A357137 Maximal run-length of the n-th composition in standard order; a(0) = 0.

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

Offset

0,4

Comments

A composition of n is a finite sequence of positive integers summing to n. 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.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

Example

Composition 92 in standard order is (2,1,1,3), so a(92) = 2.

Mathematica

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[If[n==0, 0, Max[Length/@Split[stc[n]]]], {n, 0, 100}]

Crossrefs

See link for more sequences related to standard compositions.

The version for Heinz numbers of partitions is A051903, for parts A061395.

For parts instead of run-lengths we have A333766, minimal A333768.

The opposite (minimal) version is A357138.

For first instead of maximal we have A357180, last A357181.

Cf. A000120, A001511, A003754, A029931, A051904, A055396, A056239, A070939, A286470, A356844, A357136.

Sequence in context: A135222 A285706 A357181 * A333381 A124094 A101950

Adjacent sequences: A357134 A357135 A357136 * A357138 A357139 A357140

Keyword

nonn

Author

Gus Wiseman, Sep 18 2022

Status

approved