A363484 Number of integer partitions of n covering an initial interval of positive integers with a unique mode.

0, 1, 1, 1, 2, 3, 2, 5, 6, 6, 8, 11, 12, 17, 20, 21, 27, 35, 38, 50, 56, 65, 76, 95, 105, 125, 146, 167, 198, 233, 252, 305, 351, 394, 457, 522, 585, 681, 778, 878, 994, 1135, 1269, 1446, 1638, 1828, 2067, 2339, 2613, 2940, 3301, 3684, 4143, 4634, 5156, 5771

Offset

0,5

Comments

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

Links

Table of n, a(n) for n=0..55.

Example

The a(1) = 1 through a(8) = 6 partitions:
  (1)  (11)  (111)  (211)   (221)    (21111)   (2221)     (3221)
                    (1111)  (2111)   (111111)  (3211)     (22211)
                            (11111)            (22111)    (32111)
                                               (211111)   (221111)
                                               (1111111)  (2111111)
                                                          (11111111)

Mathematica

Table[If[n==0, 0, Length[Select[IntegerPartitions[n], Union[#]==Range[Max@@#]&&Length[Commonest[#]]==1&]]], {n, 0, 30}]

Crossrefs

For parts instead of multiplicities we have A096765, complement A025147.

For multisets instead of partitions we have A097979, complement A363262.

For co-mode we have A363263, complement A363264.

The complement is counted by A363485.

A000041 counts integer partitions, A000009 covering an initial interval.

A362607 counts partitions with multiple modes, co-modes A362609.

A362608 counts partitions with a unique mode, co-mode A362610.

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

Cf. A002865, A008284, A105039, A117989, A243737, A362612.

Sequence in context: A088861 A088631 A086184 * A056552 A049273 A053590

Adjacent sequences: A363481 A363482 A363483 * A363485 A363486 A363487

Keyword

nonn

Author

Gus Wiseman, Jun 05 2023

Status

approved