cp's OEIS Frontend

A332672 Number of non-unimodal permutations of a multiset whose multiplicities are the prime indices of n.

%I A332672 #6 Feb 16 2025 08:33:59
%S A332672 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 A332672 0,104,15,0,30,162,0,0,21,104,0,90,0,30,198,0,0,336,65,124,28,42,0,
%U A332672 603,50,190,36,0,0,396,0,0,405,688,77,150,0,56,45,260,0
%N A332672 Number of non-unimodal permutations of a multiset whose multiplicities are the prime indices of n.
%C A332672 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 A332672 A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%H A332672 MathWorld, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%F A332672 a(n) = A332671(A181821(n)).
%F A332672 a(n) + A332294(n) = A318762(n).
%e A332672 The a(n) permutations for n = 8, 9, 12, 15, 16:
%e A332672   213   1212   1213   11212   1324
%e A332672   312   2112   1312   12112   1423
%e A332672         2121   2113   12121   2134
%e A332672                2131   21112   2143
%e A332672                3112   21121   2314
%e A332672                3121   21211   2413
%e A332672                               3124
%e A332672                               3142
%e A332672                               3214
%e A332672                               3241
%e A332672                               3412
%e A332672                               4123
%e A332672                               4132
%e A332672                               4213
%e A332672                               4231
%e A332672                               4312
%t A332672 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A332672 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t A332672 Table[Length[Select[Permutations[nrmptn[n]],!unimodQ[#]&]],{n,30}]
%Y A332672 Positions of zeros are one and A001751.
%Y A332672 Support is A264828 without one.
%Y A332672 Dominated by A318762.
%Y A332672 The complement is counted by A332294.
%Y A332672 A less interesting version is A332671.
%Y A332672 The opposite version is A332742.
%Y A332672 Unimodal compositions are A001523.
%Y A332672 Non-unimodal permutations are A059204.
%Y A332672 Non-unimodal compositions are A115981.
%Y A332672 Non-unimodal normal sequences are A328509.
%Y A332672 Heinz numbers of partitions with non-unimodal run-lengths are A332282.
%Y A332672 Compositions whose negation is not unimodal are A332669.
%Y A332672 Cf. A007052, A008480, A056239, A112798, A124010, A181819, A181821, A332281, A332287, A332294, A332642, A332741.
%K A332672 nonn
%O A332672 1,8
%A A332672 _Gus Wiseman_, Feb 23 2020