A353506 - OEIS

%I #12 May 20 2022 08:50:53

%S 1,1,0,0,1,0,2,0,1,0,1,1,2,1,2,0,3,3,2,3,2,0,2,3,2,1,3,1,6,3,2,3,3,2,

%T 3,4,1,2,3,6,3,2,2,3,3,1,2,6,6,4,7,2,3,6,4,3,3,0,4,5,3,5,5,6,5,3,3,3,

%U 6,5,5,6,6,3,3,3,4,4,4,6,7,2,5,7,6,2,3,4,6,11,9,4,4,1,5,6,4,7,9,6,4

%N Number of integer partitions of n whose parts have the same product as their multiplicities.

%e The a(0) = 1 through a(18) = 2 partitions:

%e   n= 0: ()

%e   n= 1: (1)

%e   n= 2:

%e   n= 3:

%e   n= 4: (211)

%e   n= 5:

%e   n= 6: (3111) (2211)

%e   n= 7:

%e   n= 8: (41111)

%e   n= 9:

%e   n=10: (511111)

%e   n=11: (32111111)

%e   n=12: (6111111) (22221111)

%e   n=13: (322111111)

%e   n=14: (71111111) (4211111111)

%e   n=15:

%e   n=16: (811111111) (4411111111) (42211111111)

%e   n=17: (521111111111) (332111111111) (322211111111)

%e   n=18: (9111111111) (333111111111)

%e For example, the partition y = (322111111) has multiplicities (1,2,6) with product 12, and the product of parts is also 3*2*2*1*1*1*1*1*1 = 12, so y is counted under a(13).

%t Table[Length[Select[IntegerPartitions[n], Times@@#==Times@@Length/@Split[#]&]],{n,0,30}]

%o (PARI) a(n) = {my(nb=0); forpart(p=n, my(s=Set(p), v=Vec(p)); if (vecprod(vector(#s, i, #select(x->(x==s[i]), v))) == vecprod(v), nb++);); nb;} \\ _Michel Marcus_, May 20 2022

%Y LHS (product of parts) is ranked by A003963, counted by A339095.

%Y RHS (product of multiplicities) is ranked by A005361, counted by A266477.

%Y For shadows instead of prime exponents we have A008619, ranked by A003586.

%Y Taking sum instead of product of parts gives A266499.

%Y For shadows instead of prime indices we have A353398, ranked by A353399.

%Y These partitions are ranked by A353503.

%Y Taking sum instead of product of multiplicities gives A353698.

%Y A008284 counts partitions by length.

%Y A098859 counts partitions with distinct multiplicities, ranked by A130091.

%Y A353507 gives product of multiplicities (of exponents) in prime signature.

%Y Cf. A085629, A114640, A116608, A118914, A124010, A319000, A325702, A353394, A353500.

%Y Cf. A000792, A266480.

%K nonn

%O 0,7

%A _Gus Wiseman_, May 17 2022

%E a(71)-a(100) from _Alois P. Heinz_, May 20 2022