A317088 Number of normal integer partitions of n whose multiset of multiplicities is also normal.

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

Author

Gus Wiseman, Jul 21 2018

Keywords

• nonn

Comments

A multiset is normal if it spans an initial interval of positive integers.

Examples

			The a(18) = 7 integer partitions are (543321), (5432211), (4433211), (4432221), (44322111), (4333221), (43322211).
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..192

Crossrefs

Cf. A000009, A000041, A000837, A055932, A296150, A317081, A317082, A317086.

Programs

Mathematica

normalQ[m_]:=Union[m]==Range[Max[m]];
Table[Length[Select[IntegerPartitions[n],And[normalQ[#],normalQ[Length/@Split[#]]]&]],{n,30}]

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