A347707 Number of distinct possible integer reverse-alternating products of integer partitions of n.

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

Offset: 0

Author

Gus Wiseman, Oct 13 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)). The reverse-alternating product is the alternating product of the reversed sequence.

Examples

			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
		

Crossrefs

The even-length version is A000035.

The non-reverse version is A028310.

The version for factorizations has special cases:

- no changes: A046951

- non-reverse: A046951

- non-integer: A038548

- odd-length: A046951 + A010052

- non-reverse non-integer: A347460

- non-integer odd-length: A347708

- non-reverse odd-length: A046951 + A010052

- non-reverse non-integer odd-length: A347708

The odd-length version is a(n) - A059841(n).

These partitions are counted by A347445, non-reverse A347446.

Counting non-integers gives A347462, non-reverse A347461.

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.

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.

A325534 counts separable partitions, complement A325535.

A345926 counts possible alternating sums of permutations of prime indices.

Cf. A000070, A002033, A002219, A108917, A122768, A325765, A344654, A344740, A347443, A347444, A347448, A347449.

Programs

Mathematica

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