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A256553 Triangle T(n,k) in which the n-th row contains the increasing list of distinct orders of degree-n permutations; n>=0, 1<=k<=A009490(n).

%I A256553 #26 Jul 15 2021 19:35:15
%S A256553 1,1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,6,1,2,3,4,5,6,1,2,3,4,5,6,7,10,12,1,
%T A256553 2,3,4,5,6,7,8,10,12,15,1,2,3,4,5,6,7,8,9,10,12,14,15,20,1,2,3,4,5,6,
%U A256553 7,8,9,10,12,14,15,20,21,30
%N A256553 Triangle T(n,k) in which the n-th row contains the increasing list of distinct orders of degree-n permutations; n>=0, 1<=k<=A009490(n).
%H A256553 Alois P. Heinz, <a href="/A256553/b256553.txt">Rows n = 0..60, flattened</a>
%F A256553 Sum_{k>=0} T(n,k)*A256554(n,k) = A181844(n).
%F A256553 T(n,k) = k for n>0 and 1<=k<=n.
%e A256553 Triangle T(n,k) begins:
%e A256553   1;
%e A256553   1;
%e A256553   1, 2;
%e A256553   1, 2, 3;
%e A256553   1, 2, 3, 4;
%e A256553   1, 2, 3, 4, 5, 6;
%e A256553   1, 2, 3, 4, 5, 6;
%e A256553   1, 2, 3, 4, 5, 6, 7, 10, 12;
%e A256553   1, 2, 3, 4, 5, 6, 7,  8, 10, 12, 15;
%e A256553   1, 2, 3, 4, 5, 6, 7,  8,  9, 10, 12, 14, 15, 20;
%e A256553   1, 2, 3, 4, 5, 6, 7,  8,  9, 10, 12, 14, 15, 20, 21, 30;
%p A256553 b:= proc(n, i) option remember; `if`(n=0 or i=1, x,
%p A256553       b(n, i-1)+(p-> add(coeff(p, x, t)*x^ilcm(t, i),
%p A256553       t=1..degree(p)))(add(b(n-i*j, i-1), j=1..n/i)))
%p A256553     end:
%p A256553 T:= n->(p->seq((h->`if`(h=0, [][], i))(coeff(p, x, i))
%p A256553      , i=1..degree(p)))(b(n$2)):
%p A256553 seq(T(n), n=0..12);
%t A256553 b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x,
%t A256553      b[n, i - 1] + Function[p, Sum[Coefficient[p, x, t]*x^LCM[t, i],
%t A256553      {t, 1, Exponent[p, x]}]][Sum[b[n - i*j, i - 1], {j, 1, n/i}]]];
%t A256553 T[n_] := Function[p, Table[Function[h, If[h == 0, Nothing, i]][
%t A256553      Coefficient[p, x, i]], {i, 1, Exponent[p, x]}]][b[n, n]];
%t A256553 Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Jul 15 2021, after _Alois P. Heinz_ *)
%Y A256553 Row sums give A060179.
%Y A256553 Row lengths give A009490.
%Y A256553 Last elements of rows give A000793.
%Y A256553 Main diagonal gives A000027.
%Y A256553 Cf. A181844, A256067, A256554.
%K A256553 nonn,look,tabf
%O A256553 0,4
%A A256553 _Alois P. Heinz_, Apr 01 2015