OFFSET
0,3
COMMENTS
a(n) is the number of integer partitions finer than (n, ..., 3, 2, 1) in the poset of integer partitions of 1 + 2 + ... + n ordered by refinement.
a(n+1)/a(n) appears to converge as n -> oo. - Chai Wah Wu, Nov 14 2018
FORMULA
a(n) <= A173519(n). - David A. Corneth, Sep 20 2023
EXAMPLE
The a(1) = 1 through a(4) = 16 partitions:
(1) (21) (321) (4321)
(111) (2211) (32221)
(3111) (33211)
(21111) (42211)
(111111) (43111)
(222211)
(322111)
(331111)
(421111)
(2221111)
(3211111)
(4111111)
(22111111)
(31111111)
(211111111)
(1111111111)
The partition (222211) is the combination of (22)(21)(2)(1), so is counted under a(4). The partition (322111) is the combination of (22)(3)(11)(1), (31)(21)(2)(1), or (211)(3)(2)(1), so is also counted under a(4).
MATHEMATICA
Table[Length[Union[Sort/@Join@@@Tuples[IntegerPartitions/@Range[1, n]]]], {n, 6}]
PROG
(Python)
from collections import Counter
from itertools import count, islice
from sympy.utilities.iterables import partitions
def A321470_gen(): # generator of terms
aset = {(1, )}
yield 1
for n in count(2):
yield len(aset)
aset = {tuple(sorted(p+q)) for p in aset for q in (tuple(sorted(Counter(q).elements())) for q in partitions(n))}
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 11 2018
EXTENSIONS
a(9)-a(11) from Alois P. Heinz, Nov 12 2018
a(12)-a(13) from Chai Wah Wu, Nov 13 2018
a(14) from Chai Wah Wu, Sep 20 2023
STATUS
approved