A364062 Number of integer partitions of n with unique co-mode 1.

0, 1, 1, 1, 1, 2, 1, 3, 2, 3, 3, 6, 2, 8, 6, 9, 6, 16, 7, 21, 12, 23, 18, 39, 17, 47, 32, 59, 40, 86, 44, 110, 72, 131, 95, 188, 103, 233, 166, 288, 201, 389, 244, 490, 347, 587, 440, 794, 524, 974, 727, 1187, 903, 1547, 1106, 1908, 1459, 2303, 1826, 2979, 2198

Offset: 0

Author

Gus Wiseman, Jul 12 2023

Keywords

• nonn

Comments

These are partitions with at least one 1 but with fewer 1's than each of the other parts.

We define a co-mode in a multiset to be an element that appears at most as many times as each of the other elements. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.

Examples

			The a(n) partitions for n = 5, 7, 11, 13, 15:
  (221)    (331)      (551)          (661)            (771)
  (11111)  (2221)     (33221)        (4441)           (44331)
           (1111111)  (33311)        (33331)          (55221)
                      (222221)       (44221)          (442221)
                      (2222111)      (332221)         (3322221)
                      (11111111111)  (2222221)        (3333111)
                                     (22222111)       (22222221)
                                     (1111111111111)  (222222111)
                                                      (111111111111111)
		

Crossrefs

For high (or unique) mode we have A241131, ranks A360013.

For low mode we have A241131, ranks A360015.

Allowing any unique co-mode gives A362610, ranks A359178.

These partitions have ranks A364061.

Adding all 1-free partitions gives A364159, ranks A364158.

A000041 counts integer partitions, strict A000009.

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

A237984 counts partitions containing their mean, ranks A327473.

A327472 counts partitions not containing their mean, ranks A327476.

A362608 counts partitions w/ unique mode, ranks A356862, complement A362605.

A362611 counts modes in prime indices, triangle A362614.

A362613 counts co-modes in prime indices, triangle A362615.

A363486 gives least mode in prime indices, A363487 greatest.

Cf. A002865, A027336, A325347, A098859, A360014, A362562, A362606, A363720, A363723, A363725, A363726.

Programs

Mathematica

comodes[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],comodes[#]=={1}&]],{n,0,30}]