A353506 - OEIS

A353506

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

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, 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, 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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,7

LINKS

Table of n, a(n) for n=0..100.

EXAMPLE

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

n= 0: ()

n= 1: (1)

n= 2:

n= 3:

n= 4: (211)

n= 5:

n= 6: (3111) (2211)

n= 7:

n= 8: (41111)

n= 9:

n=10: (511111)

n=11: (32111111)

n=12: (6111111) (22221111)

n=13: (322111111)

n=14: (71111111) (4211111111)

n=15:

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

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

n=18: (9111111111) (333111111111)

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).

MATHEMATICA

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

PROG

(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

CROSSREFS

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

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

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

Taking sum instead of product of parts gives A266499.

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

These partitions are ranked by A353503.

Taking sum instead of product of multiplicities gives A353698.

A008284 counts partitions by length.

A098859 counts partitions with distinct multiplicities, ranked by A130091.

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

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