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A323584
Second Moebius transform of A000219. Number of plane partitions of n whose multiset of rows is aperiodic and whose multiset of columns is also aperiodic.
4
1, 1, 1, 4, 8, 22, 34, 84, 137, 271, 450, 857, 1373, 2483, 3993, 6823, 10990, 18332, 28966, 47328, 74286, 118614, 184755, 290781, 448010, 695986, 1063773, 1632100, 2474970, 3759610, 5654233, 8512307, 12710995, 18973247, 28139285, 41690830, 61423271, 90379782
OFFSET
0,4
COMMENTS
A multiset is aperiodic if its multiplicities are relatively prime.
Also the number of plane partitions of n whose multiset of rows is aperiodic and whose parts are relatively prime.
LINKS
FORMULA
The Moebius transform T of a sequence q is T(q)(n) = Sum_{d|n} mu(n/d) * q(d) where mu = A008683. The first Moebius transform of A000219 is A300275 and the third is A323585.
EXAMPLE
The a(4) = 8 plane partitions with aperiodic multisets of rows and columns:
4 31 211
.
3 21 111
1 1 1
.
2 11
1 1
1 1
The a(4) = 8 plane partitions with aperiodic multiset of rows and relatively prime parts:
31 211 1111
.
3 21 111
1 1 1
.
2 11
1 1
1 1
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[Reverse@*primeMS, Join@@Permutations/@facs[n], {2}]];
Table[Sum[Length[Select[ptnplane[Times@@Prime/@y], And[GCD@@Length/@Split[#]==1, And@@GreaterEqual@@@#, And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]], {y, Select[IntegerPartitions[n], GCD@@#==1&]}], {n, 10}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 19 2019
STATUS
approved