A261764 Triangle read by rows: T(n,k) is the number of nilpotent subpermutations on an n-set, each of nilpotency index less than or equal to k.

1, 0, 1, 0, 1, 3, 0, 1, 7, 13, 0, 1, 25, 49, 73, 0, 1, 81, 261, 381, 501, 0, 1, 331, 1531, 2611, 3331, 4051, 0, 1, 1303, 9073, 19993, 27553, 32593, 37633, 0, 1, 5937, 63393, 165873, 253233, 313713, 354033, 394353, 0, 1, 26785, 465769, 1436473, 2540233, 3326473, 3870793, 4233673, 4596553

Offset: 0

Author

Samira Stitou, Sep 21 2015

Keywords

• nonn
• tabl

Examples

			T(3,2) = 7 because there are 7 nilpotent subpermutations on {1,2,3}, each of nilpotency index less than or equal to 2, namely: empty map, 1-->2, 1-->3, 2-->1, 2-->3, 3-->1, 3-->2.
Triangle starts:
1;
0, 1;
0, 1,   3;
0, 1,   7,   13;
0, 1,  25,   49,   73;
0, 1,  81,  261,  381,  501;
0, 1, 331, 1531, 2611, 3331, 4051;
...
		

References

• A. Laradji and A. Umar, On the number of subpermutations with fixed orbit size, Ars Combinatoria, 109 (2013), 447-460.

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Cf. A000262, A157400, A261762, A261763, A261765, A261766, A261767.

Programs

Maple

egf:= k-> exp(add(x^j, j=1..k)):
T:= (n, k)-> n!*coeff(series(egf(k), x, n+1), x, n):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Oct 10 2015
# second Maple program:
T:= proc(n, k) option remember; `if`(n=0, 1, add(
      T(n-j, k)*binomial(n-1, j-1)*j!, j=1..min(n,k)))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Sep 29 2017

Mathematica

Table[n!*SeriesCoefficient[Exp[x*(x^k-1)/(x-1)], {x, 0, n}], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 18 2016 *)

Formula

T(n, k) = T(n-1, k) + 2(n-1)T(n-2, k) + ... + k(n-1) ... (n-k+1)T(n-k, k), with T(n, 1) = 1 and T(n, n+r) = T(n, n) for every nonnegative integer r.

T(n,n) = A000262(n).

E.g.f. of column k: exp(x + x^2 + ... + x^k).

T(n,k) = Sum_{i=0..k} A157400(n,i).

Extensions

More terms from Alois P. Heinz, Oct 10 2015