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A347707
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Number of distinct possible integer reverse-alternating products of integer partitions of n.
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4
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1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 8, 8, 9, 9, 11, 11, 13, 12, 14, 14, 15, 15, 18, 17, 19, 18, 20, 20, 22, 21, 25, 23, 26, 25, 28, 26, 29, 27, 31, 29, 32, 31, 34, 33, 35, 34, 38, 35, 41, 37, 42, 40, 43, 41, 45, 42, 46, 44, 48, 45, 50, 46, 52, 49, 53
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OFFSET
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0,3
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COMMENTS
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We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). The reverse-alternating product is the alternating product of the reversed sequence.
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LINKS
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EXAMPLE
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Representative partitions for each of the a(16) = 11 alternating products:
(16) -> 16
(14,1,1) -> 14
(12,2,2) -> 12
(10,3,3) -> 10
(8,4,4) -> 8
(9,3,2,1,1) -> 6
(10,4,2) -> 5
(12,3,1) -> 4
(6,4,2,2,2) -> 3
(10,5,1) -> 2
(8,8) -> 1
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MATHEMATICA
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revaltprod[q_]:=Product[Reverse[q][[i]]^(-1)^(i-1), {i, Length[q]}];
Table[Length[Select[Union[revaltprod/@IntegerPartitions[n]], IntegerQ]], {n, 0, 30}]
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CROSSREFS
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The even-length version is A000035.
The non-reverse version is A028310.
The version for factorizations has special cases:
- non-reverse non-integer odd-length: A347708
The odd-length version is a(n) - A059841(n).
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A103919 counts partitions by sum and alternating sum, reverse A344612.
A119620 counts partitions with alternating product 1, ranked by A028982.
A276024 counts distinct positive subset-sums of partitions, strict A284640.
A304792 counts distinct subset-sums of partitions.
A345926 counts possible alternating sums of permutations of prime indices.
Cf. A000070, A002033, A002219, A108917, A122768, A325765, A344654, A344740, A347443, A347444, A347448, A347449.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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