

A332741


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


10



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
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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..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 nonnegated 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 nonunimodal negated prime signature are A332642.
Partitions whose 0appended first differences are unimodal are A332283.
Compositions whose negation is unimodal are A332578.
Partitions with unimodal negated runlengths 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



