The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A353506

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A353506

Number of integer partitions of n whose parts have the same product as their multiplicities.
(history; published version)

#12 by Michael De Vlieger at Fri May 20 08:50:53 EDT 2022

STATUS

proposed

approved

#11 by Michel Marcus at Fri May 20 04:19:15 EDT 2022

STATUS

editing

proposed

#10 by Michel Marcus at Fri May 20 04:19:06 EDT 2022

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

STATUS

proposed

editing

#9 by Alois P. Heinz at Fri May 20 02:57:26 EDT 2022

STATUS

editing

proposed

#8 by Alois P. Heinz at Fri May 20 02:57:13 EDT 2022

CROSSREFS

Cf. A000792, A266480.

STATUS

proposed

editing

#7 by Alois P. Heinz at Fri May 20 02:55:17 EDT 2022

STATUS

editing

proposed

#6 by Alois P. Heinz at Fri May 20 02:54:55 EDT 2022

DATA

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

KEYWORD

nonn,more,changed

EXTENSIONS

a(71)-a(100) from Alois P. Heinz, May 20 2022

STATUS

proposed

editing

#5 by Gus Wiseman at Fri May 20 01:55:08 EDT 2022

STATUS

editing

proposed

#4 by Gus Wiseman at Fri May 20 01:53:57 EDT 2022

MATHEMATICA

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

CROSSREFS

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

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

#3 by Gus Wiseman at Fri May 20 01:52:24 EDT 2022

EXAMPLE

The a(28) = 6 partitions:

(14,1,1,1,1,1,1,1,1,1,1,1,1,1,1)

(8,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)

(6,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)

(4,4,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)

(4,3,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)

(4,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)

The a(39) = 6 partitions:

(9,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)

(9,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)

(7,4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)

(6,3,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)

(5,5,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)

(3,3,3,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)

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).

CROSSREFS

The LHS (product of parts) is ranked by A003963, counted by A339095 (partial transpose A319000).

The RHS (product of multiplicities) is ranked by A005361, firsts A353500 (sorted A085629), counted by A266477.

The version for For shadows instead of prime exponents is we have A008619, ranked by A003586.

The version for For shadows instead of prime indices is we have A353398, ranked by A353399.

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