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A290959
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Matrix rank of the number of dots in the pairwise intersections of Ferrers diagrams.
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0
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1, 2, 3, 5, 7, 11, 13, 17, 20, 24, 26, 32, 34, 38, 42, 47, 49, 55, 57, 63, 67, 71, 73, 81, 84, 88
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OFFSET
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1,2
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COMMENTS
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Let f(q, r) be the number of dots in the intersection of the Ferrers diagrams of the integer partitions q and r of n. Let a(n) be the matrix rank of the p(n) by p(n) matrix of f(q, r) as q and r range over the partitions of n. Conjecture: For n > 3, a(n+1) - a(n) = A000005(n+2), the number of divisors of n. The same is true empirically for the union, complement, and set difference. Note that A000005 count rectangular partitions.
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LINKS
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MATHEMATICA
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intersection[{p_, q_}] := Module[{min},
min = Min[Length /@ {p, q}];
Total[Min /@ Transpose@{Take[p, min], Take[q, min]}]
];
intersections@k_ := intersections@k = Module[{ip = IntegerPartitions[k]},
Table[intersection@{ip[[m]], ip[[n]]}, {m, PartitionsP@k}, {n,
PartitionsP@k}]];
a[n_]:=MatrixRank@intersections@n;
Table[MatrixRank@intersections@n, {n, 20}]
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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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