cp's OEIS Frontend

A363721 Number of odd-length integer partitions of n satisfying (mean) = (median) = (mode), assuming there is a unique mode.

%I A363721 #7 Jun 22 2023 23:34:48
%S A363721 1,1,2,1,2,2,2,1,3,3,2,2,2,5,7,1,2,8,2,9,16,11,2,2,15,16,37,33,2,44,2,
%T A363721 1,79,33,103,127,2,47,166,39,2,214,2,384,738,90,2,2,277,185,631,1077,
%U A363721 2,1065,1560,477,1156,223,2,2863
%N A363721 Number of odd-length integer partitions of n satisfying (mean) = (median) = (mode), assuming there is a unique mode.
%C A363721 The median of an odd-length partition is the middle part.
%C A363721 A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.
%e A363721 The a(n) partitions for n = {1, 3, 9, 14, 15, 18, 20, 22} (A..M = 10..22):
%e A363721   1  3    9          E        F                I          K      M
%e A363721      111  333        2222222  555              666        44444  22222222222
%e A363721           111111111  3222221  33333            222222222  54443  32222222221
%e A363721                      3322211  43332            322222221  64442  33222222211
%e A363721                      4222211  53331            332222211  65441  33322222111
%e A363721                               63321            422222211  74432  42222222211
%e A363721                               111111111111111  432222111  74441  43222222111
%e A363721                                                522222111  84431  44222221111
%e A363721                                                           94421  52222222111
%e A363721                                                                  53222221111
%e A363721                                                                  62222221111
%t A363721 modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
%t A363721 Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&{Mean[#]}=={Median[#]}==modes[#]&]],{n,30}]
%Y A363721 All odd-length partitions are counted by A027193.
%Y A363721 For just (mean) = (median) we have A359895, also A240219, A359899, A359910.
%Y A363721 For just (mean) != (median) we have A359896, also A359894, A359900.
%Y A363721 Allowing any length gives A363719, ranks A363727, non-constant A363728.
%Y A363721 A000041 counts partitions, strict A000009.
%Y A363721 A008284 counts partitions by length (or negative mean), strict A008289.
%Y A363721 A359893 and A359901 count partitions by median, odd-length A359902.
%Y A363721 A362608 counts partitions with a unique mode.
%Y A363721 A363726 counts odd-length partitions with a unique mode.
%Y A363721 Cf. A237984, A325347, A326567/A326568, A327472, A363720, A363740, A363741.
%K A363721 nonn
%O A363721 1,3
%A A363721 _Gus Wiseman_, Jun 21 2023