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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. 6
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, May 18 2016 *)

CROSSREFS

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

Sequence in context: A199606 A182472 A285867 * A216806 A290776 A172249

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

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Last modified October 19 03:10 EDT 2021. Contains 348073 sequences. (Running on oeis4.)