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

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

Author

Olivier Gérard, Nov 08 2012

Keywords

• nonn
• tabf
• look

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

• Alois P. Heinz, Rows n = 1..26, flattened

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 *)