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.

1, 2, 2, 4, 5, 7, 10, 15, 18, 26, 35, 46, 61, 82, 102, 136, 174, 224, 283, 360, 449, 569, 708, 883, 1089, 1352, 1659, 2042, 2492, 3039, 3695, 4492, 5426, 6555, 7889, 9482, 11360, 13602, 16231, 19348, 23005, 27313, 32364, 38303, 45227, 53341, 62800, 73829

Offset

1,2

Comments

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

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=1..48.

Example

The a(1) = 1 through a(8) = 15 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (221)    (33)      (322)      (44)
                    (211)   (311)    (222)     (331)      (332)
                    (1111)  (2111)   (411)     (511)      (422)
                            (11111)  (3111)    (2221)     (611)
                                     (21111)   (4111)     (2222)
                                     (111111)  (22111)    (3221)
                                               (31111)    (5111)
                                               (211111)   (22211)
                                               (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

Mathematica

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

Crossrefs

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

Including mean also gives A363719, ranks A363727.

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

A000041 counts integer partitions, strict A000009.

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

A359893 and A359901 count partitions by median.

A362608 counts partitions with a unique mode, ranks A356862.

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

Sequence in context: A238594 A350837 A239945 * A238875 A344740 A241391

Adjacent sequences: A363737 A363738 A363739 * A363741 A363742 A363743

Keyword

nonn

Author

Gus Wiseman, Jun 26 2023

Status

approved