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A375407 Numbers k such that the k-th composition in standard order (row k of A066099) matches both of the dashed patterns 23-1 and 1-32.

%I A375407 #5 Aug 27 2024 09:14:27
%S A375407 421,649,802,809,837,843,933,1289,1299,1330,1445,1577,1602,1605,1617,
%T A375407 1619,1669,1673,1675,1685,1686,1687,1701,1826,1833,1861,1867,1957,
%U A375407 2469,2569,2577,2579,2597,2598,2599,2610,2658,2661,2674,2697,2850,2857,2885,2891
%N A375407 Numbers k such that the k-th composition in standard order (row k of A066099) matches both of the dashed patterns 23-1 and 1-32.
%C A375407 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 A375407 These are also numbers k such that:
%C A375407 (1) the maximal weakly increasing runs in the reverse of the k-th composition in standard order do not have weakly decreasing leaders, and
%C A375407 (2) the maximal weakly increasing runs in the k-th composition in standard order do not have weakly decreasing leaders.
%H A375407 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>.
%H A375407 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%F A375407 Intersection of A375138 and A375137.
%e A375407 Composition 89 is (2,1,3,1), which matches 2-3-1 but not 23-1.
%e A375407 Composition 165 is (2,3,2,1), which matches 23-1 but not 231.
%e A375407 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 A375407 The sequence together with corresponding compositions begins:
%e A375407    421: (1,2,3,2,1)
%e A375407    649: (2,4,3,1)
%e A375407    802: (1,3,4,2)
%e A375407    809: (1,3,2,3,1)
%e A375407    837: (1,2,4,2,1)
%e A375407    843: (1,2,3,2,1,1)
%e A375407    933: (1,1,2,3,2,1)
%e A375407   1289: (2,5,3,1)
%e A375407   1299: (2,4,3,1,1)
%e A375407   1330: (2,3,1,3,2)
%e A375407   1445: (2,1,2,3,2,1)
%e A375407   1577: (1,4,2,3,1)
%e A375407   1602: (1,3,5,2)
%e A375407   1605: (1,3,4,2,1)
%e A375407   1617: (1,3,2,4,1)
%e A375407   1619: (1,3,2,3,1,1)
%t A375407 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A375407 Select[Range[0,1000],MatchQ[stc[#],{___,y_,z_,___,x_,___}/;x<y<z] && MatchQ[stc[#],{___,x_,___,z_,y_,___}/;x<y<z]&]
%Y A375407 The non-dashed version is the intersection of A335482 and A335480.
%Y A375407 Compositions of this type are counted by A375297.
%Y A375407 For leaders of identical runs we have A375408, counted by A332834.
%Y A375407 A003242 counts anti-runs, ranks A333489.
%Y A375407 A011782 counts compositions.
%Y A375407 A238130, A238279, A333755 count compositions by number of runs.
%Y A375407 A335486 ranks compositions matching 21, reverse A335485.
%Y A375407 All of the following pertain to compositions in standard order:
%Y A375407 - Length is A000120.
%Y A375407 - Sum is A029837(n+1).
%Y A375407 - Leader is A065120.
%Y A375407 - Parts are listed by A066099, reverse A228351.
%Y A375407 - Number of adjacent equal pairs is A124762, unequal A333382.
%Y A375407 - Strict compositions are A233564.
%Y A375407 - Constant compositions are A272919.
%Y A375407 - Run-length transform is A333627, sum A070939.
%Y A375407 - Run-counts: A124766, A124765, A124768, A124769, A333381, A124767.
%Y A375407 - Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.
%Y A375407 Cf. A188919, A189076, A238343, A333213, A335466, A373948, A373953, A374633, A375123, A375137, A375139, A374768.
%K A375407 nonn
%O A375407 1,1
%A A375407 _Gus Wiseman_, Aug 23 2024