cp's OEIS Frontend

A363724 Number of integer partitions of n whose mean is a mode, i.e., partitions whose mean appears at least as many times as each of the other parts.

%I A363724 #8 Jun 25 2023 18:20:23
%S A363724 1,2,2,3,2,5,2,5,5,6,2,15,2,8,15,17,2,30,2,43,30,15,2,112,36,21,60,
%T A363724 119,2,251,2,201,126,41,271,655,2,57,250,1060,2,1099,2,844,1508,107,2,
%U A363724 3484,802,2068,900,2136,2,4558,3513,7071,1630,259,2,20260
%N A363724 Number of integer partitions of n whose mean is a mode, i.e., partitions whose mean appears at least as many times as each of the other parts.
%C A363724 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 A363724 The a(n) partitions for n = 6, 10, 12:
%e A363724   (6)            (10)                   (12)
%e A363724   (3,3)          (5,5)                  (6,6)
%e A363724   (2,2,2)        (2,2,2,2,2)            (4,4,4)
%e A363724   (3,2,1)        (3,2,2,2,1)            (5,4,3)
%e A363724   (1,1,1,1,1,1)  (4,2,2,1,1)            (6,4,2)
%e A363724                  (1,1,1,1,1,1,1,1,1,1)  (7,4,1)
%e A363724                                         (3,3,3,3)
%e A363724                                         (4,3,3,2)
%e A363724                                         (5,3,3,1)
%e A363724                                         (6,3,2,1)
%e A363724                                         (2,2,2,2,2,2)
%e A363724                                         (3,2,2,2,2,1)
%e A363724                                         (3,3,2,2,1,1)
%e A363724                                         (4,2,2,2,1,1)
%e A363724                                         (1,1,1,1,1,1,1,1,1,1,1,1)
%t A363724 modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
%t A363724 Table[Length[Select[IntegerPartitions[n],MemberQ[modes[#],Mean[#]]&]],{n,30}]
%Y A363724 For parts instead of modes we have A237984, complement A327472.
%Y A363724 The case of a unique mode is A363723, non-constant A362562.
%Y A363724 The case of more than one mode is A363731.
%Y A363724 A000041 counts partitions, strict A000009.
%Y A363724 A008284 counts partitions by length (or decreasing mean), strict A008289.
%Y A363724 A362608 counts partitions with a unique mode.
%Y A363724 A363719 = all three averages equal, ranks A363727, non-constant A363728.
%Y A363724 A363720 = all three averages different, ranks A363730, unique mode A363725.
%Y A363724 Cf. A240219, A326567/A326568, A363726, A363740.
%K A363724 nonn
%O A363724 1,2
%A A363724 _Gus Wiseman_, Jun 24 2023