cp's OEIS Frontend

A374683 Irregular triangle read by rows where row n lists the leaders of strictly increasing runs in the n-th composition in standard order.

%I A374683 #6 Jul 27 2024 09:42:59
%S A374683 1,2,1,1,3,2,1,1,1,1,1,4,3,1,2,2,2,1,1,1,1,1,1,1,1,1,1,1,5,4,1,3,2,3,
%T A374683 1,1,2,2,2,1,2,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,6,
%U A374683 5,1,4,2,4,1,1,3,3,3,2,1,3,1,3,1,1,1
%N A374683 Irregular triangle read by rows where row n lists the leaders of strictly increasing runs in the n-th composition in standard order.
%C A374683 The leaders of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.
%C A374683 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 A374683 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374683 The maximal strictly increasing subsequences of the 1234567th composition in standard order are ((3),(2),(1,2),(2),(1,2,5),(1),(1),(1)), so row 1234567 is (3,2,1,2,1,1,1,1).
%e A374683 The nonnegative integers, corresponding compositions, and leaders of strictly increasing runs begin:
%e A374683    0:      () -> ()         15: (1,1,1,1) -> (1,1,1,1)
%e A374683    1:     (1) -> (1)        16:       (5) -> (5)
%e A374683    2:     (2) -> (2)        17:     (4,1) -> (4,1)
%e A374683    3:   (1,1) -> (1,1)      18:     (3,2) -> (3,2)
%e A374683    4:     (3) -> (3)        19:   (3,1,1) -> (3,1,1)
%e A374683    5:   (2,1) -> (2,1)      20:     (2,3) -> (2)
%e A374683    6:   (1,2) -> (1)        21:   (2,2,1) -> (2,2,1)
%e A374683    7: (1,1,1) -> (1,1,1)    22:   (2,1,2) -> (2,1)
%e A374683    8:     (4) -> (4)        23: (2,1,1,1) -> (2,1,1,1)
%e A374683    9:   (3,1) -> (3,1)      24:     (1,4) -> (1)
%e A374683   10:   (2,2) -> (2,2)      25:   (1,3,1) -> (1,1)
%e A374683   11: (2,1,1) -> (2,1,1)    26:   (1,2,2) -> (1,2)
%e A374683   12:   (1,3) -> (1)        27: (1,2,1,1) -> (1,1,1)
%e A374683   13: (1,2,1) -> (1,1)      28:   (1,1,3) -> (1,1)
%e A374683   14: (1,1,2) -> (1,1)      29: (1,1,2,1) -> (1,1,1)
%t A374683 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A374683 Table[First/@Split[stc[n],Less],{n,0,100}]
%Y A374683 Row-leaders are A065120.
%Y A374683 Row-lengths are A124768.
%Y A374683 Other types of runs: A374251, A374515, A374740.
%Y A374683 The weak version is A374629, sum A374630, length A124766.
%Y A374683 Row-sums are A374684.
%Y A374683 Positions of identical rows are A374685, counted by A374686.
%Y A374683 Positions of distinct (strict) rows are A374698, counted by A374687.
%Y A374683 The opposite version is A374757, sum A374758, length A124769.
%Y A374683 All of the following pertain to compositions in standard order:
%Y A374683 - Length is A000120.
%Y A374683 - Sum is A029837(n+1) (or sometimes A070939).
%Y A374683 - Parts are listed by A066099.
%Y A374683 - Number of adjacent equal pairs is A124762, unequal A333382.
%Y A374683 - Number of max runs: A124765, A124767, A333381.
%Y A374683 - Run-length transform is A333627, sum A070939.
%Y A374683 - Run-compression transform is A373948, sum A373953, excess A373954.
%Y A374683 - Ranks of contiguous compositions are A374249, counted by A274174.
%Y A374683 - Ranks of non-contiguous compositions are A374253, counted by A335548.
%Y A374683 Cf. A106356, A188920, A189076, A238343, A272919, A333213, A373949, A374700.
%K A374683 nonn,tabf
%O A374683 0,2
%A A374683 _Gus Wiseman_, Jul 26 2024