A381544 Number of integer partitions of n not containing more ones than any other part.

0, 0, 1, 2, 3, 4, 7, 8, 13, 17, 24, 30, 45, 54, 75, 97, 127, 160, 212, 263, 342, 427, 541, 672, 851, 1046, 1307, 1607, 1989, 2428, 2993, 3631, 4443, 5378, 6533, 7873, 9527, 11424, 13752, 16447, 19701, 23470, 28016, 33253, 39537, 46801, 55428, 65408, 77238

Offset: 0

Author

Gus Wiseman, Mar 24 2025

Keywords

• nonn

Examples

			The a(2) = 1 through a(9) = 17 partitions:
  (2)  (3)   (4)   (5)    (6)     (7)     (8)      (9)
       (21)  (22)  (32)   (33)    (43)    (44)     (54)
             (31)  (41)   (42)    (52)    (53)     (63)
                   (221)  (51)    (61)    (62)     (72)
                          (222)   (322)   (71)     (81)
                          (321)   (331)   (332)    (333)
                          (2211)  (421)   (422)    (432)
                                  (2221)  (431)    (441)
                                          (521)    (522)
                                          (2222)   (531)
                                          (3221)   (621)
                                          (3311)   (3222)
                                          (22211)  (3321)
                                                   (4221)
                                                   (22221)
                                                   (32211)
                                                   (222111)
		

Crossrefs

The complement is counted by A241131, ranks A360013 = 2*A360015 (if we prepend 1).

The Heinz numbers of these partitions are A381439.

The case of equality is A382303, ranks A360014.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A047993 counts partitions with max part = length, ranks A106529.

A091602 counts partitions by the greatest multiplicity, rank statistic A051903.

A116598 counts ones in partitions, rank statistic A007814.

A239964 counts partitions with max multiplicity = length, ranks A212166.

A240312 counts partitions with max part = max multiplicity, ranks A381542.

A382302 counts partitions with max = max multiplicity = distinct length, ranks A381543.

Cf. A047966, A091605, A116861, A232697, A237984, A362608, A363724, A381079, A381437, A381438.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Count[#,1]<=Max@@Length/@Split[DeleteCases[#,1]]&]],{n,0,30}]