A347449 Number of integer partitions of n with reverse-alternating product > 1.

0, 0, 1, 1, 2, 2, 5, 5, 10, 11, 20, 22, 37, 41, 66, 75, 113, 129, 190, 218, 310, 358, 497, 576, 782, 908, 1212, 1411, 1851, 2156, 2793, 3255, 4163, 4853, 6142, 7159, 8972, 10451, 12989, 15123, 18646, 21689, 26561, 30867, 37556, 43599, 52743, 61161, 73593

Offset: 0

Author

Gus Wiseman, Sep 16 2021

Keywords

• nonn

Comments

All such partitions have odd length.

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(2) = 1 through a(9) = 11 partitions:
  (2)  (3)  (4)    (5)    (6)      (7)      (8)        (9)
            (211)  (311)  (222)    (322)    (332)      (333)
                          (321)    (421)    (422)      (432)
                          (411)    (511)    (431)      (522)
                          (21111)  (31111)  (521)      (531)
                                            (611)      (621)
                                            (22211)    (711)
                                            (32111)    (32211)
                                            (41111)    (42111)
                                            (2111111)  (51111)
                                                       (3111111)
		

Crossrefs

The strict case is A067659, except that a(0) = a(1) = 0.

The even bisection is A236559.

The non-reverse multiplicative version is A339890, weak A347456.

The case of >= 1 instead of > 1 is A344607.

The opposite version is A344608, also the non-reverse even-length case.

The complement is counted by A347443, non-reverse A119620.

Allowing any integer reverse-alternating product gives A347445.

Allowing any integer alternating product gives A347446.

Reverse version of A347448; also the odd-length case.

The Heinz numbers of these partitions are the complement of A347450.

The multiplicative version (factorizations) is A347705.

A000041 counts partitions.

A027187 counts partitions of even length.

A027193 counts partitions of odd length.

A100824 counts partitions of n with alternating sum <= 1.

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

A347462 counts possible reverse-alternating products of partitions.

Cf. A000070, A008549, A086543, A182616, A236913, A325534, A325535, A344611, A347442, A347444, A347447, A347453, A347461, A347465.

Programs

Mathematica

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

Formula

a(n) = A344607(n) - A119620(n).