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A374515 Irregular triangle read by rows where row n lists the leaders of anti-runs in the n-th composition in standard order.

%I A374515 #6 Aug 02 2024 08:58:25
%S A374515 1,2,1,1,3,2,1,1,1,1,4,3,2,2,2,1,1,1,1,1,1,1,1,1,5,4,3,3,1,2,2,2,2,2,
%T A374515 1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,6,5,4,4,1,3,3,3,3,3,1,1,2,2,
%U A374515 2,2,2,2,2,1,2,2,2,1,2,1,1,1,1,1,1
%N A374515 Irregular triangle read by rows where row n lists the leaders of anti-runs in the n-th composition in standard order.
%C A374515 Anti-runs summing to n are counted by A003242(n).
%C A374515 The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.
%C A374515 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 A374515 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374515 The maximal anti-runs of the 1234567th composition in standard order are ((3,2,1,2),(2,1,2,5,1),(1),(1)), so row 1234567 is (3,2,1,1).
%e A374515 The nonnegative integers, corresponding compositions, and leaders of anti-runs begin:
%e A374515     0:      () -> ()        15: (1,1,1,1) -> (1,1,1,1)
%e A374515     1:     (1) -> (1)       16:       (5) -> (5)
%e A374515     2:     (2) -> (2)       17:     (4,1) -> (4)
%e A374515     3:   (1,1) -> (1,1)     18:     (3,2) -> (3)
%e A374515     4:     (3) -> (3)       19:   (3,1,1) -> (3,1)
%e A374515     5:   (2,1) -> (2)       20:     (2,3) -> (2)
%e A374515     6:   (1,2) -> (1)       21:   (2,2,1) -> (2,2)
%e A374515     7: (1,1,1) -> (1,1,1)   22:   (2,1,2) -> (2)
%e A374515     8:     (4) -> (4)       23: (2,1,1,1) -> (2,1,1)
%e A374515     9:   (3,1) -> (3)       24:     (1,4) -> (1)
%e A374515    10:   (2,2) -> (2,2)     25:   (1,3,1) -> (1)
%e A374515    11: (2,1,1) -> (2,1)     26:   (1,2,2) -> (1,2)
%e A374515    12:   (1,3) -> (1)       27: (1,2,1,1) -> (1,1)
%e A374515    13: (1,2,1) -> (1)       28:   (1,1,3) -> (1,1)
%e A374515    14: (1,1,2) -> (1,1)     29: (1,1,2,1) -> (1,1)
%t A374515 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A374515 Table[First/@Split[stc[n],UnsameQ],{n,0,100}]
%Y A374515 Row-leaders of nonempty rows are A065120.
%Y A374515 Row-lengths are A333381.
%Y A374515 Row-sums are A374516.
%Y A374515 Positions of identical rows are A374519 (counted by A374517).
%Y A374515 Positions of distinct (strict) rows are A374638 (counted by A374518).
%Y A374515 A106356 counts compositions by number of maximal anti-runs.
%Y A374515 A238279 counts compositions by number of maximal runs
%Y A374515 A238424 counts partitions whose first differences are an anti-run.
%Y A374515 All of the following pertain to compositions in standard order:
%Y A374515 - Length is A000120.
%Y A374515 - Sum is A029837(n+1).
%Y A374515 - Parts are listed by A066099.
%Y A374515 - Number of adjacent equal pairs is A124762, unequal A333382.
%Y A374515 - Anti-runs are ranked by A333489, counted by A003242.
%Y A374515 - Run-length transform is A333627, sum A070939.
%Y A374515 - Run-compression is A373948 or A374251, sum A373953, excess A373954.
%Y A374515 - Ranks of contiguous compositions are A374249, counted by A274174.
%Y A374515 Six types of maximal runs:
%Y A374515 - Count: A124766, A124765, A124768, A124769, A333381, A124767.
%Y A374515 - Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
%Y A374515 - Rank: A375123, A375124, A375125, A375126, A375127, A373948.
%Y A374515 Cf. A029931, A059893, A228351, A233564, A272919, A333213, A373949.
%K A374515 nonn,tabf
%O A374515 0,2
%A A374515 _Gus Wiseman_, Jul 31 2024