A347446 Number of integer partitions of n with integer alternating product.

1, 1, 2, 3, 5, 6, 10, 12, 18, 22, 31, 37, 54, 62, 84, 100, 134, 157, 207, 241, 314, 363, 463, 537, 685, 785, 985, 1138, 1410, 1616, 1996, 2286, 2801, 3201, 3885, 4434, 5363, 6098, 7323, 8329, 9954, 11293, 13430, 15214, 18022, 20383, 24017, 27141, 31893, 35960

Offset: 0

Author

Gus Wiseman, Sep 15 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

			The a(1) = 1 through a(7) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (41)     (33)      (61)
             (111)  (31)    (221)    (42)      (322)
                    (211)   (311)    (51)      (331)
                    (1111)  (2111)   (222)     (421)
                            (11111)  (411)     (511)
                                     (2211)    (2221)
                                     (3111)    (4111)
                                     (21111)   (22111)
                                     (111111)  (31111)
                                               (211111)
                                               (1111111)
		

Crossrefs

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

Allowing any alternating product <= 1 gives A119620, reverse A347443.

Allowing any reverse-alternating product < 1 gives A344608.

The multiplicative version (factorizations) is A347437, reverse A347442.

The odd-length case is A347444, ranked by A347453.

The reverse version is A347445, ranked by A347454.

Allowing any alternating product > 1 gives A347448, reverse A347449.

Ranked by A347457.

The even-length case is A347704.

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).

A325534 counts separable partitions, ranked by A335433.

A325535 counts inseparable partitions, ranked by A335448.

A347461 counts possible alternating products of partitions.

Cf. A025047, A067661, A339890, A347450, A344654, A344740, A347439, A347440, A347451, A347463, A347705.

Programs

Mathematica

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