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%I #13 Aug 16 2017 23:48:02
%S 1,2,3,5,7,11,13,17,20,24,26,32,34,38,42,47,49,55,57,63,67,71,73,81,
%T 84,88
%N Matrix rank of the number of dots in the pairwise intersections of Ferrers diagrams.
%C 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.
%t intersection[{p_, q_}] := Module[{min},
%t min = Min[Length /@ {p, q}];
%t Total[Min /@ Transpose@{Take[p, min], Take[q, min]}]
%t ];
%t intersections@k_ := intersections@k = Module[{ip = IntegerPartitions[k]},
%t Table[intersection@{ip[[m]], ip[[n]]}, {m, PartitionsP@k}, {n,
%t PartitionsP@k}]];
%t a[n_]:=MatrixRank@intersections@n;
%t Table[MatrixRank@intersections@n, {n, 20}]
%Y Cf. A000005, A218904, A218905, A218906, A218907, A246581.
%K nonn,more
%O 1,2
%A _George Beck_, Aug 14 2017
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