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A290959 Matrix rank of the number of dots in the pairwise intersections of Ferrers diagrams. 0
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 21 03:24 EDT 2019. Contains 328291 sequences. (Running on oeis4.)