

A332294


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


14



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
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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.
Nonunimodal permutations are A059204.
Partitions whose runlengths 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 A079880
Adjacent sequences: A332291 A332292 A332293 * A332295 A332296 A332297


KEYWORD

nonn


AUTHOR

Gus Wiseman, Feb 21 2020


STATUS

approved



