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A374757 Irregular triangle read by rows where row n lists the leaders of strictly decreasing runs in the n-th composition in standard order.

%I A374757 #12 Feb 19 2025 11:57:20
%S A374757 1,2,1,1,3,2,1,2,1,1,1,4,3,2,2,2,1,1,3,1,2,1,1,2,1,1,1,1,5,4,3,3,1,2,
%T A374757 3,2,2,2,2,2,1,1,1,4,1,3,1,2,2,1,2,1,1,1,3,1,1,2,1,1,1,2,1,1,1,1,1,6,
%U A374757 5,4,4,1,3,3,3,3,2,3,1,1,2,4,2,3
%N A374757 Irregular triangle read by rows where row n lists the leaders of strictly decreasing runs in the n-th composition in standard order.
%C A374757 The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly decreasing subsequences and taking the first term of each.
%C A374757 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 A374757 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374757 the 1234567th composition in standard order is (3,2,1,2,2,1,2,5,1,1,1), with strictly decreasing runs ((3,2,1),(2),(2,1),(2),(5,1),(1),(1)), so row 1234567 is (3,2,2,2,5,1,1).
%e A374757 The nonnegative integers, corresponding compositions, and leaders of strictly decreasing runs begin:
%e A374757     0:      () -> ()        15: (1,1,1,1) -> (1,1,1,1)
%e A374757     1:     (1) -> (1)       16:       (5) -> (5)
%e A374757     2:     (2) -> (2)       17:     (4,1) -> (4)
%e A374757     3:   (1,1) -> (1,1)     18:     (3,2) -> (3)
%e A374757     4:     (3) -> (3)       19:   (3,1,1) -> (3,1)
%e A374757     5:   (2,1) -> (2)       20:     (2,3) -> (2,3)
%e A374757     6:   (1,2) -> (1,2)     21:   (2,2,1) -> (2,2)
%e A374757     7: (1,1,1) -> (1,1,1)   22:   (2,1,2) -> (2,2)
%e A374757     8:     (4) -> (4)       23: (2,1,1,1) -> (2,1,1)
%e A374757     9:   (3,1) -> (3)       24:     (1,4) -> (1,4)
%e A374757    10:   (2,2) -> (2,2)     25:   (1,3,1) -> (1,3)
%e A374757    11: (2,1,1) -> (2,1)     26:   (1,2,2) -> (1,2,2)
%e A374757    12:   (1,3) -> (1,3)     27: (1,2,1,1) -> (1,2,1)
%e A374757    13: (1,2,1) -> (1,2)     28:   (1,1,3) -> (1,1,3)
%e A374757    14: (1,1,2) -> (1,1,2)   29: (1,1,2,1) -> (1,1,2)
%t A374757 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A374757 Table[First/@Split[stc[n],Greater],{n,0,100}]
%Y A374757 Row-leaders of nonempty rows are A065120.
%Y A374757 Row-lengths are A124769.
%Y A374757 The opposite version is A374683, sum A374684, length A124768.
%Y A374757 The weak version is A374740, sum A374741, length A124765.
%Y A374757 Row-sums are A374758.
%Y A374757 Positions of identical rows are A374759 (counted by A374760).
%Y A374757 Positions of distinct (strict) rows are A374767 (counted by A374761).
%Y A374757 All of the following pertain to compositions in standard order:
%Y A374757 - Length is A000120.
%Y A374757 - Sum is A029837(n+1).
%Y A374757 - Parts are listed by A066099.
%Y A374757 - Number of adjacent equal pairs is A124762, unequal A333382.
%Y A374757 - Run-length transform is A333627, sum A070939.
%Y A374757 - Run-compression transform is A373948, sum A373953, excess A373954.
%Y A374757 - Ranks of contiguous compositions are A374249, counted by A274174.
%Y A374757 - Ranks of non-contiguous compositions are A374253, counted by A335548.
%Y A374757 Six types of runs:
%Y A374757 - Count: A124766, A124765, A124768, A124769, A333381, A124767.
%Y A374757 - Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
%Y A374757 - Rank: A375123, A375124, A375125, A375126, A375127, A373948.
%Y A374757 Cf. A051903, A106356, A188920, A189076, A233564, A238343, A333213, A373949, A374685, A374698, A374700, A374706.
%K A374757 nonn,tabf
%O A374757 0,2
%A A374757 _Gus Wiseman_, Jul 29 2024