A218905 -id:A218905 - cp's OEIS frontend

A218904 Sum of the sizes of the kernels of all integer partitions of n.

1, 2, 5, 12, 21, 40, 65, 108, 170, 264, 392, 590, 847, 1222, 1720, 2418, 3323, 4574, 6180, 8350, 11124, 14790, 19443, 25532, 33186, 43052, 55418, 71170, 90769, 115542, 146164, 184520, 231743, 290396, 362245, 450950, 559035, 691624, 852583, 1048870, 1286109

Offset: 1

Olivier Gérard, Nov 08 2012

nonn

The kernel of an integer partition is the intersection of its Ferrers diagram and of the Ferrers diagram of its conjugate. It includes the Durfee square, which is the largest square fitting into the kernel.

Alois P. Heinz, Table of n, a(n) for n = 1..70

Row sums of A218905.

Extended beyond a(20) by Alois P. Heinz, Nov 08 2012

A218907 Triangle, read by rows, of integer partitions of n by kernel size k.

Original entry on oeis.org

1, 2, 0, 2, 0, 1, 2, 0, 2, 1, 2, 0, 2, 2, 1, 2, 0, 2, 4, 2, 1, 2, 0, 2, 4, 2, 4, 1, 2, 0, 2, 6, 2, 6, 2, 2, 2, 0, 2, 6, 2, 8, 2, 6, 2, 2, 0, 2, 8, 2, 8, 2, 12, 4, 2, 2, 0, 2, 8, 2, 10, 2, 14, 6, 8, 2, 2, 0, 2, 10, 2, 10, 2, 18, 8, 14, 6, 3, 2, 0, 2, 10, 2, 12, 2, 18, 10, 20, 10, 10, 3, 2, 0, 2, 12, 2, 12, 2, 22, 12, 22, 14, 20, 10, 3, 2, 0, 2, 12, 2, 14, 2, 22, 16, 26, 16, 26, 20, 12, 4

Offset: 1

Olivier Gérard, Nov 08 2012

Row sum is A000041.
Sum k*T(n,k) = A208914(n).
The kernel of an integer partition is the intersection of its Ferrers diagram and of the Ferrers diagram of its conjugate.
  Its size is between 1 (for an all-1 partition) and n (for a self-conjugate partition).

			Triangle begins:
1;
2, 0;
2, 0, 1;
2, 0, 2,  1;
2, 0, 2,  2, 1;
2, 0, 2,  4, 2,  1;
2, 0, 2,  4, 2,  4, 1;
2, 0, 2,  6, 2,  6, 2,  2;
2, 0, 2,  6, 2,  8, 2,  6,  2;
2, 0, 2,  8, 2,  8, 2, 12,  4,  2;
2, 0, 2,  8, 2, 10, 2, 14,  6,  8,  2;
2, 0, 2, 10, 2, 10, 2, 18,  8, 14,  6,  3;
2, 0, 2, 10, 2, 12, 2, 18, 10, 20, 10, 10, 3;
2, 0, 2, 12, 2, 12, 2, 22, 12, 22, 14, 20, 10, 3;
2, 0, 2, 12, 2, 14, 2, 22, 16, 26, 16, 26, 20, 12, 4;

Cf. A218904, A218905, A218906.
Cf. A115720, A115994, A246581.
Main diagonal gives A000700.

A290959 Matrix rank of the number of dots in the pairwise intersections of Ferrers diagrams.

Original entry on oeis.org

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

George Beck, Aug 14 2017

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.

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

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}]

cp's OEIS Frontend

A218904 Sum of the sizes of the kernels of all integer partitions of n.

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A218907 Triangle, read by rows, of integer partitions of n by kernel size k.

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A290959 Matrix rank of the number of dots in the pairwise intersections of Ferrers diagrams.

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