login
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.
19
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, 119, 2, 251, 2, 201, 126, 41, 271, 655, 2, 57, 250, 1060, 2, 1099, 2, 844, 1508, 107, 2, 3484, 802, 2068, 900, 2136, 2, 4558, 3513, 7071, 1630, 259, 2, 20260
OFFSET
1,2
COMMENTS
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}.
EXAMPLE
The a(n) partitions for n = 6, 10, 12:
(6) (10) (12)
(3,3) (5,5) (6,6)
(2,2,2) (2,2,2,2,2) (4,4,4)
(3,2,1) (3,2,2,2,1) (5,4,3)
(1,1,1,1,1,1) (4,2,2,1,1) (6,4,2)
(1,1,1,1,1,1,1,1,1,1) (7,4,1)
(3,3,3,3)
(4,3,3,2)
(5,3,3,1)
(6,3,2,1)
(2,2,2,2,2,2)
(3,2,2,2,2,1)
(3,3,2,2,1,1)
(4,2,2,2,1,1)
(1,1,1,1,1,1,1,1,1,1,1,1)
MATHEMATICA
modes[ms_]:=Select[Union[ms], Count[ms, #]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n], MemberQ[modes[#], Mean[#]]&]], {n, 30}]
CROSSREFS
For parts instead of modes we have A237984, complement A327472.
The case of a unique mode is A363723, non-constant A362562.
The case of more than one mode is A363731.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length (or decreasing mean), strict A008289.
A362608 counts partitions with a unique mode.
A363719 = all three averages equal, ranks A363727, non-constant A363728.
A363720 = all three averages different, ranks A363730, unique mode A363725.
Sequence in context: A325250 A062830 A322366 * A345268 A164941 A328673
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 24 2023
STATUS
approved