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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
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.
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.
Sequence in context: A353612 A368243 A259657 * A184362 A354443 A011311
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 23 2020
STATUS
approved