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

Author

Geoffrey Critzer, Apr 01 2013

Keywords

• nonn

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.

Examples

			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}}.
		

Links

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

Crossrefs

Cf. A224219, A224244.

Programs

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]

Formula

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