cp's OEIS Frontend

A333145 Number of unimodal negated permutations of the multiset of prime indices of n.

%I A333145 #15 Feb 16 2025 08:33:59
%S A333145 1,1,1,1,1,2,1,1,1,2,1,2,1,2,2,1,1,3,1,2,2,2,1,2,1,2,1,2,1,4,1,1,2,2,
%T A333145 2,3,1,2,2,2,1,4,1,2,2,2,1,2,1,3,2,2,1,4,2,2,2,2,1,4,1,2,2,1,2,4,1,2,
%U A333145 2,4,1,3,1,2,3,2,2,4,1,2,1,2,1,4,2,2,2
%N A333145 Number of unimodal negated permutations of the multiset of prime indices of n.
%C A333145 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 A333145 A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%C A333145 Also permutations of the multiset of prime indices of n avoiding the patterns (1,2,1), (1,3,2), and (2,3,1).
%C A333145 Also the number divisors of n not divisible by the least prime factor of n. The other divisors are counted by A069157. - _Gus Wiseman_, Apr 12 2022
%H A333145 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%H A333145 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>
%F A333145 a(n) + A333146(n) = A008480(n).
%F A333145 a(n) = A000005(A028234(n)). - _Gus Wiseman_, Apr 14 2022
%F A333145 a(n) = A000005(n) - A069157(n). - _Gus Wiseman_, Apr 14 2022
%e A333145 The a(n) permutations for n = 2, 6, 18, 30, 90, 162, 210, 450:
%e A333145   (1)  (12)  (122)  (123)  (1223)  (12222)  (1234)  (12233)
%e A333145        (21)  (212)  (213)  (2123)  (21222)  (2134)  (21233)
%e A333145              (221)  (312)  (2213)  (22122)  (3124)  (22133)
%e A333145                     (321)  (3122)  (22212)  (3214)  (31223)
%e A333145                            (3212)  (22221)  (4123)  (32123)
%e A333145                            (3221)           (4213)  (32213)
%e A333145                                             (4312)  (33122)
%e A333145                                             (4321)  (33212)
%e A333145                                                     (33221)
%t A333145 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A333145 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t A333145 Table[Length[Select[Permutations[primeMS[n]],unimodQ[-#]&]],{n,30}]
%Y A333145 Dominated by A008480.
%Y A333145 The complementary divisors are counted by A069157.
%Y A333145 The non-negated version is A332288.
%Y A333145 A more interesting version is A332741.
%Y A333145 The complement is counted by A333146.
%Y A333145 A001523 counts unimodal compositions.
%Y A333145 A007052 counts unimodal normal sequences.
%Y A333145 A028233 gives the highest power of the least prime factor, quotient A028234.
%Y A333145 A332578 counts compositions whose negation is unimodal.
%Y A333145 A332638 counts partitions with unimodal negated run-lengths.
%Y A333145 A332642 lists numbers with non-unimodal negated unsorted prime signature.
%Y A333145 Cf. A056239, A112798, A115981, A124010, A328509, A332283, A332294, A332639, A332669, A332670, A332671.
%K A333145 nonn
%O A333145 1,6
%A A333145 _Gus Wiseman_, Mar 09 2020