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A050211
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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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3
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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
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OFFSET
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3,1
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COMMENTS
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Generalizes Stirling numbers of the first kind
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 257.
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LINKS
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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.
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FORMULA
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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]
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EXAMPLE
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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
..
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A008275, A008306, A050212, A050213.
Sequence in context: A008336 A360298 A033643 * A248766 A263713 A242427
Adjacent sequences: A050208 A050209 A050210 * A050212 A050213 A050214
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KEYWORD
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nonn,tabf
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AUTHOR
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Eric W. Weisstein
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STATUS
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approved
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