A333146 Number of non-unimodal negated permutations of the multiset of prime indices of n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 2, 0, 1, 1, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 8, 0, 0, 1, 0, 0, 2, 0, 1, 0, 2, 0, 7, 0, 0, 0, 1, 0, 2, 0, 3, 0, 0, 0, 8, 0, 0, 0

Offset

1,24

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.

Links

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

Eric Weisstein's World of Mathematics, Unimodal Sequence

Formula

a(n) + A333145(n) = A008480(n).

Example

The a(n) permutations for n = 12, 24, 36, 60, 72, 90, 96:
  (121)  (1121)  (1212)  (1132)  (11212)  (1232)  (111121)
         (1211)  (1221)  (1213)  (11221)  (1322)  (111211)
                 (2121)  (1231)  (12112)  (2132)  (112111)
                         (1312)  (12121)  (2231)  (121111)
                         (1321)  (12211)  (2312)
                         (2131)  (21121)  (2321)
                         (2311)  (21211)
                         (3121)

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 A332671.

A more interesting version is A332742.

The complement is counted by A333145.

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, A332288, A332294, A332639, A332669, A332670, A332741.

Sequence in context: A003475 A248639 A293959 * A135767 A208575 A355037

Adjacent sequences: A333143 A333144 A333145 * A333147 A333148 A333149

Keyword

nonn

Author

Gus Wiseman, Mar 09 2020

Status

approved