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A332741 Number of unimodal negated permutations of a multiset whose multiplicities are the prime indices of n.

%I A332741 #9 Feb 16 2025 08:33:59
%S A332741 1,1,1,2,1,2,1,4,3,2,1,4,1,2,3,8,1,6,1,4,3,2,1,8,4,2,9,4,1,6,1,16,3,2,
%T A332741 4,12,1,2,3,8,1,6,1,4,9,2,1,16,5,8,3,4,1,18,4,8,3,2,1,12,1,2,9,32,4,6,
%U A332741 1,4,3,8,1,24,1,2,12,4,5,6,1,16,27,2,1
%N A332741 Number of unimodal negated permutations of a multiset whose multiplicities are the prime indices of n.
%C A332741 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 A332741 A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%H A332741 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%F A332741 a(n) + A332742(n) = A318762(n).
%e A332741 The a(12) = 4 permutations:
%e A332741   {1,1,2,3}
%e A332741   {2,1,1,3}
%e A332741   {3,1,1,2}
%e A332741   {3,2,1,1}
%t A332741 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A332741 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t A332741 Table[Length[Select[Permutations[nrmptn[n]],unimodQ[-#]&]],{n,30}]
%Y A332741 Dominated by A318762.
%Y A332741 The non-negated version is A332294.
%Y A332741 The complement is counted by A332742.
%Y A332741 A less interesting version is A333145.
%Y A332741 Unimodal compositions are A001523.
%Y A332741 Unimodal normal sequences are A007052.
%Y A332741 Numbers with non-unimodal negated prime signature are A332642.
%Y A332741 Partitions whose 0-appended first differences are unimodal are A332283.
%Y A332741 Compositions whose negation is unimodal are A332578.
%Y A332741 Partitions with unimodal negated run-lengths are A332638.
%Y A332741 Cf. A056239, A112798, A115981, A124010, A181819, A181821, A304660, A332280, A332288, A332639, A332669, A332672.
%K A332741 nonn
%O A332741 1,4
%A A332741 _Gus Wiseman_, Mar 09 2020