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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.

%I A218905 #20 Nov 04 2024 19:24:06
%S A218905 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,
%T A218905 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,
%U A218905 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
%N A218905 Irregular triangle, read by rows, of kernel sizes of the integer partitions of n taken in graded reverse lexicographic ordering.
%C A218905 The kernel of an integer partition is the intersection of its Ferrers diagram and of the Ferrers diagram of its conjugate.
%C A218905 See comments in A080577 for the graded reverse lexicographic ordering.
%C A218905 Row length is A000041(n).
%C A218905 Row sum is A218904(n).
%H A218905 Alois P. Heinz, <a href="/A218905/b218905.txt">Rows n = 1..26, flattened</a>
%e A218905 Triangle begins:
%e A218905   1;
%e A218905   1, 1;
%e A218905   1, 3, 1;
%e A218905   1, 3, 4, 3, 1;
%e A218905   1, 3, 4, 5, 4, 3, 1;
%e A218905   1, 3, 4, 5, 4, 6, 5, 4, 4, 3, 1;
%e A218905   1, 3, 4, 5, 4, 6, 7, 6, 6, 6, 5, 4, 4, 3, 1;
%e A218905   1, 3, 4, 5, 4, 6, 7, 4, 6, 6, 8, 7, 8, 6, 6, 6, 5, 4, 4, 4, 3, 1;
%e A218905   ...
%p A218905 h:= proc(l) local ll; ll:= [seq(add(
%p A218905        `if`(l[j]>=i, 1, 0), j=1..nops(l)), i=1..l[1])];
%p A218905        add(min(l[i], ll[i]), i=1..min(nops(l), nops(ll)))
%p A218905     end:
%p A218905 g:= (n, i, l)-> `if`(n=0 or i=1, [h([l[], 1$n])],
%p A218905     [`if`(i>n, [], g(n-i, i, [l[], i]))[], g(n, i-1, l)[]]):
%p A218905 T:= n-> g(n, n, [])[]:
%p A218905 seq(T(n), n=1..10);  # _Alois P. Heinz_, Dec 14 2012
%t A218905 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_ *)
%Y A218905 Cf. A218904.
%K A218905 nonn,tabf,look
%O A218905 1,5
%A A218905 _Olivier Gérard_, Nov 08 2012