A332288 - OEIS

A332288

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

%I #5 Feb 24 2020 21:56:26

%S 1,1,1,1,1,2,1,1,1,2,1,3,1,2,2,1,1,2,1,3,2,2,1,4,1,2,1,3,1,4,1,1,2,2,

%T 2,3,1,2,2,4,1,4,1,3,3,2,1,5,1,2,2,3,1,2,2,4,2,2,1,6,1,2,3,1,2,4,1,3,

%U 2,4,1,4,1,2,2,3,2,4,1,5,1,2,1,6,2,2,2

%N Number of unimodal permutations of the multiset of prime indices of n.

%C 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 A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

%C Also permutations of the multiset of prime indices of n avoiding the patterns (2,1,2), (2,1,3), and (3,1,2).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>

%H MathWorld, <a href="http://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>

%e The a(n) permutations for n = 2, 6, 12, 24, 48, 60, 120, 180:

%e (1) (12) (112) (1112) (11112) (1123) (11123) (11223)

%e (21) (121) (1121) (11121) (1132) (11132) (11232)

%e (211) (1211) (11211) (1231) (11231) (11322)

%e (2111) (12111) (1321) (11321) (12231)

%e (21111) (2311) (12311) (12321)

%e (3211) (13211) (13221)

%e (23111) (22311)

%e (32111) (23211)

%e (32211)

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];

%t Table[Length[Select[Permutations[primeMS[n]],unimodQ]],{n,30}]

%Y Dominated by A008480.

%Y A more interesting version is A332294.

%Y The complement is counted by A332671.

%Y Unimodal compositions are A001523.

%Y Unimodal normal sequences appear to be A007052.

%Y Unimodal permutations are A011782.

%Y Non-unimodal permutations are A059204.

%Y Numbers with non-unimodal unsorted prime signature are A332282.

%Y Partitions with unimodal 0-appended first differences are A332283.

%Y Cf. A056239, A112798, A115981, A124010, A227038, A304660, A328509, A332280, A332284, A332294, A332578, A332672.

%K nonn

%O 1,6

%A _Gus Wiseman_, Feb 22 2020