A224245 Number of n-permutations in which there is a unique smallest cycle.

1, 1, 5, 14, 89, 474, 3499, 27040, 253161, 2426300, 27596051, 323960856, 4277055925, 59041067344, 898062119655, 14172430400864, 243919993681649, 4347177953716080, 83224487266425811, 1653277176082392040, 34961357216796300381, 763702067489722288136

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