cp's OEIS Frontend

A332294 Number of unimodal permutations of a multiset whose multiplicities are the prime indices of n.

%I A332294 #8 Feb 16 2025 08:33:59
%S A332294 1,1,1,2,1,3,1,4,3,4,1,6,1,5,4,8,1,9,1,8,5,6,1,12,4,7,9,10,1,12,1,16,
%T A332294 6,8,5,18,1,9,7,16,1,15,1,12,12,10,1,24,5,16,8,14,1,27,6,20,9,11,1,24,
%U A332294 1,12,15,32,7,18,1,16,10,20,1,36,1,13,16,18,6
%N A332294 Number of unimodal permutations of a multiset whose multiplicities are the prime indices of n.
%C A332294 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 A332294 A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%H A332294 MathWorld, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%F A332294 a(n) + A332672(n) = A318762(n).
%F A332294 a(n) = A332288(A181821(n)).
%e A332294 The a(12) = 6 permutations:
%e A332294   {1,1,2,3}
%e A332294   {1,1,3,2}
%e A332294   {1,2,3,1}
%e A332294   {1,3,2,1}
%e A332294   {2,3,1,1}
%e A332294   {3,2,1,1}
%t A332294 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A332294 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t A332294 Table[Length[Select[Permutations[nrmptn[n]],unimodQ]],{n,0,30}]
%Y A332294 Dominated by A318762.
%Y A332294 A less interesting version is A332288.
%Y A332294 The complement is counted by A332672.
%Y A332294 The opposite/negative version is A332741.
%Y A332294 Unimodal compositions are A001523.
%Y A332294 Non-unimodal permutations are A059204.
%Y A332294 Partitions whose run-lengths are unimodal are A332280.
%Y A332294 Cf. A007052, A056239, A112798, A115981, A124010, A304660, A328509, A332283, A332578, A332638, A332671, A332742.
%K A332294 nonn
%O A332294 1,4
%A A332294 _Gus Wiseman_, Feb 21 2020