cp's OEIS Frontend

A375138 Numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed pattern 23-1.

%I A375138 #8 Aug 11 2024 10:35:24
%S A375138 41,81,83,105,145,161,163,165,166,167,169,209,211,233,289,290,291,297,
%T A375138 321,323,325,326,327,329,331,332,333,334,335,337,339,361,401,417,419,
%U A375138 421,422,423,425,465,467,489,545,553,577,578,579,581,582,583,593,595,617
%N A375138 Numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed pattern 23-1.
%C A375138 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 A375138 These are also numbers k such that the maximal weakly increasing runs in the reverse of 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 A375138 The reverse version (A375137) ranks compositions matching the dashed pattern 1-32.
%H A375138 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>.
%e A375138 Composition 89 is (2,1,3,1), which matches 2-3-1 but not 23-1.
%e A375138 Composition 165 is (2,3,2,1), which matches 23-1 but not 231.
%e A375138 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 A375138 The sequence together with corresponding compositions begins:
%e A375138    41: (2,3,1)
%e A375138    81: (2,4,1)
%e A375138    83: (2,3,1,1)
%e A375138   105: (1,2,3,1)
%e A375138   145: (3,4,1)
%e A375138   161: (2,5,1)
%e A375138   163: (2,4,1,1)
%e A375138   165: (2,3,2,1)
%e A375138   166: (2,3,1,2)
%e A375138   167: (2,3,1,1,1)
%e A375138   169: (2,2,3,1)
%e A375138   209: (1,2,4,1)
%e A375138   211: (1,2,3,1,1)
%e A375138   233: (1,1,2,3,1)
%t A375138 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A375138 Select[Range[0,100],MatchQ[stc[#],{___,y_,z_,___,x_,___}/;x<y<z]&] (*23-1*)
%Y A375138 The complement is too dense, but counted by A189076.
%Y A375138 The non-dashed version is A335482, reverse A335480.
%Y A375138 For leaders of identical runs we have A335486, reverse A335485.
%Y A375138 Compositions of this type are counted by A374636.
%Y A375138 The reverse version is A375137, counted by A374636.
%Y A375138 Matching 12-1 also gives A375296, counted by A375140 (complement A188920).
%Y A375138 A003242 counts anti-runs, ranks A333489.
%Y A375138 A011782 counts compositions.
%Y A375138 A238130, A238279, A333755 count compositions by number of runs.
%Y A375138 All of the following pertain to compositions in standard order:
%Y A375138 - Length is A000120.
%Y A375138 - Sum is A029837(n+1).
%Y A375138 - Leader is A065120.
%Y A375138 - Parts are listed by A066099, reverse A228351.
%Y A375138 - Number of adjacent equal pairs is A124762, unequal A333382.
%Y A375138 - Strict compositions are A233564.
%Y A375138 - Constant compositions are A272919.
%Y A375138 - Run-length transform is A333627, sum A070939.
%Y A375138 - Run-counts: A124766, A124765, A124768, A124769, A333381, A124767.
%Y A375138 - Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.
%Y A375138 Cf. A056823, A106356, A188919, A238343, A333213, A335466, A373948, A373953, A374633, A375123, A375139, A374768.
%K A375138 nonn
%O A375138 1,1
%A A375138 _Gus Wiseman_, Aug 09 2024