A239312 Number of condensed integer partitions of n.

1, 1, 1, 2, 3, 3, 5, 6, 9, 10, 14, 16, 23, 27, 33, 41, 51, 62, 75, 93, 111, 134, 159, 189, 226, 271, 317, 376, 445, 520, 609, 714, 832, 972, 1129, 1304, 1520, 1753, 2023, 2326, 2692, 3077, 3540, 4050, 4642, 5298, 6054, 6887, 7854, 8926, 10133, 11501, 13044

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

The strict case is A000009.

These partitions have ranks A368110, complement A355740.

The complement is counted by A370320.

The version for prime factors (not all divisors) is A370592, ranks A368100.

The complement for prime factors is A370593, ranks A355529.

For a unique choice we have A370595, ranks A370810.

For multiple choices we have A370803, ranks A370811.

The case without ones is A370805, complement A370804.

The version for factorizations is A370814, complement A370813.

A000005 counts divisors.

A000041 counts integer partitions.

A237685 counts partitions of depth 1, or A353837 if we include depth 0.

A355731 counts choices of a divisor of each prime index, firsts A355732.

Cf. A355535, A355733, A355739, A367867, A368097, A368414, A370583, A370584, A370594, A370806, A370808.

Sequence in context: A084338 A300446 A039876 * A317167 A070830 A039862

Adjacent sequences: A239309 A239310 A239311 * A239313 A239314 A239315

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