A357180 First run-length of the n-th composition in standard order.

0, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 3, 5, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 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 87 in standard order is (2,2,1,1,1), so a(87) = 2.

Mathematica

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

Crossrefs

See link for sequences related to standard compositions.

For parts instead of run-lengths we have A065120, last A001511.

The version for Heinz numbers of partitions is A067029, last A071178.

This is the first part of row n of A333769.

For minimal instead of first we have A357138, maximal A357137.

The last instead of first run-length is A357181.

A051903 gives maximal part in prime signature.

A061395 gives maximal prime index.

A124767 counts runs in standard compositions.

A286470 gives maximal difference of prime indices.

A333766 gives maximal part of standard compositions, minimal A333768.

A353847 ranks run-sums of standard compositions.

Cf. A000120, A003754, A029931, A070939, A329395, A356841, A356844, A357134, A357135, A357136.

Sequence in context: A324247 A138904 A357138 * A196660 A342323 A374433

Adjacent sequences: A357177 A357178 A357179 * A357181 A357182 A357183

Keyword

nonn

Author

Gus Wiseman, Sep 24 2022

Status

approved