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A345919 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum < 0.

%I A345919 #6 Jul 10 2021 07:59:46
%S A345919 6,12,20,24,25,27,30,40,48,49,51,54,60,72,80,81,83,86,92,96,97,98,99,
%T A345919 101,102,103,106,108,109,111,116,120,121,123,126,144,160,161,163,166,
%U A345919 172,184,192,193,194,195,197,198,199,202,204,205,207,212,216,217,219
%N A345919 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum < 0.
%C A345919 The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
%C A345919 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 A345919 The initial terms and the corresponding compositions:
%e A345919       6: (1,2)         81: (2,4,1)
%e A345919      12: (1,3)         83: (2,3,1,1)
%e A345919      20: (2,3)         86: (2,2,1,2)
%e A345919      24: (1,4)         92: (2,1,1,3)
%e A345919      25: (1,3,1)       96: (1,6)
%e A345919      27: (1,2,1,1)     97: (1,5,1)
%e A345919      30: (1,1,1,2)     98: (1,4,2)
%e A345919      40: (2,4)         99: (1,4,1,1)
%e A345919      48: (1,5)        101: (1,3,2,1)
%e A345919      49: (1,4,1)      102: (1,3,1,2)
%e A345919      51: (1,3,1,1)    103: (1,3,1,1,1)
%e A345919      54: (1,2,1,2)    106: (1,2,2,2)
%e A345919      60: (1,1,1,3)    108: (1,2,1,3)
%e A345919      72: (3,4)        109: (1,2,1,2,1)
%e A345919      80: (2,5)        111: (1,2,1,1,1,1)
%t A345919 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A345919 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A345919 Select[Range[0,100],ats[stc[#]]<0&]
%Y A345919 The version for Heinz numbers of partitions is A119899.
%Y A345919 These are the positions of terms < 0 in A124754.
%Y A345919 These compositions are counted by A294175 (even bisection: A008549).
%Y A345919 The complement is A345913.
%Y A345919 The weak (k <= 0) version is A345915.
%Y A345919 The opposite (k < 0) version is A345917.
%Y A345919 The version for reversed alternating sum is A345920.
%Y A345919 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A345919 A011782 counts compositions.
%Y A345919 A097805 counts compositions by alternating (or reverse-alternating) sum.
%Y A345919 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A345919 A236913 counts partitions of 2n with reverse-alternating sum <= 0.
%Y A345919 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A345919 A345197 counts compositions by sum, length, and alternating sum.
%Y A345919 Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
%Y A345919 Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
%Y A345919 - k = 0:  counted by A088218, ranked by A344619/A344619.
%Y A345919 - k = 1:  counted by A000984, ranked by A345909/A345911.
%Y A345919 - k = -1: counted by A001791, ranked by A345910/A345912.
%Y A345919 - k = 2:  counted by A088218, ranked by A345925/A345922.
%Y A345919 - k = -2: counted by A002054, ranked by A345924/A345923.
%Y A345919 - k >= 0: counted by A116406, ranked by A345913/A345914.
%Y A345919 - k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
%Y A345919 - k > 0:  counted by A027306, ranked by A345917/A345918.
%Y A345919 - k < 0:  counted by A294175, ranked by A345919/A345920.
%Y A345919 - k != 0: counted by A058622, ranked by A345921/A345921.
%Y A345919 - k even: counted by A081294, ranked by A053754/A053754.
%Y A345919 - k odd:  counted by A000302, ranked by A053738/A053738.
%Y A345919 Cf. A000070, A000346, A025047, A028260, A032443, A034871, A106356, A114121, A163493, A344608, A344610, A344611, A345908.
%K A345919 nonn
%O A345919 1,1
%A A345919 _Gus Wiseman_, Jul 09 2021