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Heinz numbers of partitions Numbers with non-unimodal negated run-lengths prime signature are A332642.

Heinz numbers of partitions with non-unimodal negated run-lengths are A332642.

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

allocated for Gus WisemanNumber 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

1,4

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.

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>

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

The a(12) = 4 permutations:

{1,1,2,3}

{2,1,1,3}

{3,1,1,2}

{3,2,1,1}

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}]

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.

Heinz numbers of partitions with unimodal negated run-lengths are A332642.

Partitions whose first differences with zero appended 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.

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Gus Wiseman, Mar 09 2020

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