A218906 -id:A218906 - cp's OEIS frontend

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

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.

A304631 Expansion of (1/(1 - x))Product_{k>=1} (1 + x^(2k-1)).

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 18, 21, 24, 28, 33, 38, 43, 49, 56, 64, 72, 81, 92, 104, 116, 130, 146, 163, 181, 201, 224, 249, 275, 304, 337, 372, 409, 450, 496, 545, 597, 654, 717, 785, 857, 935, 1022, 1115, 1213, 1320, 1437, 1562, 1695, 1839, 1996, 2164, 2342, 2534

Offset: 0

Ilya Gutkovskiy, May 15 2018

nonn

Partial sums of A000700.

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
Eric Weisstein's World of Mathematics, Self-Conjugate Partition
Index entries for sequences related to partitions

Cf. A000070, A000700, A036469, A218906, A304632.

nmax = 59; CoefficientList[Series[1/(1 - x) Product[(1 + x^(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 59; CoefficientList[Series[1/(1 - x) Product[1/(1 + (-x)^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: (1/(1 - x))*Product_{k>=1} 1/(1 + (-x)^k).

a(n) ~ exp(Pi*sqrt(n/6)) * 3^(1/4) / (Pi * 2^(1/4) * n^(1/4)). - Vaclav Kotesovec, May 19 2018

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

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

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A304631 Expansion of (1/(1 - x))Product_{k>=1} (1 + x^(2k-1)).

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Formula

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

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Author

Keywords

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Mathematica

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

A304631 Expansion of (1/(1 - x))*Product_{k>=1} (1 + x^(2*k-1)).

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A304631 Expansion of (1/(1 - x))Product_{k>=1} (1 + x^(2k-1)).