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A375295 Numbers k such that the leaders of maximal weakly increasing runs in the k-th composition in standard order (row k of A066099) are not strictly decreasing.

%I A375295 #7 Aug 13 2024 09:09:37
%S A375295 13,25,27,29,45,49,50,51,53,54,55,57,59,61,77,82,89,91,93,97,98,99,
%T A375295 101,102,103,105,107,108,109,110,111,113,114,115,117,118,119,121,123,
%U A375295 125,141,153,155,157,162,165,173,177,178,179,181,182,183,185,187,189
%N A375295 Numbers k such that the leaders of maximal weakly increasing runs in the k-th composition in standard order (row k of A066099) are not strictly decreasing.
%C A375295 First differs from the non-dashed version in lacking 166, corresponding to the composition (2,3,1,2).
%C A375295 The leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.
%C A375295 Also numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed patterns 1-32 or 1-21.
%H A375295 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>.
%H A375295 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A375295 The sequence together with corresponding compositions begins:
%e A375295   13: (1,2,1)
%e A375295   25: (1,3,1)
%e A375295   27: (1,2,1,1)
%e A375295   29: (1,1,2,1)
%e A375295   45: (2,1,2,1)
%e A375295   49: (1,4,1)
%e A375295   50: (1,3,2)
%e A375295   51: (1,3,1,1)
%e A375295   53: (1,2,2,1)
%e A375295   54: (1,2,1,2)
%e A375295   55: (1,2,1,1,1)
%e A375295   57: (1,1,3,1)
%e A375295   59: (1,1,2,1,1)
%e A375295   61: (1,1,1,2,1)
%e A375295   77: (3,1,2,1)
%e A375295   82: (2,3,2)
%e A375295   89: (2,1,3,1)
%e A375295   91: (2,1,2,1,1)
%e A375295   93: (2,1,1,2,1)
%t A375295 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A375295 Select[Range[0,100],!Greater@@First/@Split[stc[#],LessEqual]&]
%t A375295 - or -
%t A375295 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A375295 Select[Range[0,100],MatchQ[stc[#],{___,x_,___,z_,y_,___}/;x<=y<z]&] (*1-32 or 1-21*)
%Y A375295 For leaders of identical runs we have A335485.
%Y A375295 Positions of non-strictly decreasing rows in A374629 (sums A374630).
%Y A375295 For identical leaders we have A374633, counted by A374631.
%Y A375295 Matching 1-32 only gives A375137, reverse A375138, both counted by A374636.
%Y A375295 Interchanging weak/strict gives A375139, counted by A375135.
%Y A375295 Compositions of this type are counted by A375140, complement A188920.
%Y A375295 The reverse version is A375296.
%Y A375295 A003242 counts anti-runs, ranks A333489.
%Y A375295 A011782 counts compositions.
%Y A375295 A238130, A238279, A333755 count compositions by number of runs.
%Y A375295 A274174 counts contiguous compositions, ranks A374249.
%Y A375295 A374637 counts compositions by sum of leaders of weakly increasing runs.
%Y A375295 All of the following pertain to compositions in standard order:
%Y A375295 - Length is A000120.
%Y A375295 - Sum is A029837(n+1).
%Y A375295 - Leader is A065120.
%Y A375295 - Parts are listed by A066099, reverse A228351.
%Y A375295 - Number of adjacent equal pairs is A124762, unequal A333382.
%Y A375295 - Strict compositions are A233564.
%Y A375295 - Constant compositions are A272919.
%Y A375295 - Run-length transform is A333627, sum A070939.
%Y A375295 - Run-counts: A124766, A124765, A124768, A124769, A333381, A124767.
%Y A375295 - Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.
%Y A375295 Cf. A056823, A106356, A188919, A189076, A238343, A261982, A333213, A335480, A335482, A373948, A374746, A374768, A375123.
%K A375295 nonn
%O A375295 1,1
%A A375295 _Gus Wiseman_, Aug 12 2024