A317088 - OEIS

%I #16 Jun 30 2020 18:51:12

%S 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,

%T 29,34,36,47,45,59,60,72,77,93,95,112,121,129,149,169,176,202,228,247,

%U 268,305,334,372,405,452,496,544,594,663,724,802

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

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

%H Chai Wah Wu, <a href="/A317088/b317088.txt">Table of n, a(n) for n = 0..192</a>

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

%t normalQ[m_]:=Union[m]==Range[Max[m]];

%t Table[Length[Select[IntegerPartitions[n],And[normalQ[#],normalQ[Length/@Split[#]]]&]],{n,30}]

%o (Python)

%o from sympy.utilities.iterables import partitions

%o from sympy import integer_nthroot

%o def A317088(n):

%o     if n == 0:

%o         return 1

%o     c = 0

%o     for d in partitions(n,k=integer_nthroot(2*n,2)[0]):

%o         l = len(d)

%o         if l > 0 and l == max(d):

%o             v = set(d.values())

%o             if len(v) == max(v):

%o                 c += 1

%o     return c # _Chai Wah Wu_, Jun 23 2020

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

%K nonn

%O 0,10

%A _Gus Wiseman_, Jul 21 2018