cp's OEIS Frontend

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

%I A357180 #9 Sep 26 2022 08:20:37
%S A357180 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,
%T A357180 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,
%U A357180 1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,2,3,2
%N A357180 First run-length of the n-th composition in standard order.
%C A357180 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 A357180 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 A357180 Composition 87 in standard order is (2,2,1,1,1), so a(87) = 2.
%t A357180 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A357180 Table[If[n==0,0,First[Length/@Split[stc[n]]]],{n,0,100}]
%Y A357180 See link for sequences related to standard compositions.
%Y A357180 For parts instead of run-lengths we have A065120, last A001511.
%Y A357180 The version for Heinz numbers of partitions is A067029, last A071178.
%Y A357180 This is the first part of row n of A333769.
%Y A357180 For minimal instead of first we have A357138, maximal A357137.
%Y A357180 The last instead of first run-length is A357181.
%Y A357180 A051903 gives maximal part in prime signature.
%Y A357180 A061395 gives maximal prime index.
%Y A357180 A124767 counts runs in standard compositions.
%Y A357180 A286470 gives maximal difference of prime indices.
%Y A357180 A333766 gives maximal part of standard compositions, minimal A333768.
%Y A357180 A353847 ranks run-sums of standard compositions.
%Y A357180 Cf. A000120, A003754, A029931, A070939, A329395, A356841, A356844, A357134, A357135, A357136.
%K A357180 nonn
%O A357180 0,4
%A A357180 _Gus Wiseman_, Sep 24 2022