A375407 - OEIS

%I #5 Aug 27 2024 09:14:27

%S 421,649,802,809,837,843,933,1289,1299,1330,1445,1577,1602,1605,1617,

%T 1619,1669,1673,1675,1685,1686,1687,1701,1826,1833,1861,1867,1957,

%U 2469,2569,2577,2579,2597,2598,2599,2610,2658,2661,2674,2697,2850,2857,2885,2891

%N 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 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:

%C (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 (2) the maximal weakly increasing runs in the k-th composition in standard order do not have weakly decreasing leaders.

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

%H Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.

%F Intersection of A375138 and A375137.

%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    421: (1,2,3,2,1)

%e    649: (2,4,3,1)

%e    802: (1,3,4,2)

%e    809: (1,3,2,3,1)

%e    837: (1,2,4,2,1)

%e    843: (1,2,3,2,1,1)

%e    933: (1,1,2,3,2,1)

%e   1289: (2,5,3,1)

%e   1299: (2,4,3,1,1)

%e   1330: (2,3,1,3,2)

%e   1445: (2,1,2,3,2,1)

%e   1577: (1,4,2,3,1)

%e   1602: (1,3,5,2)

%e   1605: (1,3,4,2,1)

%e   1617: (1,3,2,4,1)

%e   1619: (1,3,2,3,1,1)

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

%t Select[Range[0,1000],MatchQ[stc[#],{___,y_,z_,___,x_,___}/;x<y<z] && MatchQ[stc[#],{___,x_,___,z_,y_,___}/;x<y<z]&]

%Y The non-dashed version is the intersection of A335482 and A335480.

%Y Compositions of this type are counted by A375297.

%Y For leaders of identical runs we have A375408, counted by A332834.

%Y A003242 counts anti-runs, ranks A333489.

%Y A011782 counts compositions.

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

%Y A335486 ranks compositions matching 21, reverse A335485.

%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. A188919, A189076, A238343, A333213, A335466, A373948, A373953, A374633, A375123, A375137, A375139, A374768.

%K nonn

%O 1,1

%A _Gus Wiseman_, Aug 23 2024