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

Author

Gus Wiseman, Apr 30 2023

Keywords

• nonn

Comments

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

Examples

			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)
		

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.

Programs

Mathematica

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