cp's OEIS Frontend

A375137 Numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed pattern 1-32.

%I A375137 #13 Aug 11 2024 10:35:37
%S A375137 50,98,101,114,178,194,196,197,202,203,210,226,229,242,306,324,354,
%T A375137 357,370,386,388,389,393,394,395,402,404,405,406,407,418,421,434,450,
%U A375137 452,453,458,459,466,482,485,498,562,610,613,626,644,649,690,706,708,709
%N A375137 Numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed pattern 1-32.
%C A375137 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 A375137 These are also numbers k such that the maximal weakly increasing runs in the k-th composition in standard order do not have weakly decreasing leaders, where 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 A375137 The reverse version (A375138) ranks compositions matching the dashed pattern 23-1.
%H A375137 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>.
%e A375137 Composition 102 is (1,3,1,2), which matches 1-3-2 but not 1-32.
%e A375137 Composition 210 is (1,2,3,2), which matches 1-32 but not 132.
%e A375137 Composition 358 is (2,1,3,1,2), which matches 2-3-1 and 1-3-2 but not 23-1 or 1-32.
%e A375137 The terms together with corresponding compositions begin:
%e A375137    50: (1,3,2)
%e A375137    98: (1,4,2)
%e A375137   101: (1,3,2,1)
%e A375137   114: (1,1,3,2)
%e A375137   178: (2,1,3,2)
%e A375137   194: (1,5,2)
%e A375137   196: (1,4,3)
%e A375137   197: (1,4,2,1)
%e A375137   202: (1,3,2,2)
%e A375137   203: (1,3,2,1,1)
%e A375137   210: (1,2,3,2)
%e A375137   226: (1,1,4,2)
%e A375137   229: (1,1,3,2,1)
%e A375137   242: (1,1,1,3,2)
%t A375137 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A375137 Select[Range[0,100],MatchQ[stc[#],{___,x_,___,z_,y_,___}/;x<y<z]&] (*1-32*)
%Y A375137 The complement is too dense, but counted by A189076.
%Y A375137 The non-dashed version is A335480, reverse A335482.
%Y A375137 For leaders of identical runs we have A335485, reverse A335486.
%Y A375137 For identical leaders we have A374633, counted by A374631.
%Y A375137 Compositions of this type are counted by A374636.
%Y A375137 For distinct leaders we have A374768, counted by A374632.
%Y A375137 The reverse version is A375138, counted by A374636.
%Y A375137 For leaders of strictly increasing runs we have A375139, counted by A375135.
%Y A375137 Matching 1-21 also gives A375295, counted by A375140 (complement A188920).
%Y A375137 A003242 counts anti-runs, ranks A333489.
%Y A375137 A011782 counts compositions.
%Y A375137 A238130, A238279, A333755 count compositions by number of runs.
%Y A375137 All of the following pertain to compositions in standard order:
%Y A375137 - Length is A000120.
%Y A375137 - Sum is A029837(n+1).
%Y A375137 - Leader is A065120.
%Y A375137 - Parts are listed by A066099, reverse A228351.
%Y A375137 - Number of adjacent equal pairs is A124762, unequal A333382.
%Y A375137 - Strict compositions are A233564.
%Y A375137 - Constant compositions are A272919.
%Y A375137 - Run-length transform is A333627, sum A070939.
%Y A375137 - Run-counts: A124766, A124765, A124768, A124769, A333381, A124767.
%Y A375137 - Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.
%Y A375137 Cf. A056823, A106356, A188919, A238343, A333213, A373948, A373953, A374634, A374635, A374637, A375123, A375296.
%K A375137 nonn
%O A375137 1,1
%A A375137 _Gus Wiseman_, Aug 09 2024