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

0,6

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}.

Links

Table of n, a(n) for n=0..60.

Example

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)

Mathematica

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

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.

Sequence in context: A213424 A174427 A158206 * A164988 A322391 A201592

Adjacent sequences: A364059 A364060 A364061 * A364063 A364064 A364065

Keyword

nonn

Author

Gus Wiseman, Jul 12 2023

Status

approved