cp's OEIS Frontend

A317081 Number of integer partitions of n whose multiplicities cover an initial interval of positive integers.

%I A317081 #24 Apr 25 2025 08:45:31
%S A317081 1,1,1,2,3,5,5,9,11,16,20,30,34,50,58,79,96,129,152,203,243,307,375,
%T A317081 474,563,707,850,1042,1246,1532,1815,2215,2632,3173,3765,4525,5323,
%U A317081 6375,7519,8916,10478,12414,14523,17133,20034,23488,27422,32090,37285,43511,50559
%N A317081 Number of integer partitions of n whose multiplicities cover an initial interval of positive integers.
%C A317081 Also the number of integer partitions of n with distinct section-sums, where the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. - _Gus Wiseman_, Apr 21 2025
%H A317081 Chai Wah Wu, <a href="/A317081/b317081.txt">Table of n, a(n) for n = 0..160</a>
%e A317081 The a(1) = 1 through a(9) = 16 partitions:
%e A317081  (1) (2) (3)  (4)   (5)   (6)   (7)    (8)    (9)
%e A317081          (21) (31)  (32)  (42)  (43)   (53)   (54)
%e A317081               (211) (41)  (51)  (52)   (62)   (63)
%e A317081                     (221) (321) (61)   (71)   (72)
%e A317081                     (311) (411) (322)  (332)  (81)
%e A317081                                 (331)  (422)  (432)
%e A317081                                 (421)  (431)  (441)
%e A317081                                 (511)  (521)  (522)
%e A317081                                 (3211) (611)  (531)
%e A317081                                        (3221) (621)
%e A317081                                        (4211) (711)
%e A317081                                               (3321)
%e A317081                                               (4221)
%e A317081                                               (4311)
%e A317081                                               (5211)
%e A317081                                               (32211)
%t A317081 normalQ[m_]:=Union[m]==Range[Max[m]];
%t A317081 Table[Length[Select[IntegerPartitions[n],normalQ[Length/@Split[#]]&]],{n,30}]
%o A317081 (Python)
%o A317081 from sympy.utilities.iterables import partitions
%o A317081 def A317081(n):
%o A317081     if n == 0:
%o A317081         return 1
%o A317081     c = 0
%o A317081     for d in partitions(n):
%o A317081         s = set(d.values())
%o A317081         if len(s) == max(s):
%o A317081             c += 1
%o A317081     return c # _Chai Wah Wu_, Jun 22 2020
%Y A317081 The case with parts also covering an initial interval is A317088.
%Y A317081 These partitions are ranked by A317090.
%Y A317081 A000041 counts integer partitions, strict A000009.
%Y A317081 A008284 counts partitions by length, strict A008289.
%Y A317081 A047966 counts partitions with constant section-sums.
%Y A317081 A048767 interchanges prime indices and prime multiplicities (Look-and-Say), see A048768.
%Y A317081 A055932 lists numbers whose prime indices cover an initial interval.
%Y A317081 A116540 counts normal set multipartitions.
%Y A317081 A304442 counts partitions with equal run-sums, ranks A353833.
%Y A317081 A381436 lists the section-sum partition of prime indices.
%Y A317081 A381440 lists the Look-and-Say partition of prime indices.
%Y A317081 Cf. A000837, A069799, A217605, A317082, A317084, A317085, A317087.
%Y A317081 Cf. A003242, A116861, A239455, A353837, A364916, A381431, A381432, A381438.
%K A317081 nonn
%O A317081 0,4
%A A317081 _Gus Wiseman_, Jul 21 2018