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

1,5

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).

Links

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

Example

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;

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 *)

Crossrefs

Cf. A218904.

Sequence in context: A124794 A206496 A097560 * A027960 A319182 A247282

Adjacent sequences: A218902 A218903 A218904 * A218906 A218907 A218908

Keyword

nonn,tabf,look

Author

Olivier Gérard, Nov 08 2012

Status

approved