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

0,10

Comments

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

Links

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

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

Crossrefs

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

Sequence in context: A325242 A257053 A129717 * A276380 A352724 A248723

Adjacent sequences: A317085 A317086 A317087 * A317089 A317090 A317091

Keyword

nonn

Author

Gus Wiseman, Jul 21 2018

Status

approved