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A332672 Number of non-unimodal permutations of a multiset whose multiplicities are the prime indices of n. 10
0, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 6, 0, 0, 6, 16, 0, 21, 0, 12, 10, 0, 0, 48, 16, 0, 81, 20, 0, 48, 0, 104, 15, 0, 30, 162, 0, 0, 21, 104, 0, 90, 0, 30, 198, 0, 0, 336, 65, 124, 28, 42, 0, 603, 50, 190, 36, 0, 0, 396, 0, 0, 405, 688, 77, 150, 0, 56, 45, 260, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

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 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..71.

MathWorld, Unimodal Sequence

FORMULA

a(n) = A332671(A181821(n)).

a(n) + A332294(n) = A318762(n).

EXAMPLE

The a(n) permutations for n = 8, 9, 12, 15, 16:

  213   1212   1213   11212   1324

  312   2112   1312   12112   1423

        2121   2113   12121   2134

               2131   21112   2143

               3112   21121   2314

               3121   21211   2413

                              3124

                              3142

                              3214

                              3241

                              3412

                              4123

                              4132

                              4213

                              4231

                              4312

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

Positions of zeros are one and A001751.

Support is A264828 without one.

Dominated by A318762.

The complement is counted by A332294.

A less interesting version is A332671.

The opposite version is A332742.

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, A008480, A056239, A112798, A124010, A181819, A181821, A332281, A332287, A332294, A332642, A332741.

Sequence in context: A213714 A242011 A259657 * A184362 A011311 A240658

Adjacent sequences:  A332669 A332670 A332671 * A332673 A332674 A332675

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 23 2020

STATUS

approved

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Last modified July 28 14:19 EDT 2021. Contains 346335 sequences. (Running on oeis4.)