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.

A218905 Irregular triangle, read by rows, of kernel sizes of the integer partitions of n taken in graded reverse lexicographic ordering.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 3, 4, 3, 1, 1, 3, 4, 5, 4, 3, 1, 1, 3, 4, 5, 4, 6, 5, 4, 4, 3, 1, 1, 3, 4, 5, 4, 6, 7, 6, 6, 6, 5, 4, 4, 3, 1, 1, 3, 4, 5, 4, 6, 7, 4, 6, 6, 8, 7, 8, 6, 6, 6, 5, 4, 4, 4, 3, 1, 1, 3, 4, 5, 4, 6, 7, 4, 6, 6, 8, 9, 6, 8, 8, 8, 8, 7, 9, 8, 6, 6, 6, 6, 5, 4, 4, 4, 3, 1, 1, 3, 4, 5, 4, 6, 7, 4, 6, 6, 8, 9, 4, 6, 8, 8, 8, 10, 9, 8, 8, 9, 10, 8, 8, 8, 8, 7, 9, 8, 8, 6, 6, 6, 6, 5, 4, 4, 4, 4, 3, 1
Offset: 1

Views

Author

Olivier Gérard, Nov 08 2012

Keywords

Comments

The kernel of an integer partition is the intersection of its Ferrers diagram and of the Ferrers diagram of its conjugate.
See comments in A080577 for the graded reverse lexicographic ordering.
Row length is A000041(n).
Row sum is A218904(n).

Examples

			Triangle begins:
  1;
  1, 1;
  1, 3, 1;
  1, 3, 4, 3, 1;
  1, 3, 4, 5, 4, 3, 1;
  1, 3, 4, 5, 4, 6, 5, 4, 4, 3, 1;
  1, 3, 4, 5, 4, 6, 7, 6, 6, 6, 5, 4, 4, 3, 1;
  1, 3, 4, 5, 4, 6, 7, 4, 6, 6, 8, 7, 8, 6, 6, 6, 5, 4, 4, 4, 3, 1;
  ...

Links

Crossrefs

Cf. A218904.

Programs

  • Maple
    h:= proc(l) local ll; ll:= [seq(add(
       `if`(l[j]>=i, 1, 0), j=1..nops(l)), i=1..l[1])];
       add(min(l[i], ll[i]), i=1..min(nops(l), nops(ll)))
    end:
g:= (n, i, l)-> `if`(n=0 or i=1, [h([l[], 1$n])],
    [`if`(i>n, [], g(n-i, i, [l[], i]))[], g(n, i-1, l)[]]):
T:= n-> g(n, n, [])[]:
seq(T(n), n=1..10);  # Alois P. Heinz, Dec 14 2012
  • Mathematica
    h[l_List] := Module[{ll}, ll = Flatten[Table[Sum[If[l[[j]] >= i, 1, 0], {j, 1, Length[l]}], {i, 1, l[[1]]}]]; Sum[Min[l[[i]], ll[[i]]], {i, 1, Min[ Length[l], Length[ll]]}]]; g[n_, i_, l_List] := If[n==0 || i==1, Join[ {h[Join[l, Array[1&, n]]]}], Join[If[i>n, {}, g[n-i, i, Join [l, {i}]]], g[n, i-1, l]]]; T[n_] := g[n, n, {}]; Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Dec 23 2015, after Alois P. Heinz *)

A218906 Number of different kernels of integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 17, 20, 23, 27, 32, 37, 42, 48, 55, 63, 71, 80, 91, 103, 115, 129, 145, 162, 180, 200, 223, 248, 274, 303, 336, 371, 408, 449, 495, 544, 596, 653, 716, 784, 856, 934, 1021, 1114, 1212, 1319, 1436, 1561, 1694, 1838, 1995
Offset: 1

Views

Author

Olivier Gérard, Nov 08 2012

Keywords

Comments

The kernel of an integer partition is the intersection of its Ferrers diagram and of the Ferrers diagram of its conjugate.
It is also a partition of an integer (called the size of the kernel), always self-conjugate.
In fact, this sequence is the cumulative sum of A000700, the number of self-conjugate partitions of n.

Links

Crossrefs

Cf. A218904.
Cf. A000700.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-2)+`if`(i>n, 0, b(n-i, i-2))))
    end:
a:= proc(n) a(n):= b(n, n-1+irem(n, 2))+`if`(n=1, 0, a(n-1)) end:
seq (a(n), n=1..100);  # Alois P. Heinz, Nov 09 2012
  • Mathematica
    b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-2] + If[i>n, 0, b[n-i, i-2]]]]; a[n_] := b[n, n-1 + Mod[n, 2]] + If[n==1, 0, a[n-1]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 12 2015, after Alois P. Heinz *)

Formula

G.f.: -1/(1 - x) + (1/(1 - x))*Product_{k>=1} (1 + x^(2*k-1)). - Ilya Gutkovskiy, Dec 25 2016

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.

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.

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