login
Irregular triangle, read by rows, of kernel sizes of the integer partitions of n taken in graded reverse lexicographic ordering.
4

%I #20 Nov 04 2024 19:24:06

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

%C The kernel of an integer partition is the intersection of its Ferrers diagram and of the Ferrers diagram of its conjugate.

%C See comments in A080577 for the graded reverse lexicographic ordering.

%C Row length is A000041(n).

%C Row sum is A218904(n).

%H Alois P. Heinz, <a href="/A218905/b218905.txt">Rows n = 1..26, flattened</a>

%e Triangle begins:

%e 1;

%e 1, 1;

%e 1, 3, 1;

%e 1, 3, 4, 3, 1;

%e 1, 3, 4, 5, 4, 3, 1;

%e 1, 3, 4, 5, 4, 6, 5, 4, 4, 3, 1;

%e 1, 3, 4, 5, 4, 6, 7, 6, 6, 6, 5, 4, 4, 3, 1;

%e 1, 3, 4, 5, 4, 6, 7, 4, 6, 6, 8, 7, 8, 6, 6, 6, 5, 4, 4, 4, 3, 1;

%e ...

%p h:= proc(l) local ll; ll:= [seq(add(

%p `if`(l[j]>=i, 1, 0), j=1..nops(l)), i=1..l[1])];

%p add(min(l[i], ll[i]), i=1..min(nops(l), nops(ll)))

%p end:

%p g:= (n, i, l)-> `if`(n=0 or i=1, [h([l[], 1$n])],

%p [`if`(i>n, [], g(n-i, i, [l[], i]))[], g(n, i-1, l)[]]):

%p T:= n-> g(n, n, [])[]:

%p seq(T(n), n=1..10); # _Alois P. Heinz_, Dec 14 2012

%t 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 Cf. A218904.

%K nonn,tabf,look,changed

%O 1,5

%A _Olivier Gérard_, Nov 08 2012