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A224245 Number of n-permutations in which there is a unique smallest cycle. 2
1, 1, 5, 14, 89, 474, 3499, 27040, 253161, 2426300, 27596051, 323960856, 4277055925, 59041067344, 898062119655, 14172430400864, 243919993681649, 4347177953716080, 83224487266425811, 1653277176082392040, 34961357216796300381, 763702067489722288136 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

In other words, if the smallest cycle in the n-permutation has length k then no other cycle in the permutation has length k.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..449

FORMULA

E.g.f.: Sum_{k>=1} x^k/k * exp(-Sum_{i=1..k}x^i/i)/(1-x).

EXAMPLE

a(4) = 14 because we have 14 such permutations of {1,2,3,4} shown in cycle notation: {{1}, {3,4,2}}, {{1}, {4,3,2}}, {{2,3,1}, {4}}, {{2,3,4,1}}, {{2,4,3,1}}, {{2,4,1}, {3}}, {{3,2,1}, {4}}, {{3,4,2,1}}, {{3,4,1}, {2}}, {{3,2,4,1}}, {{4,3,2,1}}, {{4,2,1}, {3}}, {{4,3,1}, {2}}, {{4,2,3,1}}.

MAPLE

with(combinat):

b:= proc(n, i) option remember;

      `if`(i<1, 0, `if`(n=i, (i-1)!, 0) +add(b(n-i*j, i-1)*

       multinomial(n, n-i*j, i$j)/j!*(i-1)!^j, j=0..(n-1)/i))

    end:

a:= n-> b(n$2):

seq(a(n), n=1..25);  # Alois P. Heinz, Sep 07 2020

MATHEMATICA

nn=20; Drop[Range[0, nn]! CoefficientList[Series[Sum[x^k/k Exp[-Sum[x^i/i, {i, 1, k}]]/(1-x), {k, 1, nn}], {x, 0, nn}], x], 1]

CROSSREFS

Cf. A224219, A224244.

Sequence in context: A198039 A198091 A197797 * A238380 A183307 A334547

Adjacent sequences:  A224242 A224243 A224244 * A224246 A224247 A224248

KEYWORD

nonn

AUTHOR

Geoffrey Critzer, Apr 01 2013

STATUS

approved

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Last modified November 30 16:23 EST 2021. Contains 349423 sequences. (Running on oeis4.)