A347445 - OEIS

A347445

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

%I #9 Sep 27 2021 07:55:16

%S 1,1,2,2,4,4,7,8,12,14,20,24,32,40,50,62,77,99,115,151,170,224,251,

%T 331,360,481,517,690,728,980,1020,1379,1420,1918,1962,2643,2677,3630,

%U 3651,4920,4926,6659,6625,8931,8853,11905,11781,15805,15562,20872,20518

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

%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.

%e The a(1) = 1 through a(8) = 12 partitions:

%e (1) (2) (3) (4) (5) (6) (7) (8)

%e (11) (111) (22) (221) (33) (322) (44)

%e (211) (311) (222) (331) (332)

%e (1111) (11111) (411) (421) (422)

%e (2211) (511) (611)

%e (21111) (22111) (2222)

%e (111111) (31111) (3311)

%e (1111111) (22211)

%e (41111)

%e (221111)

%e (2111111)

%e (11111111)

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

%t Table[Length[Select[IntegerPartitions[n],IntegerQ[revaltprod[#]]&]],{n,0,30}]

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

%Y Allowing any reverse-alternating product < 1 gives A344608.

%Y The multiplicative version is A347442, unreversed A347437.

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

%Y Restricting to odd length gives A347444, ranked by A347453.

%Y The unreversed version is A347446, ranked by A347457.

%Y Allowing any reverse-alternating product > 1 gives A347449.

%Y Ranked by A347454.

%Y A000041 counts partitions, with multiplicative version A001055.

%Y A027187 counts partitions of even length.

%Y A027193 counts partitions of odd length.

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

%Y A325534 counts separable partitions, ranked by A335433.

%Y A325535 counts inseparable partitions, ranked by A335448.

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

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

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

%K nonn

%O 0,3

%A _Gus Wiseman_, Sep 14 2021