A363726 Number of odd-length integer partitions of n with a unique mode.

0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 22, 26, 39, 50, 67, 86, 118, 148, 196, 245, 315, 394, 507, 629, 792, 979, 1231, 1503, 1873, 2286, 2814, 3424, 4194, 5073, 6183, 7449, 9014, 10827, 13055, 15603, 18713, 22308, 26631, 31646, 37641, 44559, 52835, 62374, 73671

Offset

0,4

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}.

Links

Table of n, a(n) for n=0..49.

Example

The a(1) = 1 through a(8) = 8 partitions:
  (1)  (2)  (3)    (4)    (5)      (6)      (7)        (8)
            (111)  (211)  (221)    (222)    (322)      (332)
                          (311)    (411)    (331)      (422)
                          (11111)  (21111)  (511)      (611)
                                            (22111)    (22211)
                                            (31111)    (32111)
                                            (1111111)  (41111)
                                                       (2111111)

Mathematica

modes[ms_]:=Select[Union[ms], Count[ms, #]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n], Length[modes[#]]==1&&OddQ[Length[#]]&]], {n, 30}]

Crossrefs

The constant case is A001227.

Allowing multiple modes gives A027193, ranks A026424.

Allowing any length gives A362608, ranks A356862.

A000041 counts partitions, strict A000009.

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

A359893 and A359901 count partitions by median.

Cf. A325347, A363719, A363720, A363731, A363724, A363725, A363740.

Sequence in context: A317785 A014810 A318771 * A239835 A361860 A026929

Adjacent sequences: A363723 A363724 A363725 * A363727 A363728 A363729

Keyword

nonn

Author

Gus Wiseman, Jun 27 2023

Status

approved