A347445 Number of integer partitions of n with integer reverse-alternating product.

1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 20, 24, 32, 40, 50, 62, 77, 99, 115, 151, 170, 224, 251, 331, 360, 481, 517, 690, 728, 980, 1020, 1379, 1420, 1918, 1962, 2643, 2677, 3630, 3651, 4920, 4926, 6659, 6625, 8931, 8853, 11905, 11781, 15805, 15562, 20872, 20518

Offset: 0

Author

Gus Wiseman, Sep 14 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

			The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (221)    (33)      (322)      (44)
                    (211)   (311)    (222)     (331)      (332)
                    (1111)  (11111)  (411)     (421)      (422)
                                     (2211)    (511)      (611)
                                     (21111)   (22111)    (2222)
                                     (111111)  (31111)    (3311)
                                               (1111111)  (22211)
                                                          (41111)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)
		

Crossrefs

Allowing any reverse-alternating product >= 1 gives A344607.

Allowing any reverse-alternating product < 1 gives A344608.

The multiplicative version is A347442, unreversed A347437.

Allowing any reverse-alternating product <= 1 gives A347443.

Restricting to odd length gives A347444, ranked by A347453.

The unreversed version is A347446, ranked by A347457.

Allowing any reverse-alternating product > 1 gives A347449.

Ranked by A347454.

A000041 counts partitions, with multiplicative version A001055.

A027187 counts partitions of even length.

A027193 counts partitions of odd length.

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

A325534 counts separable partitions, ranked by A335433.

A325535 counts inseparable partitions, ranked by A335448.

A339890 counts factorizations with alternating product > 1, reverse A347705.

A347462 counts possible reverse-alternating products of partitions.

Cf. A025047, A067661, A119620, A344654, A344740, A347439, A347440, A347448, A347450, A347451, A347461, A347463, A347704.

Programs

Mathematica

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