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A321738
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Number of ways to partition the Young diagram of the integer partition with Heinz number n into vertical sections.
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13
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1, 1, 1, 2, 1, 3, 1, 5, 7, 4, 1, 10, 1, 5, 13, 15, 1, 27, 1, 17, 21, 6, 1, 37, 34, 7, 87, 26, 1, 60, 1, 52, 31, 8, 73, 114, 1, 9, 43, 77, 1, 115, 1, 37, 235, 10, 1, 151, 209, 175, 57, 50, 1, 409, 136, 141, 73, 11, 1, 295, 1, 12, 543, 203, 229, 198, 1, 65, 91
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OFFSET
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1,4
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COMMENTS
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The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
A vertical section is a partial Young diagram with at most one square in each row. For example, a partition (shown as a coloring by positive integers) into vertical sections of the Young diagram of (322) is:
1 2 3
1 2
2 3
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LINKS
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EXAMPLE
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The a(12) = 10 partitions of the Young diagram of (211) into vertical sections:
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
3 3 2 3 2 1 1 3 2 1
4 3 3 2 2 3 2 1 1 1
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
spsu[_, {}]:={{}}; spsu[foo_, set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@spsu[Select[foo, Complement[#, Complement[set, s]]=={}&], Complement[set, s]]]/@Cases[foo, {i, ___}];
ptnpos[y_]:=Position[Table[1, {#}]&/@y, 1];
ptnverts[y_]:=Select[Rest[Subsets[ptnpos[y]]], UnsameQ@@First/@#&];
Table[With[{y=Reverse[primeMS[n]]}, Length[spsu[ptnverts[y], ptnpos[y]]]], {n, 30}]
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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