A053195 Number of level permutations of degree n.

1, 1, 2, 3, 18, 45, 360, 1575, 20790, 99225, 1332450, 9823275, 181496700, 1404728325, 26221595400, 273922023375, 7196040101250, 69850115960625, 1662139682453250, 22561587455281875, 675158520854317500, 9002073394657468125, 259715927440434465000

Offset: 0

Author

Vladeta Jovovic, Mar 02 2000

Keywords

• nonn
• nice

Comments

A permutation is level if the powers of 2 dividing its cycle lengths are all equal.

For odd n, level permutations of degree n are just permutations that have odd order, i.e., A053195(2*n+1) = A000246(2*n+1). - Vladeta Jovovic, Sep 29 2004

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..400
• L. Babai and P. J. Cameron, Automorphisms and enumeration of switching classes of tournaments, Electron. J. Combin., 7 (2000), no. 1, Research Paper 38, 25 pp.

Crossrefs

Cf. A049313, A053197.

Programs

Maple

with(combinat):
b:= proc(n, i, p) option remember; `if`(n=0, 1, `if`(i<1, 0,
       add(multinomial(n, n-i*j, i$j)/j!*(i-1)!^j*
       b(n-i*j, i-2*p, p), j=0..n/i)))
    end:
a:= n-> (m-> `if`(n=0, 1, add(b(n, (h-> h-1+irem(h, 2)
    )(iquo(n, 2^j))*2^j, 2^j), j=0..m)))(ilog2(n)):
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 11 2015

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_, p_] := b[n, i, p] = If[n == 0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!*(i-1)!^j*b[n - i*j, i-2*p, p], {j, 0, n/i}]]]; a[n_] := Function[{m}, If[n == 0, 1, Sum[ b[n, Function [{h}, h - 1 + Mod[h, 2]][Quotient[n, 2^j]]*2^j, 2^j], {j, 0, m}]]][Log[2, n] // Floor]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jul 07 2015, after Alois P. Heinz *)

Extensions

a(0)=1 prepended by Alois P. Heinz, Jun 11 2015