A375138 - OEIS

%I #8 Aug 11 2024 10:35:24

%S 41,81,83,105,145,161,163,165,166,167,169,209,211,233,289,290,291,297,

%T 321,323,325,326,327,329,331,332,333,334,335,337,339,361,401,417,419,

%U 421,422,423,425,465,467,489,545,553,577,578,579,581,582,583,593,595,617

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

%C 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 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 The reverse version (A375137) ranks compositions matching the dashed pattern 1-32.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>.

%e Composition 89 is (2,1,3,1), which matches 2-3-1 but not 23-1.

%e Composition 165 is (2,3,2,1), which matches 23-1 but not 231.

%e 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 The sequence together with corresponding compositions begins:

%e    41: (2,3,1)

%e    81: (2,4,1)

%e    83: (2,3,1,1)

%e   105: (1,2,3,1)

%e   145: (3,4,1)

%e   161: (2,5,1)

%e   163: (2,4,1,1)

%e   165: (2,3,2,1)

%e   166: (2,3,1,2)

%e   167: (2,3,1,1,1)

%e   169: (2,2,3,1)

%e   209: (1,2,4,1)

%e   211: (1,2,3,1,1)

%e   233: (1,1,2,3,1)

%t stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t Select[Range[0,100],MatchQ[stc[#],{___,y_,z_,___,x_,___}/;x<y<z]&] (*23-1*)

%Y The complement is too dense, but counted by A189076.

%Y The non-dashed version is A335482, reverse A335480.

%Y For leaders of identical runs we have A335486, reverse A335485.

%Y Compositions of this type are counted by A374636.

%Y The reverse version is A375137, counted by A374636.

%Y Matching 12-1 also gives A375296, counted by A375140 (complement A188920).

%Y A003242 counts anti-runs, ranks A333489.

%Y A011782 counts compositions.

%Y A238130, A238279, A333755 count compositions by number of runs.

%Y All of the following pertain to compositions in standard order:

%Y - Length is A000120.

%Y - Sum is A029837(n+1).

%Y - Leader is A065120.

%Y - Parts are listed by A066099, reverse A228351.

%Y - Number of adjacent equal pairs is A124762, unequal A333382.

%Y - Strict compositions are A233564.

%Y - Constant compositions are A272919.

%Y - Run-length transform is A333627, sum A070939.

%Y - Run-counts: A124766, A124765, A124768, A124769, A333381, A124767.

%Y - Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.

%Y Cf. A056823, A106356, A188919, A238343, A333213, A335466, A373948, A373953, A374633, A375123, A375139, A374768.

%K nonn

%O 1,1

%A _Gus Wiseman_, Aug 09 2024