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A333769 Irregular triangle read by rows where row k is the sequence of run-lengths of the k-th composition in standard order.

%I A333769 #8 May 28 2020 05:01:19
%S A333769 1,1,2,1,1,1,1,1,3,1,1,1,2,1,2,1,1,1,1,1,2,1,4,1,1,1,1,1,1,2,1,1,2,1,
%T A333769 1,1,1,1,3,1,1,1,1,1,1,2,1,1,2,2,1,2,1,1,3,1,5,1,1,1,1,1,1,2,2,1,1,1,
%U A333769 1,1,1,1,3,1,1,1,1,1,3,2,2,1,1,1,1,1,1
%N A333769 Irregular triangle read by rows where row k is the sequence of run-lengths of the k-th composition in standard order.
%C A333769 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.
%e A333769 The standard compositions and their run-lengths:
%e A333769    0:        () -> ()
%e A333769    1:       (1) -> (1)
%e A333769    2:       (2) -> (1)
%e A333769    3:     (1,1) -> (2)
%e A333769    4:       (3) -> (1)
%e A333769    5:     (2,1) -> (1,1)
%e A333769    6:     (1,2) -> (1,1)
%e A333769    7:   (1,1,1) -> (3)
%e A333769    8:       (4) -> (1)
%e A333769    9:     (3,1) -> (1,1)
%e A333769   10:     (2,2) -> (2)
%e A333769   11:   (2,1,1) -> (1,2)
%e A333769   12:     (1,3) -> (1,1)
%e A333769   13:   (1,2,1) -> (1,1,1)
%e A333769   14:   (1,1,2) -> (2,1)
%e A333769   15: (1,1,1,1) -> (4)
%e A333769   16:       (5) -> (1)
%e A333769   17:     (4,1) -> (1,1)
%e A333769   18:     (3,2) -> (1,1)
%e A333769   19:   (3,1,1) -> (1,2)
%e A333769 For example, the 119th composition is (1,1,2,1,1,1), so row 119 is (2,1,3).
%t A333769 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A333769 Table[Length/@Split[stc[n]],{n,0,30}]
%Y A333769 Row sums are A000120.
%Y A333769 Row lengths are A124767.
%Y A333769 Row k is the A333627(k)-th standard composition.
%Y A333769 A triangle counting compositions by runs-resistance is A329744.
%Y A333769 All of the following pertain to compositions in standard order (A066099):
%Y A333769 - Partial sums from the right are A048793.
%Y A333769 - Sum is A070939.
%Y A333769 - Adjacent equal pairs are counted by A124762.
%Y A333769 - Strict compositions are A233564.
%Y A333769 - Partial sums from the left are A272020.
%Y A333769 - Constant compositions are A272919.
%Y A333769 - Normal compositions are A333217.
%Y A333769 - Heinz number is A333219.
%Y A333769 - Runs-resistance is A333628.
%Y A333769 - First appearances of run-resistances are A333629.
%Y A333769 - Combinatory separations are A334030.
%Y A333769 Cf. A029931, A098504, A114994, A181819, A182850, A225620, A228351, A238279, A242882, A318928, A329747, A333489, A333630.
%K A333769 nonn,tabf
%O A333769 0,3
%A A333769 _Gus Wiseman_, Apr 10 2020