A353504 -id:A353504 - cp's OEIS frontend

A353698 Number of integer partitions of n whose product equals their length.

0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 2, 1, 2, 0, 2, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 3, 0, 3, 2, 2, 1, 5, 0, 1, 2, 5, 1, 4, 0, 3, 3, 2, 1, 4, 2, 3, 2, 2, 0, 5, 1, 4, 2, 2, 3, 6, 1, 2, 2, 5, 1, 4, 0, 4, 3, 3, 1, 6, 2, 3, 4, 4, 2, 4, 1, 4, 2, 3, 1, 8, 2, 4, 2, 4, 2, 5, 2, 4, 2

Offset: 0

Gus Wiseman, May 19 2022

nonn

			The a(n) partitions for selected n (A..H = 10..17):
n=9:    n=21:             n=27:                 n=33:
---------------------------------------------------------------------------
51111   B1111111111       E1111111111111        H1111111111111111
321111  72111111111111    921111111111111111    B211111111111111111111
        531111111111111   54111111111111111111  831111111111111111111111
        4221111111111111                        5511111111111111111111111
                                                333111111111111111111111111

Andrew Howroyd, Table of n, a(n) for n = 0..10000

The LHS (product of parts) is counted by A339095, rank statistic A003963.
The RHS (length) is counted by A008284, rank statistic A001222.
These partitions are ranked by A353699.
A266477 counts partitions by product of multiplicities, rank stat A005361.
A353504 counts partitions w/ product less than product of multiplicities.
A353505 counts partitions w/ product greater than product of multiplicities.
A353506 counts partitions w/ prod equal to prod of mults, ranked by A353503.
Cf. A000041, A002033, A098859, A114640, A181819, A225485, A266499, A319000, A325280, A353398, A353507.

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

a(r,m=r,p=1,k=0) = {(p==k+r) + sum(m=2, min(m, (k+r)\p),  self()(r-m, min(m,r-m), p*m, k+1))} \\ Andrew Howroyd, Jan 02 2023

Terms a(61) and beyond from Andrew Howroyd, Jan 02 2023

A353505 Number of integer partitions of n whose product is greater than the product of their multiplicities.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 7, 11, 17, 24, 35, 47, 66, 89, 121, 162, 214, 276, 362, 464, 599, 763, 971, 1219, 1537, 1918, 2393, 2966, 3668, 4512, 5549, 6784, 8287, 10076, 12238, 14807, 17898, 21556, 25931, 31094, 37243, 44486, 53075, 63158, 75069, 89025, 105447, 124636

Offset: 0

Gus Wiseman, May 19 2022

nonn

			The a(0) = 0 through a(7) = 11 partitions:
  .  .  (2)  (3)   (4)   (5)    (6)    (7)
             (21)  (22)  (32)   (33)   (43)
                   (31)  (41)   (42)   (52)
                         (221)  (51)   (61)
                         (311)  (222)  (322)
                                (321)  (331)
                                (411)  (421)
                                       (511)
                                       (2221)
                                       (3211)
                                       (4111)

RHS (product of multiplicities) is counted by A266477, ranked by A005361.
LHS (product of parts) is counted by A339095, ranked by A003963.
The version for less instead of greater is A353504.
The version for equality is A353506, ranked by A353503.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A353398 counts partitions with the same products of multiplicities as of shadows, ranked by A353399.
Cf. A002033, A008284, A008619, A085629, A097318, A098859, A114640, A116608, A130091, A304678, A353394, A353507.

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

cp's OEIS Frontend

A353698 Number of integer partitions of n whose product equals their length.

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