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A218905 Irregular triangle, read by rows, of kernel sizes of the integer partitions of n taken in graded reverse lexicographic ordering. 4
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn,tabf,look
AUTHOR
Olivier Gérard, Nov 08 2012
STATUS
approved

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Last modified May 24 16:32 EDT 2024. Contains 372781 sequences. (Running on oeis4.)