A357137 - OEIS

%I #10 Sep 24 2022 14:59:28

%S 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,

%T 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,

%U 1,1,1,3,1,2,2,2,1,1,2,4,1,1,1,2,2,3,2

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

%C 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.

%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>

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

%t stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t Table[If[n==0,0,Max[Length/@Split[stc[n]]]],{n,0,100}]

%Y See link for more sequences related to standard compositions.

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

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

%Y The opposite (minimal) version is A357138.

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

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

%K nonn

%O 0,4

%A _Gus Wiseman_, Sep 18 2022