A326332 Number of integer partitions of n with unsortable prime factors.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 5, 9, 14, 22, 33, 50, 71, 100, 140, 196, 265, 360, 480, 641, 842, 1104, 1432, 1855, 2378, 3040, 3858, 4888, 6146, 7708, 9616, 11969, 14818, 18305, 22511, 27629, 33773, 41191, 50069, 60744, 73453, 88645, 106681

Offset

0,14

Comments

An integer partition has unsortable prime factors if there is no permutation (c_1,...,c_k) of the parts such that the maximum prime factor of c_i is at most the minimum prime factor of c_{i+1}. For example, the partition (27,8,6) is sortable because the permutation (8,6,27) satisfies the condition.

Links

Table of n, a(n) for n=0..51.

Formula

A000041(n) = a(n) + A326333(n).

Example

The a(12) = 1 through a(17) = 14 partitions:
  (6,6)  (10,3)   (6,6,2)    (6,6,3)      (10,6)         (14,3)
         (6,6,1)  (10,3,1)   (10,3,2)     (6,6,4)        (6,6,5)
                  (6,6,1,1)  (6,6,2,1)    (10,3,3)       (10,4,3)
                             (10,3,1,1)   (6,6,2,2)      (10,6,1)
                             (6,6,1,1,1)  (6,6,3,1)      (6,6,3,2)
                                          (10,3,2,1)     (6,6,4,1)
                                          (6,6,2,1,1)    (10,3,2,2)
                                          (10,3,1,1,1)   (10,3,3,1)
                                          (6,6,1,1,1,1)  (6,6,2,2,1)
                                                         (6,6,3,1,1)
                                                         (10,3,2,1,1)
                                                         (6,6,2,1,1,1)
                                                         (10,3,1,1,1,1)
                                                         (6,6,1,1,1,1,1)

Mathematica

Table[Length[Select[IntegerPartitions[n], !OrderedQ[Join@@Sort[First/@FactorInteger[#]&/@#, OrderedQ[PadRight[{#1, #2}]]&]]&]], {n, 0, 20}]

Crossrefs

Sortable integer partitions are A326333.

Unsortable set partitions are A058681.

Unsortable normal multiset partitions are A326211.

MM-numbers of unsortable multiset partitions are A326258.

Cf. A000041, A000108, A001055, A056239, A112798, A326209, A326212.

Sequence in context: A280204 A240841 A056686 * A079962 A244986 A293547

Adjacent sequences: A326329 A326330 A326331 * A326333 A326334 A326335

Keyword

nonn

Author

Gus Wiseman, Jun 27 2019

Status

approved