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A333627 The a(n)-th composition in standard order is the sequence of run-lengths of the n-th composition in standard order.

%I A333627 #10 Mar 31 2020 10:54:13
%S A333627 0,1,1,2,1,3,3,4,1,3,2,6,3,7,5,8,1,3,3,6,3,5,7,12,3,7,6,14,5,11,9,16,
%T A333627 1,3,3,6,2,7,7,12,3,7,4,10,7,15,13,24,3,7,7,14,7,13,15,28,5,11,10,22,
%U A333627 9,19,17,32,1,3,3,6,3,7,7,12,3,5,6,14,7,15,13
%N A333627 The a(n)-th composition in standard order is the sequence of run-lengths of the n-th composition in standard order.
%C A333627 A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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 A333627 Claude Lenormand, <a href="/A318921/a318921.pdf">Deux transformations sur les mots</a>, Preprint, 5 pages, Nov 17 2003.
%F A333627 A000120(n) = A070939(a(n)).
%F A333627 A000120(a(n)) = A124767(n).
%e A333627 The standard compositions and their run-lengths:
%e A333627        0 ~ () -> () ~ 0
%e A333627       1 ~ (1) -> (1) ~ 1
%e A333627       2 ~ (2) -> (1) ~ 1
%e A333627      3 ~ (11) -> (2) ~ 2
%e A333627       4 ~ (3) -> (1) ~ 1
%e A333627      5 ~ (21) -> (11) ~ 3
%e A333627      6 ~ (12) -> (11) ~ 3
%e A333627     7 ~ (111) -> (3) ~ 4
%e A333627       8 ~ (4) -> (1) ~ 1
%e A333627      9 ~ (31) -> (11) ~ 3
%e A333627     10 ~ (22) -> (2) ~ 2
%e A333627    11 ~ (211) -> (12) ~ 6
%e A333627     12 ~ (13) -> (11) ~ 3
%e A333627    13 ~ (121) -> (111) ~ 7
%e A333627    14 ~ (112) -> (21) ~ 5
%e A333627   15 ~ (1111) -> (4) ~ 8
%e A333627      16 ~ (5) -> (1) ~ 1
%e A333627     17 ~ (41) -> (11) ~ 3
%e A333627     18 ~ (32) -> (11) ~ 3
%e A333627    19 ~ (311) -> (12) ~ 6
%t A333627 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A333627 Table[Total[2^(Accumulate[Reverse[Length/@Split[stc[n]]]])]/2,{n,0,30}]
%Y A333627 Positions of first appearances are A333630.
%Y A333627 All of the following pertain to compositions in standard order (A066099):
%Y A333627 - The length is A000120.
%Y A333627 - The partial sums from the right are A048793.
%Y A333627 - The sum is A070939.
%Y A333627 - Adjacent equal pairs are counted by A124762.
%Y A333627 - Equal runs are counted by A124767.
%Y A333627 - Strict compositions are ranked by A233564.
%Y A333627 - The partial sums from the left are A272020.
%Y A333627 - Constant compositions are ranked by A272919.
%Y A333627 - Normal compositions are ranked by A333217.
%Y A333627 - Heinz number is A333219.
%Y A333627 - Anti-runs are counted by A333381.
%Y A333627 - Adjacent unequal pairs are counted by A333382.
%Y A333627 - Runs-resistance is A333628.
%Y A333627 - First appearances of run-resistances are A333629.
%Y A333627 Cf. A029931, A098504, A114994, A225620, A228351, A238279, A242882, A318928, A329744, A329747, A333489.
%K A333627 nonn
%O A333627 0,4
%A A333627 _Gus Wiseman_, Mar 30 2020