A382303 Number of integer partitions of n with exactly as many ones as the next greatest multiplicity.

0, 0, 0, 1, 1, 1, 3, 2, 4, 5, 8, 6, 15, 13, 19, 25, 33, 36, 54, 58, 80, 96, 122, 141, 188, 217, 274, 326, 408, 474, 600, 695, 859, 1012, 1233, 1440, 1763, 2050, 2475, 2899, 3476, 4045, 4850, 5630, 6695, 7797, 9216, 10689, 12628, 14611, 17162, 19875, 23253

Offset: 0

Author

Gus Wiseman, Mar 24 2025

Keywords

• nonn

Examples

			The a(3) = 1 through a(10) = 8 partitions:
  (21)  (31)  (41)  (51)    (61)   (71)    (81)      (91)
                    (321)   (421)  (431)   (531)     (541)
                    (2211)         (521)   (621)     (631)
                                   (3311)  (32211)   (721)
                                           (222111)  (4321)
                                                     (4411)
                                                     (33211)
                                                     (42211)
		

Crossrefs

First differences of A241131, ranks A360013 = 2*A360015 (if we prepend 1).

The Heinz numbers of these partitions are A360014.

Equal case of A381544 (ranks A381439).

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A047993 counts partitions with max = 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 = max multiplicity, ranks A381542.

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

Cf. A047966, A051904, A091605, A116861, A237984, A239455, A362608, A363724, A381079, A381437, A381438.

Programs

Mathematica

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