A357138 Minimal run-length of the n-th composition in standard order; a(0) = 0.

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

Offset: 0

Author

Gus Wiseman, Sep 18 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 92 in standard order is (2,1,1,3), so a(92) = 1.
		

Links

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

Crossrefs

See link for more sequences related to standard compositions.

The version for Heinz numbers of partitions is A051904, for parts A055396.

For parts instead of run-length we have A333768, maximal A333766.

The opposite (maximal) version is A357137.

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

Cf. A000120, A001511, A003754, A029931, A051903, A056239, A061395, A070939, A297173, A356844, A357136.

Programs

Mathematica

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