A347462 - OEIS

%I #6 Oct 27 2021 22:22:58

%S 1,1,2,3,4,6,8,11,13,17,22,28,33,42,51,59,69,84,100,117,137,163,191,

%T 222,256,290,332,378,429,489,564,643,729,819,929,1040,1167,1313,1473,

%U 1647,1845,2045,2272,2521,2785,3076,3398,3744,4115,4548,5010,5524,6086

%N Number of distinct possible 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 Partitions representing each of the a(7) = 11 reverse-alternating products:

%e      (7) -> 7

%e     (61) -> 1/6

%e     (52) -> 2/5

%e    (511) -> 5

%e     (43) -> 3/4

%e    (421) -> 2

%e   (4111) -> 1/4

%e    (331) -> 1

%e    (322) -> 3

%e   (3211) -> 2/3

%e   (2221) -> 1/2

%t revaltprod[q_]:=Product[Reverse[q][[i]]^(-1)^(i-1),{i,Length[q]}];

%t Table[Length[Union[revaltprod/@IntegerPartitions[n]]],{n,0,30}]

%Y The version for non-reverse alternating sum instead of product is A004526.

%Y Counting only integers gives A028310, non-reverse A347707.

%Y The version for factorizations is A038548, non-reverse A347460.

%Y The non-reverse version is 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 A108917 counts knapsack partitions, ranked by A299702.

%Y A122768 counts distinct submultisets of partitions.

%Y A126796 counts complete partitions.

%Y A293627 counts knapsack factorizations by sum.

%Y A301957 counts distinct subset-products of prime indices.

%Y A304792 counts subset-sums of partitions, positive A276024, strict A284640.

%Y A304793 counts distinct positive subset-sums of prime indices.

%Y A325534 counts separable partitions, ranked by A335433.

%Y A325535 counts inseparable partitions, ranked by A335448.

%Y Cf. A000070, A001055, A002033, A002219, A028983, A119620, A325768, A345926, A347443, A347444, A347445, A347446.

%K nonn

%O 0,3

%A _Gus Wiseman_, Oct 06 2021