cp's OEIS Frontend

A382526 Number of integer partitions of n with fewer ones than greatest multiplicity.

%I A382526 #6 Apr 07 2025 09:26:45
%S A382526 0,0,1,1,2,3,4,6,9,12,16,24,30,41,56,72,94,124,158,205,262,331,419,
%T A382526 531,663,829,1033,1281,1581,1954,2393,2936,3584,4366,5300,6433,7764,
%U A382526 9374,11277,13548,16225,19425,23166,27623,32842,39004,46212,54719,64610,76251
%N A382526 Number of integer partitions of n with fewer ones than greatest multiplicity.
%e A382526 The a(2) = 1 through a(9) = 12 partitions:
%e A382526   (2)  (3)  (4)   (5)    (6)    (7)     (8)      (9)
%e A382526             (22)  (32)   (33)   (43)    (44)     (54)
%e A382526                   (221)  (42)   (52)    (53)     (63)
%e A382526                          (222)  (322)   (62)     (72)
%e A382526                                 (331)   (332)    (333)
%e A382526                                 (2221)  (422)    (432)
%e A382526                                         (2222)   (441)
%e A382526                                         (3221)   (522)
%e A382526                                         (22211)  (3222)
%e A382526                                                  (3321)
%e A382526                                                  (4221)
%e A382526                                                  (22221)
%t A382526 Table[Length[Select[IntegerPartitions[n],Count[#,1]<Max@@Length/@Split[DeleteCases[#,1]]&]],{n,0,30}]
%Y A382526 The complement (greater than or equal to) is A241131 except first, ranks A360015.
%Y A382526 The opposite version (greater than) is A241131 shifted except first, ranks A360013.
%Y A382526 These partitions have ranks A382856, complement A360015.
%Y A382526 The weak version (less than or equal to) is A381544, ranks A381439.
%Y A382526 For equality we have A382303, ranks A360014.
%Y A382526 A000041 counts integer partitions, strict A000009.
%Y A382526 A008284 counts partitions by length, strict A008289.
%Y A382526 A047993 counts partitions with max part = length, ranks A106529.
%Y A382526 A091602 counts partitions by the greatest multiplicity, rank statistic A051903.
%Y A382526 A116598 counts ones in partitions, rank statistic A007814.
%Y A382526 A239964 counts partitions with max multiplicity = length, ranks A212166.
%Y A382526 A240312 counts partitions with max part = max multiplicity, ranks A381542.
%Y A382526 A382302 counts partitions with max = max multiplicity = distinct length, ranks A381543.
%Y A382526 Cf. A047966, A091605, A116861, A232697, A237984, A362608, A363724, A381079, A381437, A381438.
%K A382526 nonn
%O A382526 0,5
%A A382526 _Gus Wiseman_, Apr 05 2025