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%I #5 Jun 28 2019 21:14:17
%S 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,
%T 480,641,842,1104,1432,1855,2378,3040,3858,4888,6146,7708,9616,11969,
%U 14818,18305,22511,27629,33773,41191,50069,60744,73453,88645,106681
%N Number of integer partitions of n with unsortable prime factors.
%C 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.
%F A000041(n) = a(n) + A326333(n).
%e The a(12) = 1 through a(17) = 14 partitions:
%e (6,6) (10,3) (6,6,2) (6,6,3) (10,6) (14,3)
%e (6,6,1) (10,3,1) (10,3,2) (6,6,4) (6,6,5)
%e (6,6,1,1) (6,6,2,1) (10,3,3) (10,4,3)
%e (10,3,1,1) (6,6,2,2) (10,6,1)
%e (6,6,1,1,1) (6,6,3,1) (6,6,3,2)
%e (10,3,2,1) (6,6,4,1)
%e (6,6,2,1,1) (10,3,2,2)
%e (10,3,1,1,1) (10,3,3,1)
%e (6,6,1,1,1,1) (6,6,2,2,1)
%e (6,6,3,1,1)
%e (10,3,2,1,1)
%e (6,6,2,1,1,1)
%e (10,3,1,1,1,1)
%e (6,6,1,1,1,1,1)
%t Table[Length[Select[IntegerPartitions[n],!OrderedQ[Join@@Sort[First/@FactorInteger[#]&/@#,OrderedQ[PadRight[{#1,#2}]]&]]&]],{n,0,20}]
%Y Sortable integer partitions are A326333.
%Y Unsortable set partitions are A058681.
%Y Unsortable normal multiset partitions are A326211.
%Y MM-numbers of unsortable multiset partitions are A326258.
%Y Cf. A000041, A000108, A001055, A056239, A112798, A326209, A326212.
%K nonn
%O 0,14
%A _Gus Wiseman_, Jun 27 2019
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