A374629 - OEIS

%I #14 Aug 21 2024 22:47:08

%S 1,2,1,3,2,1,1,1,4,3,1,2,2,1,1,1,1,1,1,5,4,1,3,2,3,1,2,2,1,2,1,2,1,1,

%T 1,1,1,1,1,1,1,1,1,1,6,5,1,4,2,4,1,3,3,2,1,3,1,3,1,2,2,1,2,2,1,2,1,2,

%U 1,1,2,1,2,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1

%N Irregular triangle listing the leaders of maximal weakly increasing runs in the n-th composition in standard order.

%C The leaders of maximal weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.

%C 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.

%H Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.

%e The 58654th composition in standard order is (1,1,3,2,4,1,1,1,2), with maximal weakly increasing runs ((1,1,3),(2,4),(1,1,1,2)), so row 58654 is (1,2,1).

%e The nonnegative integers, corresponding compositions, and leaders of maximal weakly increasing runs begin:

%e     0:      () -> ()      15: (1,1,1,1) -> (1)

%e     1:     (1) -> (1)     16:       (5) -> (5)

%e     2:     (2) -> (2)     17:     (4,1) -> (4,1)

%e     3:   (1,1) -> (1)     18:     (3,2) -> (3,2)

%e     4:     (3) -> (3)     19:   (3,1,1) -> (3,1)

%e     5:   (2,1) -> (2,1)   20:     (2,3) -> (2)

%e     6:   (1,2) -> (1)     21:   (2,2,1) -> (2,1)

%e     7: (1,1,1) -> (1)     22:   (2,1,2) -> (2,1)

%e     8:     (4) -> (4)     23: (2,1,1,1) -> (2,1)

%e     9:   (3,1) -> (3,1)   24:     (1,4) -> (1)

%e    10:   (2,2) -> (2)     25:   (1,3,1) -> (1,1)

%e    11: (2,1,1) -> (2,1)   26:   (1,2,2) -> (1)

%e    12:   (1,3) -> (1)     27: (1,2,1,1) -> (1,1)

%e    13: (1,2,1) -> (1,1)   28:   (1,1,3) -> (1)

%e    14: (1,1,2) -> (1)     29: (1,1,2,1) -> (1,1)

%t stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t Table[First/@Split[stc[n],LessEqual],{n,0,100}]

%Y Row-leaders are A065120.

%Y Row-lengths are A124766.

%Y Row-sums are A374630.

%Y Positions of constant rows are A374633, counted by A374631.

%Y Positions of strict rows are A374768, counted by A374632.

%Y For other types of runs we have A374251, A374515, A374683, A374740, A374757.

%Y Positions of non-weakly decreasing rows are A375137.

%Y A011782 counts compositions.

%Y A238130, A238279, A333755 count compositions by number of runs.

%Y A335456 counts patterns matched by compositions.

%Y All of the following pertain to compositions in standard order:

%Y - Length is A000120.

%Y - Sum is A029837(n+1).

%Y - Leader is A065120.

%Y - Parts are listed by A066099, reverse A228351.

%Y - Number of adjacent equal pairs is A124762, unequal A333382.

%Y - Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.

%Y - Ranks of anti-run compositions are A333489, counted by A003242.

%Y - Run-length transform is A333627, length A124767, sum A070939.

%Y - Run-compression transform is A373948, sum A373953, excess A373954.

%Y - Ranks of contiguous compositions are A374249, counted by A274174.

%Y - Ranks of non-contiguous compositions are A374253, counted by A335548.

%Y Cf. A046660, A106356, A188920, A189076, A238343, A272919, A333213, A373949, A374634, A374635, A374637, A374701, A375123.

%K nonn,tabf

%O 0,2

%A _Gus Wiseman_, Jul 20 2024