 4, 18, 4, 40, 20, 2, 2, 20, 10, 10, 2, 48, 24, 12, 12, 52, 6, 26, 6, 24, 12, 6, 6, 12, 6, 6, 1, 64, 32, 16, 16, 8, 8, 34, 8, 72, 4, 4, 4, 36

Offset: 0

Author

Gus Wiseman, Aug 02 2024

Keywords

• nonn

Comments

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

The leaders of weakly decreasing runs in a sequence are obtained by splitting it into maximal weakly decreasing 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 decreasing runs ((1),(3,2,1),(2,1)), with leaders (1,3,2). This is the 50th composition in standard order, so a(813) = 50.
		

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 A374701, counted by A374743.

Positions of elements of A272919 are A374744, counted by A374742.

Ranks of rows of A374740.

The opposite version is A375123.

The strict version is A375126.

The strict opposite version is A375125.

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) = A070939(n).

- Parts are listed by A066099.

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

- Run-length transform is A333627, sum A070939.

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

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

- Run-sum transformation is A353847.

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. A065120, A106356, A189076, A238343, A333213, A373949, A374635, A374687, A374748, A374758.

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],GreaterEqual]],{n,0,100}]

Formula

A000120(a(n)) = A124765(n).

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

A070939(a(n)) = A374741(n).