A332742 - OEIS

%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