A357181 Last run-length of the n-th composition in standard order.

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

Offset: 0

Author

Gus Wiseman, Sep 24 2022

Keywords

• nonn

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.

Examples

			Composition 87 in standard order is (2,2,1,1,1), so a(87) = 3.
		

Links

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

Crossrefs

See link for sequences related to standard compositions.

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

For Heinz numbers of partitions we have A071178, first A067029.

This is the last part of row n of A333769.

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

The first instead of last run-length is A357180.

A051903 gives maximal part of 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 composition, minimal A333768.

A353847 ranks run-sums of standard compositions.

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

Programs

Mathematica

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