cp's OEIS Frontend

A363731 Number of integer partitions of n whose mean is a mode but not the only mode.

%I A363731 #8 Jun 25 2023 18:20:17
%S A363731 0,0,0,0,0,0,1,0,0,2,1,0,5,0,1,8,5,0,12,0,19,14,2,0,52,21,3,23,59,0,
%T A363731 122,0,97,46,6,167,303,0,8,82,559,0,543,0,355,745,15,0,1685,510,1083,
%U A363731 251,840,0,2325,1832,3692,426,34,0,9599
%N A363731 Number of integer partitions of n whose mean is a mode but not the only mode.
%C A363731 A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.
%e A363731 The a(n) partitions for n = 6, 9, 12, 15, 18:
%e A363731   (3,2,1)  (4,3,2)  (5,4,3)        (6,5,4)      (7,6,5)
%e A363731            (5,3,1)  (6,4,2)        (7,5,3)      (8,6,4)
%e A363731                     (7,4,1)        (8,5,2)      (9,6,3)
%e A363731                     (6,3,2,1)      (9,5,1)      (10,6,2)
%e A363731                     (3,3,2,2,1,1)  (4,4,3,3,1)  (11,6,1)
%e A363731                                    (5,3,3,2,2)  (4,4,3,3,2,2)
%e A363731                                    (5,4,3,2,1)  (5,5,3,3,1,1)
%e A363731                                    (7,3,3,1,1)  (6,4,3,3,1,1)
%e A363731                                                 (7,3,3,2,2,1)
%e A363731                                                 (8,3,3,2,1,1)
%e A363731                                                 (3,3,3,2,2,2,1,1,1)
%e A363731                                                 (6,2,2,2,2,1,1,1,1)
%t A363731 modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
%t A363731 Table[Length[Select[IntegerPartitions[n],MemberQ[modes[#],Mean[#]]&&!{Mean[#]}==modes[#]&]],{n,30}]
%Y A363731 For a unique mode we have A363723, non-constant A362562.
%Y A363731 For any number of modes we have A363724.
%Y A363731 A000041 counts partitions, strict A000009.
%Y A363731 A008284 counts partitions by length (or decreasing mean), strict A008289.
%Y A363731 A237984 counts partitions containing their mean, ranks A327473.
%Y A363731 A327472 counts partitions not containing their mean, ranks A327476.
%Y A363731 A362608 counts partitions with a unique mode, ranks A356862.
%Y A363731 A363719 counts partitions with all three averages equal, ranks A363727.
%Y A363731 Cf. A240219, A326567/A326568, A359893, A363720, A363725, A363740.
%K A363731 nonn
%O A363731 0,10
%A A363731 _Gus Wiseman_, Jun 24 2023