

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



