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