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 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. 5
 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, 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, 1, 1, 1, 1, 2, 1, 3, 4, 2, 2, 2, 1, 1, 1, 2, 3, 3 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS An anti-run is a sequence with no adjacent equal parts. 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. LINKS EXAMPLE 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.     1:           (1) -> (1)     3:         (1,1) -> (1,1)     5:         (2,1) -> (2)     7:       (1,1,1) -> (1,1,1)    11:       (2,1,1) -> (2,1)    13:       (1,2,1) -> (3)    14:       (1,1,2) -> (1,2)    15:     (1,1,1,1) -> (1,1,1,1)    23:     (2,1,1,1) -> (2,1,1)    27:     (1,2,1,1) -> (3,1)    29:     (1,1,2,1) -> (1,3)    30:     (1,1,1,2) -> (1,1,2)    31:   (1,1,1,1,1) -> (1,1,1,1,1)    43:     (2,2,1,1) -> (1,2,1)    45:     (2,1,2,1) -> (4)    46:     (2,1,1,2) -> (2,2)    47:   (2,1,1,1,1) -> (2,1,1,1)    55:   (1,2,1,1,1) -> (3,1,1)    59:   (1,1,2,1,1) -> (1,3,1)    61:   (1,1,1,2,1) -> (1,1,3)    62:   (1,1,1,1,2) -> (1,1,1,2)    63: (1,1,1,1,1,1) -> (1,1,1,1,1,1) 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). MATHEMATICA stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse; Table[Length/@Split[stc[n], UnsameQ], {n, 0, 50}] CROSSREFS A000120 gives row sums. A333381 gives row lengths. A333769 is the version for runs. A003242 counts anti-run compositions. A011782 counts compositions. A106351 counts anti-run compositions by length. A329744 is a triangle counting compositions by runs-resistance. A333755 counts compositions by number of runs. All of the following pertain to compositions in standard order (A066099): - Sum is A070939. - Adjacent equal pairs are counted by A124762. - Runs are counted by A124767. - Strict compositions are A233564. - Constant compositions are A272919. - Patterns are A333217. - Heinz number is A333219. - Anti-runs are counted by A333381. - Anti-run compositions are A333489. - Runs-resistance is A333628. - Combinatory separations are A334030. Cf. A106356, A113835, A114994, A124767, A181819, A228351, A238279, A318928, A333216, A333627, A333630. Sequence in context: A184318 A030410 A085301 * A138385 A030614 A328615 Adjacent sequences:  A337562 A337563 A337564 * A337566 A337567 A337568 KEYWORD nonn,tabf AUTHOR Gus Wiseman, Sep 07 2020 STATUS approved

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Last modified June 27 19:29 EDT 2022. Contains 354898 sequences. (Running on oeis4.)