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A333145 Number of unimodal negated permutations of the multiset of prime indices of n. 3
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, 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, 2, 4, 1, 3, 1, 2, 3, 2, 2, 4, 1, 2, 1, 2, 1, 4, 2, 2, 2 (list; graph; refs; listen; history; text; internal format)
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 (1,2,1), (1,3,2), and (2,3,1).

LINKS

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

Eric Weisstein's World of Mathematics, Unimodal Sequence

Wikipedia, Permutation pattern

FORMULA

a(n) + A333146(n) = A008480(n).

EXAMPLE

The a(n) permutations for n = 2, 6, 18, 30, 90, 162, 210, 450:

  (1)  (12)  (122)  (123)  (1223)  (12222)  (1234)  (12233)

       (21)  (212)  (213)  (2123)  (21222)  (2134)  (21233)

             (221)  (312)  (2213)  (22122)  (3124)  (22133)

                    (321)  (3122)  (22212)  (3214)  (31223)

                           (3212)  (22221)  (4123)  (32123)

                           (3221)           (4213)  (32213)

                                            (4312)  (33122)

                                            (4321)  (33212)

                                                    (33221)

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.

The non-negated version is A332288.

A more interesting version is A332741.

The complement is counted by A333146.

Unimodal compositions are A001523.

Unimodal normal sequences are A007052.

Compositions whose negation is unimodal are A332578.

Partitions with unimodal negated run-lengths are A332638.

Numbers with non-unimodal negated unsorted prime signature are A332642.

Cf. A056239, A112798, A115981, A124010, A328509, A332283, A332294, A332639, A332669, A332670, A332671.

Sequence in context: A336570 A121382 A305150 * A335449 A318464 A051265

Adjacent sequences:  A333142 A333143 A333144 * A333146 A333147 A333148

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 09 2020

STATUS

approved

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Last modified July 28 13:15 EDT 2021. Contains 346332 sequences. (Running on oeis4.)