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The version for Heinz numbers of partitions is A344609, counted by A344607.

The version for Heinz numbers of partitions is A344609, counted by A344607.

The opposite (k <= 0) version is A345916.

The strict (k > 0) case is A345918.

The complement (< 0) is A345920, counted by A294175.

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

The version for Heinz numbers of partitions is A344609, counted by A344607; the complement is A119899, counted by A344608.

The case of non-Heinz numbers of partitions is A119899, counted by A344608.

A345197 counts compositions by sum, length, and alternating sum (anti-sums: A163493, anti: A345907, trace: A345908).

Cf. A000070, A000346, ~A007318, A008549, `A008965, A025047, A027187, `A027193, A032443, A034871, ~A058696, `A106356, A114121, `A120452, A163493, A238279, `A239830, ~A344614, A344650, A344743.

The version for non-Heinz numbers of partitions is A344609, counted by A344607; the complement is A119899, counted by A344608.

The version for Heinz numbers of partitions is A344609, counted by A344607.

A120452 counts partitions of 2n with reverse-alternating sum 2 (negative: A344741).

Cf. A000070, A000346, ~A007318, A008549, `A008965, A025047, A027187, `A027193, A032443, A034871, ~A058696, `A106356, A114121, `A120452, A238279, `A239830, ~A344614, A344650, A344743.

allocated for Gus WisemanNumbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum >= 0.

0, 1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 24, 26, 27, 28, 30, 31, 32, 35, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 67, 69, 70, 72, 73, 74, 76, 79, 80, 82, 83, 84, 86, 87, 88

1,3

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

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.

The sequence of terms together with the corresponding compositions begins:

0: () 19: (3,1,1) 40: (2,4)

1: (1) 20: (2,3) 41: (2,3,1)

2: (2) 21: (2,2,1) 42: (2,2,2)

3: (1,1) 22: (2,1,2) 43: (2,2,1,1)

4: (3) 24: (1,4) 44: (2,1,3)

6: (1,2) 26: (1,2,2) 46: (2,1,1,2)

7: (1,1,1) 27: (1,2,1,1) 47: (2,1,1,1,1)

8: (4) 28: (1,1,3) 48: (1,5)

10: (2,2) 30: (1,1,1,2) 50: (1,3,2)

11: (2,1,1) 31: (1,1,1,1,1) 51: (1,3,1,1)

12: (1,3) 32: (6) 52: (1,2,3)

13: (1,2,1) 35: (4,1,1) 53: (1,2,2,1)

14: (1,1,2) 36: (3,3) 54: (1,2,1,2)

15: (1,1,1,1) 37: (3,2,1) 55: (1,2,1,1,1)

16: (5) 38: (3,1,2) 56: (1,1,4)

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

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

Select[Range[0, 100], sats[stc[#]]>=0&]

The version for prime indices is A000027, counted by A000041.

These compositions are counted by A116406.

The version for non-Heinz numbers of partitions is A119899, counted by A344608.

The version for Heinz numbers of partitions is A344609, counted by A344607.

These are the positions of terms >= 0 in A344618.

The version for unreversed alternating sum is A345913.

The opposite (<= 0) version is A345916.

The strict (> 0) case is A345918.

The complement (< 0) is A345920.

A011782 counts compositions.

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

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

A120452 counts partitions of 2n with reverse-alternating sum 2 (negative: A344741).

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

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

A344610 counts partitions by sum and positive reverse-alternating sum.

A344611 counts partitions of 2n with reverse-alternating sum >= 0.

A345197 counts compositions by sum, length, and alternating sum (anti-sums: A163493, anti: A345907, trace: A345908).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Cf. A000070, A000346, ~A007318, A008549, `A008965, A025047, A027187, `A027193, A032443, A034871, ~A058696, `A106356, A114121, A238279, `A239830, ~A344614, A344650, A344743.

allocated

nonn

Gus Wiseman, Jul 04 2021

approved

editing

allocated for Gus Wiseman

allocated

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