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

0,3

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.

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 *)

Crossrefs

Cf. A049313, A053197.

Sequence in context: A163910 A064777 A137784 * A347429 A211867 A003693

Adjacent sequences: A053192 A053193 A053194 * A053196 A053197 A053198

Keyword

nonn,nice

Author

Vladeta Jovovic, Mar 02 2000

Extensions

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

Status

approved