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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).
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%I #26 Jul 15 2021 19:35:15

%S 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 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 7,8,9,10,12,14,15,20,21,30

%N 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 Alois P. Heinz, <a href="/A256553/b256553.txt">Rows n = 0..60, flattened</a>

%F Sum_{k>=0} T(n,k)*A256554(n,k) = A181844(n).

%F T(n,k) = k for n>0 and 1<=k<=n.

%e Triangle T(n,k) begins:

%e 1;

%e 1;

%e 1, 2;

%e 1, 2, 3;

%e 1, 2, 3, 4;

%e 1, 2, 3, 4, 5, 6;

%e 1, 2, 3, 4, 5, 6;

%e 1, 2, 3, 4, 5, 6, 7, 10, 12;

%e 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 15;

%e 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 20;

%e 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 20, 21, 30;

%p b:= proc(n, i) option remember; `if`(n=0 or i=1, x,

%p b(n, i-1)+(p-> add(coeff(p, x, t)*x^ilcm(t, i),

%p t=1..degree(p)))(add(b(n-i*j, i-1), j=1..n/i)))

%p end:

%p T:= n->(p->seq((h->`if`(h=0, [][], i))(coeff(p, x, i))

%p , i=1..degree(p)))(b(n$2)):

%p seq(T(n), n=0..12);

%t b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x,

%t b[n, i - 1] + Function[p, Sum[Coefficient[p, x, t]*x^LCM[t, i],

%t {t, 1, Exponent[p, x]}]][Sum[b[n - i*j, i - 1], {j, 1, n/i}]]];

%t T[n_] := Function[p, Table[Function[h, If[h == 0, Nothing, i]][

%t Coefficient[p, x, i]], {i, 1, Exponent[p, x]}]][b[n, n]];

%t Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Jul 15 2021, after _Alois P. Heinz_ *)

%Y Row sums give A060179.

%Y Row lengths give A009490.

%Y Last elements of rows give A000793.

%Y Main diagonal gives A000027.

%Y Cf. A181844, A256067, A256554.

%K nonn,look,tabf

%O 0,4

%A _Alois P. Heinz_, Apr 01 2015