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Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum < 0.
28

%I #6 Jul 10 2021 07:59:46

%S 6,12,20,24,25,27,30,40,48,49,51,54,60,72,80,81,83,86,92,96,97,98,99,

%T 101,102,103,106,108,109,111,116,120,121,123,126,144,160,161,163,166,

%U 172,184,192,193,194,195,197,198,199,202,204,205,207,212,216,217,219

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

%C The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

%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.

%e The initial terms and the corresponding compositions:

%e 6: (1,2) 81: (2,4,1)

%e 12: (1,3) 83: (2,3,1,1)

%e 20: (2,3) 86: (2,2,1,2)

%e 24: (1,4) 92: (2,1,1,3)

%e 25: (1,3,1) 96: (1,6)

%e 27: (1,2,1,1) 97: (1,5,1)

%e 30: (1,1,1,2) 98: (1,4,2)

%e 40: (2,4) 99: (1,4,1,1)

%e 48: (1,5) 101: (1,3,2,1)

%e 49: (1,4,1) 102: (1,3,1,2)

%e 51: (1,3,1,1) 103: (1,3,1,1,1)

%e 54: (1,2,1,2) 106: (1,2,2,2)

%e 60: (1,1,1,3) 108: (1,2,1,3)

%e 72: (3,4) 109: (1,2,1,2,1)

%e 80: (2,5) 111: (1,2,1,1,1,1)

%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];

%t Select[Range[0,100],ats[stc[#]]<0&]

%Y The version for Heinz numbers of partitions is A119899.

%Y These are the positions of terms < 0 in A124754.

%Y These compositions are counted by A294175 (even bisection: A008549).

%Y The complement is A345913.

%Y The weak (k <= 0) version is A345915.

%Y The opposite (k < 0) version is A345917.

%Y The version for reversed alternating sum is A345920.

%Y A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.

%Y A011782 counts compositions.

%Y A097805 counts compositions by alternating (or reverse-alternating) sum.

%Y A103919 counts partitions by sum and alternating sum (reverse: A344612).

%Y A236913 counts partitions of 2n with reverse-alternating sum <= 0.

%Y A316524 gives the alternating sum of prime indices (reverse: A344616).

%Y A345197 counts compositions by sum, length, and alternating sum.

%Y Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.

%Y Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:

%Y - k = 0: counted by A088218, ranked by A344619/A344619.

%Y - k = 1: counted by A000984, ranked by A345909/A345911.

%Y - k = -1: counted by A001791, ranked by A345910/A345912.

%Y - k = 2: counted by A088218, ranked by A345925/A345922.

%Y - k = -2: counted by A002054, ranked by A345924/A345923.

%Y - k >= 0: counted by A116406, ranked by A345913/A345914.

%Y - k <= 0: counted by A058622(n-1), ranked by A345915/A345916.

%Y - k > 0: counted by A027306, ranked by A345917/A345918.

%Y - k < 0: counted by A294175, ranked by A345919/A345920.

%Y - k != 0: counted by A058622, ranked by A345921/A345921.

%Y - k even: counted by A081294, ranked by A053754/A053754.

%Y - k odd: counted by A000302, ranked by A053738/A053738.

%Y Cf. A000070, A000346, A025047, A028260, A032443, A034871, A106356, A114121, A163493, A344608, A344610, A344611, A345908.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jul 09 2021