cp's OEIS Frontend

A374740 Irregular triangle read by rows where row n lists the leaders of weakly decreasing runs in the n-th composition in standard order.

%I A374740 #8 Jul 26 2024 08:58:38
%S A374740 1,2,1,3,2,1,2,1,4,3,2,2,1,3,1,2,1,2,1,5,4,3,3,2,3,2,2,2,2,1,4,1,3,1,
%T A374740 2,1,2,1,3,1,2,1,2,1,6,5,4,4,3,3,3,2,3,2,4,2,3,2,2,2,3,2,2,2,2,2,1,5,
%U A374740 1,4,1,3,1,3,1,2,3,1,2,1,2,2,1,2,1,4
%N A374740 Irregular triangle read by rows where row n lists the leaders of weakly decreasing runs in the n-th composition in standard order.
%C A374740 The leaders of weakly decreasing runs in a sequence are obtained by splitting it into maximal weakly decreasing subsequences and taking the first term of each.
%C A374740 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 A374740 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374740 The maximal weakly decreasing subsequences of the 1234567th composition in standard order are ((3,2,1),(2,2,1),(2),(5,1,1,1)), so row 1234567 is (3,2,2,5).
%e A374740 The nonnegative integers, corresponding compositions, and leaders of weakly decreasing runs begin:
%e A374740     0: () -> ()           15: (1,1,1,1) -> (1)
%e A374740     1: (1) -> (1)         16: (5) -> (5)
%e A374740     2: (2) -> (2)         17: (4,1) -> (4)
%e A374740     3: (1,1) -> (1)       18: (3,2) -> (3)
%e A374740     4: (3) -> (3)         19: (3,1,1) -> (3)
%e A374740     5: (2,1) -> (2)       20: (2,3) -> (2,3)
%e A374740     6: (1,2) -> (1,2)     21: (2,2,1) -> (2)
%e A374740     7: (1,1,1) -> (1)     22: (2,1,2) -> (2,2)
%e A374740     8: (4) -> (4)         23: (2,1,1,1) -> (2)
%e A374740     9: (3,1) -> (3)       24: (1,4) -> (1,4)
%e A374740    10: (2,2) -> (2)       25: (1,3,1) -> (1,3)
%e A374740    11: (2,1,1) -> (2)     26: (1,2,2) -> (1,2)
%e A374740    12: (1,3) -> (1,3)     27: (1,2,1,1) -> (1,2)
%e A374740    13: (1,2,1) -> (1,2)   28: (1,1,3) -> (1,3)
%e A374740    14: (1,1,2) -> (1,2)   29: (1,1,2,1) -> (1,2)
%t A374740 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A374740 Table[First/@Split[stc[n],GreaterEqual],{n,0,100}]
%Y A374740 Row-leaders are A065120.
%Y A374740 Row-lengths are A124765.
%Y A374740 Other types of runs are A374251, A374515, A374683, A374757.
%Y A374740 The opposite is A374629.
%Y A374740 Positions of distinct (strict) rows are A374701, counted by A374743.
%Y A374740 Row-sums are A374741, opposite A374630.
%Y A374740 Positions of identical rows are A374744, counted by A374742.
%Y A374740 All of the following pertain to compositions in standard order:
%Y A374740 - Length is A000120.
%Y A374740 - Sum is A029837(n+1) (or sometimes A070939).
%Y A374740 - Parts are listed by A066099.
%Y A374740 - Number of adjacent equal pairs is A124762, unequal A333382.
%Y A374740 - Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
%Y A374740 - Ranks of anti-run compositions are A333489, counted by A003242.
%Y A374740 - Run-length transform is A333627, sum A070939.
%Y A374740 - Run-compression transform is A373948, sum A373953, excess A373954.
%Y A374740 - Ranks of contiguous compositions are A374249, counted by A274174.
%Y A374740 - Ranks of non-contiguous compositions are A374253, counted by A335548.
%Y A374740 Cf. A106356, A188920, A189076, A238343, A272919, A333213, A373949, A374634, A374635, A374637.
%K A374740 nonn,tabf
%O A374740 0,2
%A A374740 _Gus Wiseman_, Jul 24 2024