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A332288
Number of unimodal permutations of the multiset of prime indices of n.
17
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).
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.
Sequence in context: A140747 A330757 A322373 * A335450 A372772 A324191
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 22 2020
STATUS
approved