cp's OEIS Frontend

A332671 Number of non-unimodal permutations of the multiset of prime indices of n.

%I A332671 #6 Feb 16 2025 08:33:59
%S A332671 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,
%T A332671 0,3,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,2,0,0,0,0,0,6,0,0,0,0,0,2,0,0,
%U A332671 0,2,0,6,0,0,1,0,0,2,0,0,0,0,0,6,0,0,0
%N A332671 Number of non-unimodal permutations of the multiset of prime indices of n.
%C A332671 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A332671 A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%H A332671 MathWorld, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%F A332671 a(n) + A332288(n) = A008480(n).
%F A332671 a(A181821(n)) = A332672(n).
%e A332671 The a(n) permutations for n = 18, 30, 36, 42, 50, 54, 60, 66, 70, 72:
%e A332671   212  213  1212  214  313  2122  1213  215  314  11212
%e A332671        312  2112  412       2212  1312  512  413  12112
%e A332671             2121                  2113            12121
%e A332671                                   2131            21112
%e A332671                                   3112            21121
%e A332671                                   3121            21211
%t A332671 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A332671 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t A332671 Table[Length[Select[Permutations[primeMS[n]],!unimodQ[#]&]],{n,100}]
%Y A332671 Dominated by A008480.
%Y A332671 The complement is counted by A332288.
%Y A332671 A more interesting version is A332672.
%Y A332671 Unimodal compositions are A001523.
%Y A332671 Non-unimodal permutations are A059204.
%Y A332671 Non-unimodal compositions are A115981.
%Y A332671 Non-unimodal normal sequences are A328509.
%Y A332671 Heinz numbers of partitions with non-unimodal run-lengths are A332282.
%Y A332671 Compositions whose negation is not unimodal are A332669.
%Y A332671 Cf. A007052, A056239, A112798, A124010, A332281, A332284, A332287, A332294, A332639, A332642.
%K A332671 nonn
%O A332671 1,30
%A A332671 _Gus Wiseman_, Feb 22 2020