A290959 - OEIS

A290959

Matrix rank of the number of dots in the pairwise intersections of Ferrers diagrams.

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

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..26.

MATHEMATICA

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}]