A347461 Number of distinct possible alternating products of integer partitions of n.

1, 1, 2, 3, 4, 6, 7, 10, 12, 16, 19, 23, 27, 34, 41, 49, 57, 67, 78, 91, 106, 125, 147, 166, 187, 215, 245, 277, 317, 357, 405, 460, 524, 592, 666, 740, 829, 928, 1032, 1147, 1273, 1399, 1555, 1713, 1892, 2087, 2298, 2523, 2783, 3070, 3383, 3724, 4104, 4504

Offset: 0

Author

Gus Wiseman, Oct 06 2021

Keywords

• nonn

Comments

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

Examples

			Partitions representing each of the a(7) = 10 alternating products are:
     (7) -> 7
    (61) -> 6
    (52) -> 5/2
   (511) -> 5
    (43) -> 4/3
   (421) -> 2
  (4111) -> 4
   (331) -> 1
   (322) -> 3
  (3211) -> 3/2
		

Crossrefs

The version for alternating sum is A004526.

Counting only integers gives A028310, reverse A347707.

The version for factorizations is A347460, reverse A038548.

The reverse version is A347462.

A000041 counts partitions.

A027187 counts partitions of even length.

A027193 counts partitions of odd length.

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

A108917 counts knapsack partitions, ranked by A299702.

A122768 counts distinct submultisets of partitions.

A126796 counts complete partitions.

A293627 counts knapsack factorizations by sum.

A301957 counts distinct subset-products of prime indices.

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

A304793 counts distinct positive subset-sums of prime indices.

A325534 counts separable partitions, ranked by A335433.

A325535 counts inseparable partitions, ranked by A335448.

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

Programs

Mathematica

altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Union[altprod/@IntegerPartitions[n]]],{n,0,30}]