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

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}]

Crossrefs

Cf. A000005, A218904, A218905, A218906, A218907, A246581.

Sequence in context: A248199 A198196 A139054 * A003309 A063884 A316787

Adjacent sequences: A290956 A290957 A290958 * A290960 A290961 A290962

Keyword

nonn,more

Author

George Beck, Aug 14 2017

Status

approved