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A349057 Numbers k such that the k-th composition in standard order is not weakly alternating.

%I A349057 #8 Dec 10 2021 11:13:03
%S A349057 37,46,52,53,69,75,78,92,93,101,104,105,107,110,116,117,133,137,139,
%T A349057 142,150,151,156,157,165,174,180,181,184,185,186,187,190,197,200,201,
%U A349057 203,206,208,209,210,211,214,215,220,221,229,232,233,235,238,244,245,261
%N A349057 Numbers k such that the k-th composition in standard order is not weakly alternating.
%C A349057 We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.
%C A349057 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.
%e A349057 The terms and corresponding compositions begin:
%e A349057    37: (3,2,1)
%e A349057    46: (2,1,1,2)
%e A349057    52: (1,2,3)
%e A349057    53: (1,2,2,1)
%e A349057    69: (4,2,1)
%e A349057    75: (3,2,1,1)
%e A349057    78: (3,1,1,2)
%e A349057    92: (2,1,1,3)
%e A349057    93: (2,1,1,2,1)
%e A349057   101: (1,3,2,1)
%e A349057   104: (1,2,4)
%e A349057   105: (1,2,3,1)
%e A349057   107: (1,2,2,1,1)
%e A349057   110: (1,2,1,1,2)
%e A349057   116: (1,1,2,3)
%e A349057   117: (1,1,2,2,1)
%t A349057 stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A349057 whkQ[y_]:=And@@Table[If[EvenQ[m], y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
%t A349057 Select[Range[0,100],!whkQ[stc[#]]&&!whkQ[-stc[#]]&]
%Y A349057 The strong case is A345168, complement A345167, counted by A345192.
%Y A349057 The strong anti-run case is A345169, counted by A345195.
%Y A349057 Including all non-anti-runs gives A348612, complement A333489.
%Y A349057 These compositions are counted by A349053, complement A349052.
%Y A349057 The directed cases are counted by A129852 (incr.) and A129853 (decr.).
%Y A349057 The complement for patterns is A349058, strong A345194.
%Y A349057 The complement for ordered factorizations is A349059, strong A348610.
%Y A349057 Partitions of this type are counted by A349061, complement A349060.
%Y A349057 Partitions of this type are ranked by A349794.
%Y A349057 Non-strict partitions of this type are counted by A349796.
%Y A349057 Permutations of prime indices of this type are counted by A349797.
%Y A349057 A001250 counts alternating permutations, complement A348615.
%Y A349057 A003242 counts Carlitz (anti-run) compositions, complement A261983.
%Y A349057 A011782 counts compositions.
%Y A349057 A025047 counts alternating/wiggly compositions, directed A025048, A025049.
%Y A349057 A345164 counts alternating permutations of prime indices, weak A349056.
%Y A349057 A345165 counts partitions w/o an alternating permutation, ranked by A345171.
%Y A349057 A345170 counts partitions w/ an alternating permutation, ranked by A345172.
%Y A349057 A349054 counts strict alternating compositions.
%Y A349057 Cf. A001700, A096441, A128761, A344615, A344654, A345173, A348613, A349051, A349794, A349795, A349799.
%K A349057 nonn
%O A349057 1,1
%A A349057 _Gus Wiseman_, Dec 04 2021