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A375296 Numbers k such that the leaders of maximal weakly increasing runs in the reverse of the k-th composition in standard order (row k of A228351) are not strictly decreasing.

%I A375296 #5 Aug 16 2024 15:08:38
%S A375296 13,25,27,29,41,45,49,51,53,54,55,57,59,61,77,81,82,83,89,91,93,97,99,
%T A375296 101,102,103,105,107,108,109,110,111,113,115,117,118,119,121,123,125,
%U A375296 141,145,153,155,157,161,162,163,165,166,167,169,173,177,179,181,182
%N A375296 Numbers k such that the leaders of maximal weakly increasing runs in the reverse of the k-th composition in standard order (row k of A228351) are not strictly decreasing.
%C A375296 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 A375296 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.
%C A375296 Also numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed patterns 23-1 or 12-1.
%H A375296 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>.
%H A375296 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A375296 The sequence together with corresponding compositions begins:
%e A375296   13: (1,2,1)
%e A375296   25: (1,3,1)
%e A375296   27: (1,2,1,1)
%e A375296   29: (1,1,2,1)
%e A375296   41: (2,3,1)
%e A375296   45: (2,1,2,1)
%e A375296   49: (1,4,1)
%e A375296   51: (1,3,1,1)
%e A375296   53: (1,2,2,1)
%e A375296   54: (1,2,1,2)
%e A375296   55: (1,2,1,1,1)
%e A375296   57: (1,1,3,1)
%e A375296   59: (1,1,2,1,1)
%e A375296   61: (1,1,1,2,1)
%t A375296 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A375296 Select[Range[0,100],!Greater@@First/@Split[Reverse[stc[#]],LessEqual]&]
%t A375296 - or -
%t A375296 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A375296 Select[Range[0,300],MatchQ[stc[#],{___,y_,z_,___,x_,___}/;x<=y<z]&] (*23-1 or 12-1*)
%Y A375296 For leaders of identical runs we have A335486, reverse A335485.
%Y A375296 Matching 1-32 only gives A375138, reverse A375137, both counted by A374636.
%Y A375296 Compositions of this type are counted by A375140, complement A188920.
%Y A375296 The reverse version is A375295.
%Y A375296 A003242 counts anti-runs, ranks A333489.
%Y A375296 A011782 counts compositions.
%Y A375296 A238130, A238279, A333755 count compositions by number of runs.
%Y A375296 A274174 counts contiguous compositions, ranks A374249.
%Y A375296 A374637 counts compositions by sum of leaders of weakly increasing runs.
%Y A375296 All of the following pertain to compositions in standard order:
%Y A375296 - Length is A000120.
%Y A375296 - Sum is A029837(n+1).
%Y A375296 - Leader is A065120.
%Y A375296 - Parts are listed by A066099, reverse A228351.
%Y A375296 - Number of adjacent equal pairs is A124762, unequal A333382.
%Y A375296 - Strict compositions are A233564.
%Y A375296 - Constant compositions are A272919.
%Y A375296 - Run-length transform is A333627, sum A070939.
%Y A375296 - Run-counts: A124766, A124765, A124768, A124769, A333381, A124767.
%Y A375296 - Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.
%Y A375296 Cf. A056823, A106356, A188919, A189076, A238343, A333213, A335480, A335482, A373948, A374630, A374633, A374768, A375123.
%K A375296 nonn
%O A375296 1,1
%A A375296 _Gus Wiseman_, Aug 13 2024