A050212 -id:A050212 - cp's OEIS frontend

A050211 Triangle of number of permutations of {1, 2, ..., n} having exactly k cycles, each of which is of length >=r for r=3.

2, 6, 24, 120, 40, 720, 420, 5040, 3948, 40320, 38304, 2240, 362880, 396576, 50400, 3628800, 4419360, 859320, 39916800, 53048160, 13665960, 246400, 479001600, 684478080, 216339552, 9609600, 6227020800, 9464307840, 3501834336

Offset: 3

Eric W. Weisstein

Generalizes Stirling numbers of the first kind

			Table begins:
   n\k |      u     u^2    u^3
  = = = = = = = = = = = = = = =
    3  |      2
    4  |      6
    5  |     24
    6  |    120     40
    7  |    720    420
    8  |   5040   3948
    9  |  40320  38304    2240
   ...

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

Alois P. Heinz, Rows n = 3..200, flattened
S. Brassesco, M. A. Méndez, The asymptotic expansion for the factorial and Lagrange inversion formula, arXiv:1002.3894v1 [math.CA], 2010.
G. Nemes, On the Coefficients of the Asymptotic Expansion of n!, J. Int. Seq. 13 (2010), 10.6.6.
Eric Weisstein's World of Mathematics, Permutation Cycle.

Cf. A008275, A008306, A050212, A050213.

b:= proc(n) option remember; expand(`if`(n=0, 1, add(
      b(n-i)*x*binomial(n-1, i-1)*(i-1)!, i=3..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
seq(T(n), n=3..15);  # Alois P. Heinz, Sep 25 2016

t[n_ /; n >= 3, k_ /; k >= 1] := t[n, k] = (n - 1)*t[n - 1, k] + (n - 2)*(n - 1)*t[n - 3, k - 1] ; t[, ] = 0; t[3, 1] = 2; Flatten[ Table[t[n, k], {n, 3, 15}, {k, 1, Floor[n/3]}]] (* Jean-François Alcover, Nov 05 2012, after Peter Bala *)

From Peter Bala, Sep 06 2011: (Start)
E.g.f.: (1-t)^(-u)*exp(-u*(t+t^2/2)) - 1 = (2*u)*t^3/3!+(6*u)*t^4/4!+(24*u)*t^5/5!+(120*u+40*u^2)*t^6/6!+(720*u+420*u^2)*t^7/7!+....
E.g.f. for column k: 1/k!*(-log(1-x)-x-x^2/2)^k.
Starting at row 3, row lengths are 1, 1, 1, 2, 2, 2, 3, 3, 3, ....
Recurrence: T(n,k) = (n-1)*T(n-1,k) + (n-1)*(n-2)*T(n-3,k-1). (End)

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

A050211 Triangle of number of permutations of {1, 2, ..., n} having exactly k cycles, each of which is of length >=r for r=3.

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