A381079 Number of integer partitions of n whose greatest multiplicity is equal to their sum of distinct parts.

0, 1, 0, 0, 1, 1, 0, 3, 1, 3, 1, 2, 0, 7, 2, 6, 7, 11, 3, 19, 8, 22, 16, 32, 17, 48, 21, 50, 39, 71, 35, 101, 58, 120, 89, 156, 97, 228, 133, 267, 203, 352, 228, 483, 322, 571, 444, 734, 524, 989, 683, 1160, 942, 1490, 1103, 1919, 1438, 2302, 1890, 2881, 2243, 3683, 2842, 4384, 3703, 5461

Offset: 0

Author

Gus Wiseman, Mar 03 2025

Keywords

• nonn

Comments

Are there only 4 zeros?

Examples

			The partition (3,2,2,1,1,1,1,1,1) has greatest multiplicity 6 and distinct parts (3,2,1) with sum 6, so is counted under a(13).
The a(1) = 1 through a(13) = 7 partitions:
  1  .  .  22  2111  .  2221   22211  333     331111  5111111   .  33331
                        22111         222111          32111111     322222
                        31111         411111                       3331111
                                                                   4411111
                                                                   61111111
                                                                   322111111
                                                                   421111111
		

Crossrefs

For greatest part instead of multiplicity we have A000005.

Counting partitions by the LHS gives A091602, rank statistic A051903.

Counting partitions by the RHS gives A116861, rank statistic A066328.

These partitions are ranked by A381632, for part instead of multiplicity A246655.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A047993 counts balanced partitions, ranks A106529.

A091605 counts partitions with greatest multiplicity 2.

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

Cf. A027193, A047966, A048767, A051904, A212166, A237984, A239455, A241131, A362608, A363724.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Max@@Length/@Split[#]==Total[Union[#]]&]],{n,0,30}]