A050212 - OEIS

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

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OFFSET

4,1

COMMENTS

Generalizes Stirling numbers of the first kind.

REFERENCES

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

LINKS

Alois P. Heinz, Rows n = 4..300, flattened

Eric Weisstein's World of Mathematics, Permutation cycle.

EXAMPLE

Triangle begins:

: 6;

: 24;

: 120;

: 720;

: 5040, 1260;

: 40320, 18144;

: 362880, 223776;

: 3628800, 2756160;

: 39916800, 35307360, 1247400;

MAPLE

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

MATHEMATICA

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 *)

CROSSREFS

Cf. A008275, A008306, A050211, A050213.

Sequence in context: A334328 A324064 A378267 * A369833 A354074 A293300

Adjacent sequences: A050209 A050210 A050211 * A050213 A050214 A050215

KEYWORD

nonn,tabf

AUTHOR

Eric W. Weisstein

EXTENSIONS

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

STATUS

approved

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