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

0,6

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

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).

Example

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;
...

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 *)

Crossrefs

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

Sequence in context: A182472 A285867 A349780 * A216806 A290776 A373086

Adjacent sequences: A261761 A261762 A261763 * A261765 A261766 A261767

Keyword

nonn,tabl

Author

Samira Stitou, Sep 21 2015

Extensions

More terms from Alois P. Heinz, Oct 10 2015

Status

approved