A347446 - OEIS

A347446

Number of integer partitions of n with integer alternating product.

%I #9 Sep 27 2021 07:55:26

%S 1,1,2,3,5,6,10,12,18,22,31,37,54,62,84,100,134,157,207,241,314,363,

%T 463,537,685,785,985,1138,1410,1616,1996,2286,2801,3201,3885,4434,

%U 5363,6098,7323,8329,9954,11293,13430,15214,18022,20383,24017,27141,31893,35960

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

%C We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

%e The a(1) = 1 through a(7) = 12 partitions:

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

%e (11) (21) (22) (41) (33) (61)

%e (111) (31) (221) (42) (322)

%e (211) (311) (51) (331)

%e (1111) (2111) (222) (421)

%e (11111) (411) (511)

%e (2211) (2221)

%e (3111) (4111)

%e (21111) (22111)

%e (111111) (31111)

%e (211111)

%e (1111111)

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

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

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

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

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

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

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

%Y The reverse version is A347445, ranked by A347454.

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

%Y Ranked by A347457.

%Y The even-length case is A347704.

%Y A000041 counts partitions.

%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 A347461 counts possible alternating products of partitions.

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

%K nonn

%O 0,3

%A _Gus Wiseman_, Sep 15 2021