OFFSET
0,4
COMMENTS
Suppose that p is a partition of n. Let x(1), x(2), ..., x(k) be the distinct parts of p, and let m(i) be the multiplicity of x(i) in p. Let c(p) be the partition {m(1)*x(1), m(2)*x(2), ..., x(k)*m(k)} of n. Call a partition q of n a condensed partition of n if q = c(p) for some partition p of n. Then a(n) is the number of distinct condensed partitions of n. Note that c(p) = p if and only if p has distinct parts and that condensed partitions can have repeated parts.
Also the number of integer partitions of n such that it is possible to choose a different divisor of each part. For example, the partition (6,4,4,1) has choices (3,2,4,1), (3,4,2,1), (6,2,4,1), (6,4,2,1) so is counted under a(15). - Gus Wiseman, Mar 12 2024
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..100 (first 84 terms from Manfred Scheucher)
Manfred Scheucher, Python Script
EXAMPLE
a(5) = 3 gives the number of partitions of 5 that result from condensations as shown here: 5 -> 5, 41 -> 41, 32 -> 32, 311 -> 32, 221 -> 41, 2111 -> 32, 11111 -> 5.
From Gus Wiseman, Mar 12 2024: (Start)
The a(1) = 1 through a(9) = 10 condensed partitions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(2,1) (2,2) (3,2) (3,3) (4,3) (4,4) (5,4)
(3,1) (4,1) (4,2) (5,2) (5,3) (6,3)
(5,1) (6,1) (6,2) (7,2)
(3,2,1) (3,2,2) (7,1) (8,1)
(4,2,1) (3,3,2) (4,3,2)
(4,2,2) (4,4,1)
(4,3,1) (5,2,2)
(5,2,1) (5,3,1)
(6,2,1)
(End)
MAPLE
b:= proc(n, i) option remember; `if`(n=0, {[]},
`if`(i=1, {[n]}, {seq(map(x-> `if`(j=0, x,
sort([x[], i*j])), b(n-i*j, i-1))[], j=0..n/i)}))
end:
a:= n-> nops(b(n$2)):
seq(a(n), n=0..50); # Alois P. Heinz, Jul 01 2019
MATHEMATICA
u[n_, k_] := u[n, k] = Map[Total, Split[IntegerPartitions[n][[k]]]]; t[n_] := t[n] = DeleteDuplicates[Table[Sort[u[n, k]], {k, 1, PartitionsP[n]}]]; Table[Length[t[n]], {n, 0, 30}]
Table[Length[Select[IntegerPartitions[n], Length[Select[Tuples[Divisors/@#], UnsameQ@@#&]]>0&]], {n, 0, 30}] (* Gus Wiseman, Mar 12 2024 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Mar 15 2014
EXTENSIONS
Typo in definition corrected by Manfred Scheucher, May 29 2015
Name edited by Gus Wiseman, Mar 13 2024
STATUS
approved