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A375408 Numbers k such that the k-th composition in standard order is not weakly increasing or weakly decreasing.

%I A375408 #11 Feb 19 2025 11:57:53
%S A375408 13,22,25,27,29,38,41,44,45,46,49,50,51,53,54,55,57,59,61,70,76,77,78,
%T A375408 81,82,83,86,88,89,90,91,92,93,94,97,98,99,101,102,103,105,107,108,
%U A375408 109,110,111,113,114,115,117,118,119,121,123,125,134,140,141,142
%N A375408 Numbers k such that the k-th composition in standard order is not weakly increasing or weakly decreasing.
%C A375408 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.
%H A375408 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>.
%F A375408 Intersection of A335485 and A335486.
%e A375408 The terms and corresponding compositions begin:
%e A375408   13: (1,2,1)
%e A375408   22: (2,1,2)
%e A375408   25: (1,3,1)
%e A375408   27: (1,2,1,1)
%e A375408   29: (1,1,2,1)
%e A375408   38: (3,1,2)
%e A375408   41: (2,3,1)
%e A375408   44: (2,1,3)
%e A375408   45: (2,1,2,1)
%e A375408   46: (2,1,1,2)
%e A375408   49: (1,4,1)
%e A375408   50: (1,3,2)
%e A375408   51: (1,3,1,1)
%e A375408   53: (1,2,2,1)
%e A375408   54: (1,2,1,2)
%e A375408   55: (1,2,1,1,1)
%e A375408   57: (1,1,3,1)
%e A375408   59: (1,1,2,1,1)
%t A375408 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A375408 Select[Range[0,100],!LessEqual@@stc[#]&&!GreaterEqual@@stc[#]&]
%Y A375408 The version for run-lengths of compositions is A332833.
%Y A375408 Compositions of this type are counted by A332834, complement maybe A329398.
%Y A375408 A001523 counts unimodal compositions, ranks too dense.
%Y A375408 A011782 counts compositions.
%Y A375408 A114994 ranks weakly decreasing compositions, complement A335485.
%Y A375408 A115981 counts non-unimodal compositions, ranked by A335373.
%Y A375408 A225620 ranks weakly increasing compositions, complement A335486.
%Y A375408 A238130, A238279, A333755 count compositions by number of runs.
%Y A375408 A332835 counts compositions with weakly incr. or weakly decr. run-lengths.
%Y A375408 All of the following pertain to compositions in standard order:
%Y A375408 - Length is A000120.
%Y A375408 - Sum is A029837(n+1).
%Y A375408 - Parts are listed by A066099.
%Y A375408 - Number of adjacent equal pairs is A124762, unequal A333382.
%Y A375408 - Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
%Y A375408 - Ranks of strict compositions are A233564.
%Y A375408 - Ranks of constant compositions are A272919.
%Y A375408 - Anti-runs are ranked by A333489, counted by A003242.
%Y A375408 - Run-length transform is A333627, sum A070939.
%Y A375408 Cf. A001511, A002051, A065120, A329744, A332745, A332836, A332870, A333218.
%K A375408 nonn
%O A375408 1,1
%A A375408 _Gus Wiseman_, Sep 18 2024