|
%I #5 Mar 09 2020 18:26:39
%S 0,0,0,0,0,1,0,2,3,2,0,8,0,3,7,16,0,24,0,16,12,4,0,52,16,5,81,26,0,54,
%T 0,104,18,6,31,168,0,7,25,112,0,99,0,38,201,8,0,344,65,132,33,52,0,
%U 612,52,202,42,9,0,408,0,10,411,688,80,162,0,68,52,272
%N Number of non-unimodal negated 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%F a(n) + A332741(n) = A318762(n).
%e The a(n) permutations for n = 6, 8, 9, 10, 12, 14, 15, 16:
%e 121 132 1212 1121 1132 11121 11212 1243
%e 231 1221 1211 1213 11211 11221 1324
%e 2121 1231 12111 12112 1342
%e 1312 12121 1423
%e 1321 12211 1432
%e 2131 21121 2143
%e 2311 21211 2314
%e 3121 2341
%e 2413
%e 2431
%e 3142
%e 3241
%e 3412
%e 3421
%e 4132
%e 4231
%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 Dominated by A318762.
%Y The complement of the non-negated version is counted by A332294.
%Y The non-negated version is A332672.
%Y The complement is counted by A332741.
%Y A less interesting version is A333146.
%Y Unimodal compositions are A001523.
%Y Unimodal normal sequences are A007052.
%Y Non-unimodal normal sequences are A328509.
%Y Partitions with non-unimodal 0-appended first differences are A332284.
%Y Compositions whose negation is unimodal are A332578.
%Y Partitions with non-unimodal negated run-lengths are A332639.
%Y Numbers whose negated prime signature is not unimodal are A332642.
%Y Cf. A056239, A112798, A115981, A124010, A181819, A181821, A304660, A332280, A332283, A332288, A332638, A332669, A333145.
%K nonn
%O 1,8
%A _Gus Wiseman_, Mar 09 2020
|
