A332671 Number of non-unimodal permutations of the multiset of prime indices of n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 6, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 6, 0, 0, 0

Offset

1,30

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.

MathWorld, Unimodal Sequence

Formula

a(n) + A332288(n) = A008480(n).

a(A181821(n)) = A332672(n).

Example

The a(n) permutations for n = 18, 30, 36, 42, 50, 54, 60, 66, 70, 72:
  212  213  1212  214  313  2122  1213  215  314  11212
       312  2112  412       2212  1312  512  413  12112
            2121                  2113            12121
                                  2131            21112
                                  3112            21121
                                  3121            21211

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, 100}]

Crossrefs

Dominated by A008480.

The complement is counted by A332288.

A more interesting version is A332672.

Unimodal compositions are A001523.

Non-unimodal permutations are A059204.

Non-unimodal compositions are A115981.

Non-unimodal normal sequences are A328509.

Heinz numbers of partitions with non-unimodal run-lengths are A332282.

Compositions whose negation is not unimodal are A332669.

Cf. A007052, A056239, A112798, A124010, A332281, A332284, A332287, A332294, A332639, A332642.

Sequence in context: A351254 A356303 A238429 * A074039 A047764 A022883

Adjacent sequences: A332668 A332669 A332670 * A332672 A332673 A332674

Keyword

nonn

Author

Gus Wiseman, Feb 22 2020

Status

approved