A332741 Number of unimodal negated permutations of a multiset whose multiplicities are the prime indices of n.

1, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 3, 8, 1, 6, 1, 4, 3, 2, 1, 8, 4, 2, 9, 4, 1, 6, 1, 16, 3, 2, 4, 12, 1, 2, 3, 8, 1, 6, 1, 4, 9, 2, 1, 16, 5, 8, 3, 4, 1, 18, 4, 8, 3, 2, 1, 12, 1, 2, 9, 32, 4, 6, 1, 4, 3, 8, 1, 24, 1, 2, 12, 4, 5, 6, 1, 16, 27, 2, 1

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..83.

Eric Weisstein's World of Mathematics, Unimodal Sequence

Formula

a(n) + A332742(n) = A318762(n).

Example

The a(12) = 4 permutations:
  {1,1,2,3}
  {2,1,1,3}
  {3,1,1,2}
  {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, 30}]

Crossrefs

Dominated by A318762.

The non-negated version is A332294.

The complement is counted by A332742.

A less interesting version is A333145.

Unimodal compositions are A001523.

Unimodal normal sequences are A007052.

Numbers with non-unimodal negated prime signature are A332642.

Partitions whose 0-appended first differences are unimodal are A332283.

Compositions whose negation is unimodal are A332578.

Partitions with unimodal negated run-lengths are A332638.

Cf. A056239, A112798, A115981, A124010, A181819, A181821, A304660, A332280, A332288, A332639, A332669, A332672.

Sequence in context: A325567 A009195 A072994 * A052126 A094521 A321757

Adjacent sequences: A332738 A332739 A332740 * A332742 A332743 A332744

Keyword

nonn

Author

Gus Wiseman, Mar 09 2020

Status

approved