cp's OEIS Frontend

A325330 Number of integer partitions of n whose multiplicities have multiplicities that cover an initial interval of positive integers.

%I A325330 #4 May 02 2019 08:52:56
%S A325330 1,1,2,2,4,5,7,11,16,22,31,44,55,77,96,127,158,208,251,329,400,501,
%T A325330 610,766,915,1141,1368,1677,2005,2454,2913,3553,4219,5110,6053,7300,
%U A325330 8644,10376,12238,14645,17216,20504,24047,28501,33336,39373,45871,53926,62745
%N A325330 Number of integer partitions of n whose multiplicities have multiplicities that cover an initial interval of positive integers.
%C A325330 Partitions whose parts cover an initial interval of positive integers are counted by A000009, with Heinz numbers A055932. Partitions whose multiplicities cover an initial interval of positive integers are counted by A317081, with Heinz numbers A317090. Partitions whose parts and multiplicities both cover an initial interval of positive integers are counted by A317088, with Heinz numbers A317089. Partitions whose multiplicities at every depth cover an initial interval of positive integers are counted by A317245, with Heinz numbers A317246.
%C A325330 The Heinz numbers of these partitions are given by A325370.
%e A325330 The a(0) = 1 through a(8) = 16 partitions:
%e A325330   ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A325330            (11)  (111)  (22)    (221)    (33)      (322)      (44)
%e A325330                         (211)   (311)    (222)     (331)      (332)
%e A325330                         (1111)  (2111)   (411)     (511)      (422)
%e A325330                                 (11111)  (3111)    (2221)     (611)
%e A325330                                          (21111)   (3211)     (2222)
%e A325330                                          (111111)  (4111)     (3221)
%e A325330                                                    (22111)    (4211)
%e A325330                                                    (31111)    (5111)
%e A325330                                                    (211111)   (22211)
%e A325330                                                    (1111111)  (32111)
%e A325330                                                               (41111)
%e A325330                                                               (221111)
%e A325330                                                               (311111)
%e A325330                                                               (2111111)
%e A325330                                                               (11111111)
%e A325330 For example, the partition (5,5,4,3,3,3,2,2) has multiplicities (2,1,3,2) with multiplicities (1,2,1) which cover the initial interval {1,2}, so (5,5,4,3,3,3,2,2) is counted under a(27).
%t A325330 normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t A325330 Table[Length[Select[IntegerPartitions[n],normQ[Length/@Split[Sort[Length/@Split[#]]]]&]],{n,0,30}]
%Y A325330 Cf. A000837, A055932, A317081, A317088, A317089, A317090, A317245, A320348, A325331, A325333, A325337, A325370.
%K A325330 nonn
%O A325330 0,3
%A A325330 _Gus Wiseman_, May 01 2019