A363740 - OEIS

A363740

Number of integer partitions of n whose median appears more times than any other part, i.e., partitions containing a unique mode equal to the median.

%I #7 Jun 27 2023 10:28:55

%S 1,2,2,4,5,7,10,15,18,26,35,46,61,82,102,136,174,224,283,360,449,569,

%T 708,883,1089,1352,1659,2042,2492,3039,3695,4492,5426,6555,7889,9482,

%U 11360,13602,16231,19348,23005,27313,32364,38303,45227,53341,62800,73829

%N Number of integer partitions of n whose median appears more times than any other part, i.e., partitions containing a unique mode equal to the median.

%C The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

%C 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 The a(1) = 1 through a(8) = 15 partitions:

%e (1) (2) (3) (4) (5) (6) (7) (8)

%e (11) (111) (22) (221) (33) (322) (44)

%e (211) (311) (222) (331) (332)

%e (1111) (2111) (411) (511) (422)

%e (11111) (3111) (2221) (611)

%e (21111) (4111) (2222)

%e (111111) (22111) (3221)

%e (31111) (5111)

%e (211111) (22211)

%e (1111111) (32111)

%e (41111)

%e (221111)

%e (311111)

%e (2111111)

%e (11111111)

%t modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];

%t Table[Length[Select[IntegerPartitions[n],{Median[#]}==modes[#]&]],{n,30}]

%Y For mean instead of mode we have A240219, see A359894, A359889, A359895, A359897, A359899.

%Y Including mean also gives A363719, ranks A363727.

%Y For mean instead of median we have A363723, see A363724, A363731.

%Y A000041 counts integer partitions, strict A000009.

%Y A008284 counts partitions by length (or decreasing mean), strict A008289.

%Y A359893 and A359901 count partitions by median.

%Y A362608 counts partitions with a unique mode, ranks A356862.

%Y Cf. A027193, A237984, A325347, A362562, A363720, A363725, A363726.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jun 26 2023