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%I #11 Dec 18 2021 14:56:30
%S 3,7,10,11,14,15,19,21,23,26,27,28,29,30,31,35,36,39,42,43,47,51,55,
%T 56,57,58,59,60,61,62,63,67,71,73,74,79,83,84,85,86,87,90,91,94,95,99,
%U 100,103,106,111,112,113,114,115,118,119,120,121,122,123,124,125
%N Numbers k such that the k-th composition in standard order is weakly alternating but has at least two adjacent equal parts.
%C We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.
%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 This sequence ranks compositions that are weakly but not strongly alternating.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Alternating_permutation">Alternating permutation</a>
%F Equals A345168 \ A349057 = A348612 \ A349057.
%e The terms and corresponding compositions begin:
%e 3: (1,1)
%e 7: (1,1,1)
%e 10: (2,2)
%e 11: (2,1,1)
%e 14: (1,1,2)
%e 15: (1,1,1,1)
%e 19: (3,1,1)
%e 21: (2,2,1)
%e 23: (2,1,1,1)
%e 26: (1,2,2)
%e 27: (1,2,1,1)
%e 28: (1,1,3)
%e 29: (1,1,2,1)
%e 30: (1,1,1,2)
%e 31: (1,1,1,1,1)
%t stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
%t Select[Range[0,100],(whkQ[stc[#]]||whkQ[-stc[#]])&&MatchQ[stc[#],{___,x_,x_,___}]&]
%Y Partitions of this type are counted by A349795, ranked by A350137.
%Y Permutations of prime indices of this type are counted by A349798.
%Y These compositions are counted by A349800.
%Y A001250 = alternating permutations, ranked by A349051, complement A348615.
%Y A003242 = Carlitz (anti-run) compositions, ranked by A333489.
%Y A025047/A025048/A025049 = alternating compositions, ranked by A345167.
%Y A261983 = non-anti-run compositions, ranked by A348612.
%Y A345164 = alternating permutations of prime indices, with twins A344606.
%Y A345165 = partitions without an alternating permutation, ranked by A345171.
%Y A345170 = partitions with an alternating permutation, ranked by A345172.
%Y A345166 = separable partitions with no alternations, ranked by A345173.
%Y A345192 = non-alternating compositions, ranked by A345168.
%Y A345195 = non-alternating anti-run compositions, ranked by A345169.
%Y A349052/A129852/A129853 = weakly alternating compositions.
%Y A349053 = non-weakly alternating compositions, ranked by A349057.
%Y A349056 = weak alternations of prime indices, complement A349797.
%Y A349060 = weak alternations of partitions, complement A349061.
%Y Cf. A005649, A049774, A096441, A128761, A344615, A345163, A349054, A349058, A349796, A349801, A350140.
%K nonn
%O 1,1
%A _Gus Wiseman_, Dec 15 2021
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