A332294 Number of unimodal permutations of a multiset whose multiplicities are the prime indices of n.

1, 1, 1, 2, 1, 3, 1, 4, 3, 4, 1, 6, 1, 5, 4, 8, 1, 9, 1, 8, 5, 6, 1, 12, 4, 7, 9, 10, 1, 12, 1, 16, 6, 8, 5, 18, 1, 9, 7, 16, 1, 15, 1, 12, 12, 10, 1, 24, 5, 16, 8, 14, 1, 27, 6, 20, 9, 11, 1, 24, 1, 12, 15, 32, 7, 18, 1, 16, 10, 20, 1, 36, 1, 13, 16, 18, 6

Offset

1,4

Comments

This multiset is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

A sequence of positive 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..77.

MathWorld, Unimodal Sequence

Formula

a(n) + A332672(n) = A318762(n).

a(n) = A332288(A181821(n)).

Example

The a(12) = 6 permutations:
  {1,1,2,3}
  {1,1,3,2}
  {1,2,3,1}
  {1,3,2,1}
  {2,3,1,1}
  {3,2,1,1}

Mathematica

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {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[nrmptn[n]], unimodQ]], {n, 0, 30}]

Crossrefs

Dominated by A318762.

A less interesting version is A332288.

The complement is counted by A332672.

The opposite/negative version is A332741.

Unimodal compositions are A001523.

Non-unimodal permutations are A059204.

Partitions whose run-lengths are unimodal are A332280.

Cf. A007052, A056239, A112798, A115981, A124010, A304660, A328509, A332283, A332578, A332638, A332671, A332742.

Sequence in context: A091420 A323906 A020952 * A079554 A247892 A366649

Adjacent sequences: A332291 A332292 A332293 * A332295 A332296 A332297

Keyword

nonn

Author

Gus Wiseman, Feb 21 2020

Status

approved