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Number of distinct possible integer reverse-alternating products of integer partitions of n.
4

%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