cp's OEIS Frontend

A337565 Irregular triangle read by rows where row k is the sequence of maximal anti-run lengths in the k-th composition in standard order.

%I A337565 #9 Sep 17 2020 20:35:20
%S A337565 1,1,1,1,1,2,2,1,1,1,1,2,1,1,2,1,2,3,1,2,1,1,1,1,1,2,2,2,1,2,1,2,3,2,
%T A337565 1,1,2,3,2,1,3,1,1,2,1,3,1,1,2,1,1,1,1,1,1,2,2,2,1,1,1,3,3,2,1,1,2,3,
%U A337565 1,1,1,1,2,1,3,4,2,2,2,1,1,1,2,3,3
%N A337565 Irregular triangle read by rows where row k is the sequence of maximal anti-run lengths in the k-th composition in standard order.
%C A337565 An anti-run is a sequence with no adjacent equal parts.
%C A337565 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 A337565 The first column below lists various selected n; the second column gives the corresponding composition; the third column gives the corresponding row of the triangle, i.e., the anti-run lengths.
%e A337565     1:           (1) -> (1)
%e A337565     3:         (1,1) -> (1,1)
%e A337565     5:         (2,1) -> (2)
%e A337565     7:       (1,1,1) -> (1,1,1)
%e A337565    11:       (2,1,1) -> (2,1)
%e A337565    13:       (1,2,1) -> (3)
%e A337565    14:       (1,1,2) -> (1,2)
%e A337565    15:     (1,1,1,1) -> (1,1,1,1)
%e A337565    23:     (2,1,1,1) -> (2,1,1)
%e A337565    27:     (1,2,1,1) -> (3,1)
%e A337565    29:     (1,1,2,1) -> (1,3)
%e A337565    30:     (1,1,1,2) -> (1,1,2)
%e A337565    31:   (1,1,1,1,1) -> (1,1,1,1,1)
%e A337565    43:     (2,2,1,1) -> (1,2,1)
%e A337565    45:     (2,1,2,1) -> (4)
%e A337565    46:     (2,1,1,2) -> (2,2)
%e A337565    47:   (2,1,1,1,1) -> (2,1,1,1)
%e A337565    55:   (1,2,1,1,1) -> (3,1,1)
%e A337565    59:   (1,1,2,1,1) -> (1,3,1)
%e A337565    61:   (1,1,1,2,1) -> (1,1,3)
%e A337565    62:   (1,1,1,1,2) -> (1,1,1,2)
%e A337565    63: (1,1,1,1,1,1) -> (1,1,1,1,1,1)
%e A337565 For example, the 119th composition is (1,1,2,1,1,1), with maximal anti-runs ((1),(1,2,1),(1),(1)), so row 119 is (1,3,1,1).
%t A337565 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A337565 Table[Length/@Split[stc[n],UnsameQ],{n,0,50}]
%Y A337565 A000120 gives row sums.
%Y A337565 A333381 gives row lengths.
%Y A337565 A333769 is the version for runs.
%Y A337565 A003242 counts anti-run compositions.
%Y A337565 A011782 counts compositions.
%Y A337565 A106351 counts anti-run compositions by length.
%Y A337565 A329744 is a triangle counting compositions by runs-resistance.
%Y A337565 A333755 counts compositions by number of runs.
%Y A337565 All of the following pertain to compositions in standard order (A066099):
%Y A337565 - Sum is A070939.
%Y A337565 - Adjacent equal pairs are counted by A124762.
%Y A337565 - Runs are counted by A124767.
%Y A337565 - Strict compositions are A233564.
%Y A337565 - Constant compositions are A272919.
%Y A337565 - Patterns are A333217.
%Y A337565 - Heinz number is A333219.
%Y A337565 - Anti-runs are counted by A333381.
%Y A337565 - Anti-run compositions are A333489.
%Y A337565 - Runs-resistance is A333628.
%Y A337565 - Combinatory separations are A334030.
%Y A337565 Cf. A106356, A113835, A114994, A124767, A181819, A228351, A238279, A318928, A333216, A333627, A333630.
%K A337565 nonn,tabf
%O A337565 0,6
%A A337565 _Gus Wiseman_, Sep 07 2020