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A326332 Number of integer partitions of n with unsortable prime factors. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 18 15:00 EDT 2019. Contains 326106 sequences. (Running on oeis4.)