A332288 Number of unimodal permutations of the multiset of prime indices of n.

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, 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, 2, 4, 1, 4, 1, 2, 2, 3, 2, 4, 1, 5, 1, 2, 1, 6, 2, 2, 2

Offset

1,6

Comments

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.

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

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

Links

Table of n, a(n) for n=1..87.

Wikipedia, Permutation pattern

MathWorld, Unimodal Sequence

Example

The a(n) permutations for n = 2, 6, 12, 24, 48, 60, 120, 180:
  (1)  (12)  (112)  (1112)  (11112)  (1123)  (11123)  (11223)
       (21)  (121)  (1121)  (11121)  (1132)  (11132)  (11232)
             (211)  (1211)  (11211)  (1231)  (11231)  (11322)
                    (2111)  (12111)  (1321)  (11321)  (12231)
                            (21111)  (2311)  (12311)  (12321)
                                     (3211)  (13211)  (13221)
                                             (23111)  (22311)
                                             (32111)  (23211)
                                                      (32211)

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]];
Table[Length[Select[Permutations[primeMS[n]], unimodQ]], {n, 30}]

Crossrefs

Dominated by A008480.

A more interesting version is A332294.

The complement is counted by A332671.

Unimodal compositions are A001523.

Unimodal normal sequences appear to be A007052.

Unimodal permutations are A011782.

Non-unimodal permutations are A059204.

Numbers with non-unimodal unsorted prime signature are A332282.

Partitions with unimodal 0-appended first differences are A332283.

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

Sequence in context: A140747 A330757 A322373 * A335450 A372772 A324191

Adjacent sequences: A332285 A332286 A332287 * A332289 A332290 A332291

Keyword

nonn

Author

Gus Wiseman, Feb 22 2020

Status

approved