A049313 -id:A049313 - cp's OEIS frontend

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

Vladeta Jovovic, Mar 02 2000

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

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.

Cf. A049313, A053197.

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

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 *)

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

A053197 Number of level partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 6, 5, 10, 8, 13, 12, 21, 18, 27, 27, 42, 38, 54, 54, 77, 76, 101, 104, 143, 142, 183, 192, 249, 256, 323, 340, 432, 448, 550, 585, 722, 760, 918, 982, 1190, 1260, 1502, 1610, 1917, 2048, 2408, 2590, 3053, 3264, 3800, 4097, 4765, 5120, 5910, 6378

Offset: 0

Vladeta Jovovic, Mar 02 2000

A partition is level if the powers of 2 dividing its parts are all equal.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A049313, A053195.

b:= proc(n, i, p) option remember; `if`(n=0, 1,
     `if`(i<1, 0, add(b(n-i*j, i-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^(1+j)), j=0..m)))(ilog2(n)):
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 11 2015

a[n_] := Sum[ PartitionsQ[n/2^k], {k, 0, IntegerExponent[n, 2]}]; Table[ a[n], {n, 1, 55}] (* Jean-François Alcover, Dec 12 2011, after Vladeta Jovovic *)

a(n) = Sum_{k=0..A007814(n)} A000009(n/2^k). a(2*n+1) = A000009(2*n+1) = A078408(n). - Vladeta Jovovic, Sep 29 2004

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

A006536 Switching classes of digraphs.

Original entry on oeis.org

2, 6, 38, 684, 50224, 13946352, 14061131152, 50947324188128, 666754350781269760, 31763991166456912086656, 5550343429517595744902558208, 3581353054755185129132443153728000

Offset: 1

N. J. A. Sloane

R. C. Read, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976

Cf. A002854, A049313.

More terms from Vladeta Jovovic, Jul 22 2000

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A053195 Number of level permutations of degree n.

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A053197 Number of level partitions of n.

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A006536 Switching classes of digraphs.

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