A347449 - OEIS

%I #10 Sep 27 2021 07:56:01

%S 0,0,1,1,2,2,5,5,10,11,20,22,37,41,66,75,113,129,190,218,310,358,497,

%T 576,782,908,1212,1411,1851,2156,2793,3255,4163,4853,6142,7159,8972,

%U 10451,12989,15123,18646,21689,26561,30867,37556,43599,52743,61161,73593

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

%C All such partitions have odd length.

%C 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.

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

%e The a(2) = 1 through a(9) = 11 partitions:

%e   (2)  (3)  (4)    (5)    (6)      (7)      (8)        (9)

%e             (211)  (311)  (222)    (322)    (332)      (333)

%e                           (321)    (421)    (422)      (432)

%e                           (411)    (511)    (431)      (522)

%e                           (21111)  (31111)  (521)      (531)

%e                                             (611)      (621)

%e                                             (22211)    (711)

%e                                             (32111)    (32211)

%e                                             (41111)    (42111)

%e                                             (2111111)  (51111)

%e                                                        (3111111)

%t altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];

%t Table[Length[Select[IntegerPartitions[n],altprod[Reverse[#]]>1&]],{n,0,30}]

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

%Y The even bisection is A236559.

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

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

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

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

%Y Allowing any integer reverse-alternating product gives A347445.

%Y Allowing any integer alternating product gives A347446.

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

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

%Y The multiplicative version (factorizations) is A347705.

%Y A000041 counts partitions.

%Y A027187 counts partitions of even length.

%Y A027193 counts partitions of odd length.

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

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

%Y A347462 counts possible reverse-alternating products of partitions.

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

%K nonn

%O 0,5

%A _Gus Wiseman_, Sep 16 2021