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%I #6 Nov 22 2021 19:59:40
%S 0,12,160,193,195,198,204,216,240,2304,2561,2563,2566,2572,2584,2608,
%T 2656,2752,2944,3074,3077,3079,3082,3085,3087,3092,3097,3099,3102,
%U 3112,3121,3123,3126,3132,3152,3169,3171,3174,3180,3192,3232,3265,3267,3270,3276
%N Numbers k such that the k-th composition in standard order has sum equal to negative twice its alternating sum.
%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 The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
%e The terms and corresponding compositions begin:
%e 0: ()
%e 12: (1,3)
%e 160: (2,6)
%e 193: (1,6,1)
%e 195: (1,5,1,1)
%e 198: (1,4,1,2)
%e 204: (1,3,1,3)
%e 216: (1,2,1,4)
%e 240: (1,1,1,5)
%e 2304: (3,9)
%e 2561: (2,9,1)
%e 2563: (2,8,1,1)
%e 2566: (2,7,1,2)
%e 2572: (2,6,1,3)
%e 2584: (2,5,1,4)
%t ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t Select[Range[0,1000],Total[stc[#]]==-2*ats[stc[#]]&]
%Y These compositions are counted by A224274 up to 0's.
%Y Except for 0, a subset of A345919.
%Y The positive version is A348614, reverse A349153.
%Y An unordered version is A348617, counted by A001523.
%Y The reverse version is A349155.
%Y A positive unordered version is A349159, counted by A000712 up to 0's.
%Y A000346 = even-length compositions with alt sum != 0, complement A001700.
%Y A003242 counts Carlitz compositions.
%Y A011782 counts compositions.
%Y A025047 counts alternating or wiggly compositions, complement A345192.
%Y A034871, A097805, and A345197 count compositions by alternating sum.
%Y A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A116406 counts compositions with alternating sum >=0, ranked by A345913.
%Y A138364 counts compositions with alternating sum 0, ranked by A344619.
%Y Cf. A000070, A000984, A008549, A027306, A058622, A088218, A114121, A120452, A262977, A294175, A345917, A349160.
%Y Statistics of standard compositions:
%Y - The compositions themselves are the rows of A066099.
%Y - Number of parts is given by A000120, distinct A334028.
%Y - Sum and product of parts are given by A070939 and A124758.
%Y - Maximum and minimum parts are given by A333766 and A333768.
%Y Classes of standard compositions:
%Y - Partitions and strict partitions are ranked by A114994 and A333256.
%Y - Multisets and sets are ranked by A225620 and A333255.
%Y - Strict and constant compositions are ranked by A233564 and A272919.
%Y - Carlitz compositions are ranked by A333489, complement A348612.
%Y - Necklaces are ranked by A065609, dual A333764, reversed A333943.
%Y - Alternating compositions are ranked by A345167, complement A345168.
%K nonn
%O 1,2
%A _Gus Wiseman_, Nov 21 2021
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