A050211 -id:A050211 - cp's OEIS frontend

A050212 Triangle of number of permutations of {1, 2, ..., n} having exactly k cycles, each of which is of length >=r for r=4.

6, 24, 120, 720, 5040, 1260, 40320, 18144, 362880, 223776, 3628800, 2756160, 39916800, 35307360, 1247400, 479001600, 476910720, 38918880, 6227020800, 6822541440, 889945056, 87178291200, 103440879360, 18478684224

Offset: 4

Eric W. Weisstein

Generalizes Stirling numbers of the first kind.

			Triangle begins:
:        6;
:       24;
:      120;
:      720;
:     5040,     1260;
:    40320,    18144;
:   362880,   223776;
:  3628800,  2756160;
: 39916800, 35307360, 1247400;

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 257.

Alois P. Heinz, Rows n = 4..300, flattened
Eric Weisstein's World of Mathematics, Permutation cycle.

Cf. A008275, A008306, A050211, A050213.

b:= proc(n) option remember; expand(`if`(n=0, 1, add(
      b(n-i)*x*binomial(n-1, i-1)*(i-1)!, i=4..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
seq(T(n), n=4..20);  # Alois P. Heinz, Sep 25 2016

b[n_] := b[n] = Expand[If[n==0, 1, Sum[b[n-i] x Binomial[n-1, i-1] (i-1)!, {i, 4, n}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, Exponent[p, x]}]][b[n]];
Table[T[n], {n, 4, 20}] // Flatten (* Jean-François Alcover, Dec 07 2019, after Alois P. Heinz *)

Offset changed from 1 to 4 by Alois P. Heinz, Sep 25 2016

A050213 Triangle of number of permutations of {1, 2, ..., n} having exactly k cycles, each of which is of length >=r for r=5.

Original entry on oeis.org

24, 120, 720, 5040, 40320, 362880, 72576, 3628800, 1330560, 39916800, 20338560, 479001600, 303937920, 6227020800, 4643084160, 87178291200, 73721007360, 1743565824, 1307674368000, 1224694598400, 69742632960, 20922789888000

Offset: 5

Eric W. Weisstein

Generalizes Stirling numbers of the first kind.

			Triangle begins:
05:       24;
06:      120;
07:      720;
08:     5040;
09:    40320;
10:   362880,    72576;
11:  3628800,  1330560;
12: 39916800, 20338560;

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 257.

Alois P. Heinz, Rows n = 5..300, flattened
Eric Weisstein's World of Mathematics, Permutation Cycle.

Cf. A008275, A008306, A050211, A050212.

b:= proc(n) option remember; expand(`if`(n=0, 1, add(
      b(n-i)*x*binomial(n-1, i-1)*(i-1)!, i=5..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
seq(T(n), n=5..20);  # Alois P. Heinz, Sep 25 2016

b[n_] := b[n] = Expand[If[n == 0, 1, Sum[b[n - i]*x*Binomial[n - 1, i - 1]* (i - 1)!, {i, 5, n}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][ b[n]];
T /@ Range[5, 20] // Flatten (* Jean-François Alcover, Dec 08 2019, after Alois P. Heinz *)

Offset changed from 1 to 5 by Alois P. Heinz, Sep 25 2016

cp's OEIS Frontend

A050212 Triangle of number of permutations of {1, 2, ..., n} having exactly k cycles, each of which is of length >=r for r=4.

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