A266503 Number of symmetric difference-closed 4-sets consisting of sets consisting of an even number of pairwise disjoint 2-subsets of {1,2,...,n}.

0, 0, 0, 0, 0, 15, 105, 735, 4095, 26775, 162855, 1105335, 7187895, 51126075, 356831475, 2676468795, 19890931515, 156769986555, 1232704469115, 10178240218875, 84190426730235, 725667326178795

Offset

1,6

Comments

A set of this form forms a group (isomorphic to the Klein four-group) under the symmetric difference operation. Such sets may be regarded in a natural way as Klein four-subgroups of the alternating group A_n.

Links

Table of n, a(n) for n=1..22.

John Campbell, A class of symmetric difference-closed sets related to commuting involutions, Discrete Mathematics & Theoretical Computer Science, Vol 19 no. 1, 2017.

Formula

a(n) = n!*sum(sum(sum((2^(k-2*i-2*j))/(k!*(2*i-k)!*(2*j-k)!*(n-4*i-4*j+2*k)!*(delta(i, j)+delta(i, k)+1)!), k=max(i, 2*i+2*j-[n/2])..min(2*j, [1/4*(4*i+4*j-1)])), j=1..i), i=1..[n/2]).

From Vaclav Kotesovec, Apr 10 2016: (Start)

Recurrence: (n-6)*(n-4)*(n-2)*a(n) = (2*n - 7)*(2*n^2 - 14*n + 15)*a(n-1) + 3*(n-7)*(n-1)*(n^2 - 7*n + 11)*a(n-2) - (n-2)*(n-1)*(9*n^2 - 85*n + 189)*a(n-3) + (n-3)*(n-2)*(n-1)*(n^2 - n - 22)*a(n-4) - 2*(n-4)^2*(n-3)*(n-2)*(n-1)*a(n-5) - (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(3*n - 19)*a(n-6) + 3*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-7).

E.g.f.: exp(x)/3 - exp(-x*(x-2)/2)/8 - exp(x*(x+2)/2)/4 + exp(x*(3*x+2)/2)/24.

a(n) ~ 2^(-7/2) * 3^(n/2 - 1) * exp(sqrt(n/3) - n/2 - 1/12) * n^(n/2).

(End)

a(n) = 1/3 + (-1)^n*A001464(n)/8 - A000085(n)/4 + A115327(n)/24. - Vaclav Kotesovec, May 28 2016

Example

For example, there are a(n) = 15 sets of this form in the case whereby n=6:
{{{12}, {34}}, {{34}, {56}}, {{12}, {56}}, {}}
{{{12}, {35}}, {{35}, {46}}, {{12}, {46}}, {}}
{{{12}, {36}}, {{36}, {45}}, {{12}, {45}}, {}}
{{{13}, {24}}, {{24}, {56}}, {{13}, {56}}, {}}
{{{13}, {25}}, {{25}, {46}}, {{13}, {46}}, {}}
{{{13}, {26}}, {{26}, {45}}, {{13}, {45}}, {}}
{{{14}, {23}}, {{23}, {56}}, {{14}, {56}}, {}}
{{{14}, {25}}, {{25}, {36}}, {{14}, {36}}, {}}
{{{14}, {26}}, {{26}, {35}}, {{14}, {35}}, {}}
{{{15}, {23}}, {{23}, {46}}, {{15}, {46}}, {}}
{{{15}, {24}}, {{24}, {36}}, {{15}, {36}}, {}}
{{{15}, {26}}, {{26}, {34}}, {{15}, {34}}, {}}
{{{16}, {23}}, {{23}, {45}}, {{16}, {45}}, {}}
{{{16}, {24}}, {{24}, {35}}, {{16}, {35}}, {}}
{{{16}, {25}}, {{25}, {34}}, {{16}, {34}}, {}}

Mathematica

a[n_] := n!*Sum[Sum[Sum[(2^(k-2*i-2*j))/(k!*(2*i-k)!*(2*j-k)!*(n-4*i-4*j+2*k)!*(KroneckerDelta[i, j]+KroneckerDelta[i, k]+1)!), {k, Max[i, 2*i+2*j-Floor[n/2]], Min[2*j, Floor[1/4*(4*i+4*j-1)]]}], {j, 1, i}], {i, 1, Floor[n/2]}] ; Print[Table[a[n], {n, 1, 22}]] ;
Rest[CoefficientList[Series[E^x/3 - E^(-x*(x-2)/2)/8 - E^(x*(x+2)/2)/4 + E^(x*(3*x+2)/2)/24, {x, 0, 30}], x] * Range[0, 30]!] (* Vaclav Kotesovec, Apr 10 2016 *)

Crossrefs

Cf. A267840.

Sequence in context: A055848 A202493 A200852 * A058085 A060218 A226109

Adjacent sequences: A266500 A266501 A266502 * A266504 A266505 A266506

Keyword

nonn,easy

Author

John M. Campbell, Jan 24 2016

Status

approved