cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Olivier Gérard, Nov 08 2012

Keywords

Comments

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).

Examples

			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;

Crossrefs

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

A246582 G.f.: x^((k^2+k)/2)/(mul(1-x^i,i=1..k)*mul(1+x^r,r=1..oo)) with k = 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, -1, 1, -2, 1, -3, 1, -4, 2, -5, 3, -6, 5, -7, 7, -8, 10, -10, 13, -12, 17, -15, 21, -19, 26, -24, 31, -30, 38, -38, 45, -47, 54, -58, 64, -71, 77, -86, 91, -103, 109, -124, 129, -147, 154, -174, 182, -205, 216, -241, 254, -282, 300, -330, 351, -384, 412, -447, 480, -519, 560, -602, 649, -696, 753, -805
Offset: 0

Views

Author

N. J. A. Sloane, Aug 31 2014

Keywords

References

  • Fulman, Jason. Random matrix theory over finite fields. Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 3, with k=3.

Links

Crossrefs

For k=0 and 1 we get A081362, A027349 (apart from signs). Cf. A246581, A246583.

Programs

  • Maple
    fSp:=proc(k) local a,i,r;
a:=x^((k^2+k)/2)/mul(1-x^i,i=1..k);
a:=a/mul(1+x^r,r=1..101);
series(a,x,101);
seriestolist(%);
end;
fSp(3);
  • Mathematica
    nmax = 100; CoefficientList[Series[x^6/((1-x)*(1-x^2)*(1-x^3)) * Product[1/(1+x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 11 2016 *)

Formula

a(n) ~ (-1)^n * 3^(1/4) * exp(sqrt(n/6)*Pi) / (2^(13/4)*Pi*n^(1/4)). - Vaclav Kotesovec, Mar 12 2016

A246583 G.f.: x^((k^2+k)/2)/(mul(1-x^i,i=1..k)*mul(1+x^r,r=1..oo)) with k = 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, -1, 2, -2, 3, -4, 3, -6, 5, -9, 6, -12, 10, -16, 13, -20, 20, -26, 26, -32, 37, -41, 47, -51, 63, -65, 78, -81, 101, -103, 123, -128, 155, -161, 187, -199, 232, -247, 278, -302, 341, -371, 407, -449, 495, -545, 589, -654, 711, -786, 843, -936, 1011, -1116, 1194, -1320, 1423, -1563, 1674
Offset: 0

Views

Author

N. J. A. Sloane, Aug 31 2014

Keywords

References

  • Fulman, Jason. Random matrix theory over finite fields. Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 3, with k=4.

Links

Crossrefs

For k=0 and 1 we get A081362, A027349 (apart from signs). Cf. A246581, A246582.

Programs

  • Maple
    fSp:=proc(k) local a,i,r;
a:=x^((k^2+k)/2)/mul(1-x^i,i=1..k);
a:=a/mul(1+x^r,r=1..101);
series(a,x,101);
seriestolist(%);
end;
fSp(4);
  • Mathematica
    nmax = 100; CoefficientList[Series[x^10/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) * Product[1/(1+x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 11 2016 *)

Formula

a(n) ~ (-1)^n * 3^(3/4) * n^(1/4) * exp(sqrt(n/6)*Pi) / (2^(15/4)*Pi^2). - Vaclav Kotesovec, Mar 12 2016

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

Views

Author

George Beck, Aug 14 2017

Keywords

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.

Crossrefs

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

Programs

  • 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}]
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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