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%I #5 Feb 24 2020 21:56:51
%S 0,0,0,0,0,0,0,2,3,0,0,6,0,0,6,16,0,21,0,12,10,0,0,48,16,0,81,20,0,48,
%T 0,104,15,0,30,162,0,0,21,104,0,90,0,30,198,0,0,336,65,124,28,42,0,
%U 603,50,190,36,0,0,396,0,0,405,688,77,150,0,56,45,260,0
%N Number of non-unimodal permutations of a multiset whose multiplicities are the prime indices of n.
%C This multiset is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.
%C A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%H MathWorld, <a href="http://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%F a(n) = A332671(A181821(n)).
%F a(n) + A332294(n) = A318762(n).
%e The a(n) permutations for n = 8, 9, 12, 15, 16:
%e 213 1212 1213 11212 1324
%e 312 2112 1312 12112 1423
%e 2121 2113 12121 2134
%e 2131 21112 2143
%e 3112 21121 2314
%e 3121 21211 2413
%e 3124
%e 3142
%e 3214
%e 3241
%e 3412
%e 4123
%e 4132
%e 4213
%e 4231
%e 4312
%t nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t Table[Length[Select[Permutations[nrmptn[n]],!unimodQ[#]&]],{n,30}]
%Y Positions of zeros are one and A001751.
%Y Support is A264828 without one.
%Y Dominated by A318762.
%Y The complement is counted by A332294.
%Y A less interesting version is A332671.
%Y The opposite version is A332742.
%Y Unimodal compositions are A001523.
%Y Non-unimodal permutations are A059204.
%Y Non-unimodal compositions are A115981.
%Y Non-unimodal normal sequences are A328509.
%Y Heinz numbers of partitions with non-unimodal run-lengths are A332282.
%Y Compositions whose negation is not unimodal are A332669.
%Y Cf. A007052, A008480, A056239, A112798, A124010, A181819, A181821, A332281, A332287, A332294, A332642, A332741.
%K nonn
%O 1,8
%A _Gus Wiseman_, Feb 23 2020
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