A362609 Number of integer partitions of n with more than one part of least multiplicity.

0, 0, 0, 1, 1, 2, 4, 5, 9, 14, 19, 26, 42, 51, 74, 103, 136, 174, 246, 303, 411, 523, 674, 844, 1114, 1364, 1748, 2174, 2738, 3354, 4247, 5139, 6413, 7813, 9613, 11630, 14328, 17169, 20958, 25180, 30497, 36401, 44025, 52285, 62834, 74626, 89111, 105374, 125662

Offset

0,6

Comments

These are partitions where no part appears fewer times than all of the others.

Links

Table of n, a(n) for n=0..48.

Example

The partition (4,2,2,1) has least multiplicity 1, and two parts of multiplicity 1 (namely 1 and 4), so is counted under a(9).
The a(3) = 1 through a(9) = 14 partitions:
  (21)  (31)  (32)  (42)    (43)    (53)     (54)
              (41)  (51)    (52)    (62)     (63)
                    (321)   (61)    (71)     (72)
                    (2211)  (421)   (431)    (81)
                            (3211)  (521)    (432)
                                    (3221)   (531)
                                    (3311)   (621)
                                    (4211)   (3321)
                                    (32111)  (4221)
                                             (4311)
                                             (5211)
                                             (42111)
                                             (222111)
                                             (321111)

Mathematica

Table[Length[Select[IntegerPartitions[n], Count[Length/@Split[#], Min@@Length/@Split[#]]>1&]], {n, 0, 30}]

Crossrefs

For parts instead of multiplicities we have A117989, ranks A283050.

For median instead of co-mode we have A238479, complement A238478.

These partitions have ranks A362606.

For mode instead of co-mode we have A362607, ranks A362605.

For mode complement instead of co-mode we have A362608, ranks A356862.

The complement is counted by A362610, ranks A359178.

A000041 counts integer partitions.

A275870 counts collapsible partitions.

A359893 counts partitions by median.

A362611 counts modes in prime factorization, co-modes A362613.

A362614 counts partitions by number of modes, co-modes A362615.

Cf. A002865, A008284, A053263, A098859, A304442, A353864, A360071, A362612.

Sequence in context: A129282 A073153 A073154 * A238656 A077882 A351293

Adjacent sequences: A362606 A362607 A362608 * A362610 A362611 A362612

Keyword

nonn

Author

Gus Wiseman, Apr 30 2023

Status

approved