login
Irregular triangle read by rows where row k is the sequence of run-lengths of the k-th composition in standard order.
16

%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