A364193 Number of integer partitions of n where the least part is the unique mode.

0, 1, 2, 2, 4, 4, 7, 9, 13, 17, 24, 32, 43, 58, 75, 97, 130, 167, 212, 274, 346, 438, 556, 695, 865, 1082, 1342, 1655, 2041, 2511, 3067, 3756, 4568, 5548, 6728, 8130, 9799, 11810, 14170, 16980, 20305, 24251, 28876, 34366, 40781, 48342, 57206, 67597, 79703

Offset

0,3

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 in {a,a,b,b,b,c,d,d,d} are {b,d}.

Links

Table of n, a(n) for n=0..48.

Example

The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (211)   (2111)   (222)     (511)      (422)
                    (1111)  (11111)  (411)     (3211)     (611)
                                     (3111)    (4111)     (2222)
                                     (21111)   (22111)    (4211)
                                     (111111)  (31111)    (5111)
                                               (211111)   (32111)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

Mathematica

Table[If[n==0, 0, Length[Select[IntegerPartitions[n], Last[Length/@Split[#]]>Max@@Most[Length/@Split[#]]&]]], {n, 0, 30}]

Crossrefs

For greatest part and multiple modes we have A171979.

Allowing multiple modes gives A240303.

For greatest instead of least part we have A362612, ranks A362616.

For mean instead of least part we have A363723.

These partitions have ranks A364160.

A000041 counts integer partitions.

A362611 counts modes in prime factorization, A362613 co-modes.

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

A363486 gives least mode in prime indices, A363487 greatest.

A363952 counts partitions by low mode, A363953 high.

Ranking and counting partitions:

- A356862 = unique mode, counted by A362608

- A359178 = unique co-mode, counted by A362610

- A362605 = multiple modes, counted by A362607

- A362606 = multiple co-modes, counted by A362609

Cf. A002865, A008284, A070003, A098859, A102750, A237984, A327472, A360015.

Sequence in context: A183566 A222709 A034396 * A253412 A291148 A032190

Adjacent sequences: A364190 A364191 A364192 * A364194 A364195 A364196

Keyword

nonn

Author

Gus Wiseman, Jul 16 2023

Status

approved