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%I #6 Oct 27 2021 22:24:08
%S 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,
%T 20,22,21,25,23,26,25,28,26,29,27,31,29,32,31,34,33,35,34,38,35,41,37,
%U 42,40,43,41,45,42,46,44,48,45,50,46,52,49,53
%N Number of distinct possible integer reverse-alternating products of integer partitions of n.
%C 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.
%e Representative partitions for each of the a(16) = 11 alternating products:
%e (16) -> 16
%e (14,1,1) -> 14
%e (12,2,2) -> 12
%e (10,3,3) -> 10
%e (8,4,4) -> 8
%e (9,3,2,1,1) -> 6
%e (10,4,2) -> 5
%e (12,3,1) -> 4
%e (6,4,2,2,2) -> 3
%e (10,5,1) -> 2
%e (8,8) -> 1
%t revaltprod[q_]:=Product[Reverse[q][[i]]^(-1)^(i-1),{i,Length[q]}];
%t Table[Length[Select[Union[revaltprod/@IntegerPartitions[n]],IntegerQ]],{n,0,30}]
%Y The even-length version is A000035.
%Y The non-reverse version is A028310.
%Y The version for factorizations has special cases:
%Y - no changes: A046951
%Y - non-reverse: A046951
%Y - non-integer: A038548
%Y - odd-length: A046951 + A010052
%Y - non-reverse non-integer: A347460
%Y - non-integer odd-length: A347708
%Y - non-reverse odd-length: A046951 + A010052
%Y - non-reverse non-integer odd-length: A347708
%Y The odd-length version is a(n) - A059841(n).
%Y These partitions are counted by A347445, non-reverse A347446.
%Y Counting non-integers gives A347462, non-reverse A347461.
%Y A000041 counts partitions.
%Y A027187 counts partitions of even length.
%Y A027193 counts partitions of odd length.
%Y A103919 counts partitions by sum and alternating sum, reverse A344612.
%Y A119620 counts partitions with alternating product 1, ranked by A028982.
%Y A276024 counts distinct positive subset-sums of partitions, strict A284640.
%Y A304792 counts distinct subset-sums of partitions.
%Y A325534 counts separable partitions, complement A325535.
%Y A345926 counts possible alternating sums of permutations of prime indices.
%Y Cf. A000070, A002033, A002219, A108917, A122768, A325765, A344654, A344740, A347443, A347444, A347448, A347449.
%K nonn
%O 0,3
%A _Gus Wiseman_, Oct 13 2021
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