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

Author

Eric W. Weisstein

Keywords

• nonn
• tabf

Comments

Generalizes Stirling numbers of the first kind

Examples

			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
   ...
		

References

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

Links

• 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.

Crossrefs

Cf. A008275, A008306, A050212, A050213.

Programs

Maple

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

Mathematica

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 *)

Formula

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)