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A317088
Number of normal integer partitions of n whose multiset of multiplicities is also normal.
11
1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 3, 4, 1, 4, 4, 5, 4, 6, 7, 9, 10, 13, 13, 15, 15, 17, 23, 22, 29, 29, 34, 36, 47, 45, 59, 60, 72, 77, 93, 95, 112, 121, 129, 149, 169, 176, 202, 228, 247, 268, 305, 334, 372, 405, 452, 496, 544, 594, 663, 724, 802
OFFSET
0,10
COMMENTS
A multiset is normal if it spans an initial interval of positive integers.
LINKS
EXAMPLE
The a(18) = 7 integer partitions are (543321), (5432211), (4433211), (4432221), (44322111), (4333221), (43322211).
MATHEMATICA
normalQ[m_]:=Union[m]==Range[Max[m]];
Table[Length[Select[IntegerPartitions[n], And[normalQ[#], normalQ[Length/@Split[#]]]&]], {n, 30}]
PROG
(Python)
from sympy.utilities.iterables import partitions
from sympy import integer_nthroot
def A317088(n):
if n == 0:
return 1
c = 0
for d in partitions(n, k=integer_nthroot(2*n, 2)[0]):
l = len(d)
if l > 0 and l == max(d):
v = set(d.values())
if len(v) == max(v):
c += 1
return c # Chai Wah Wu, Jun 23 2020
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 21 2018
STATUS
approved