A266503 - OEIS

%I #39 Jun 17 2019 02:01:27

%S 0,0,0,0,0,15,105,735,4095,26775,162855,1105335,7187895,51126075,

%T 356831475,2676468795,19890931515,156769986555,1232704469115,

%U 10178240218875,84190426730235,725667326178795

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

%C 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.

%H John Campbell, <a href="https://dmtcs.episciences.org/3210">A class of symmetric difference-closed sets related to commuting involutions</a>, Discrete Mathematics & Theoretical Computer Science, Vol 19 no. 1, 2017.

%F 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]).

%F From _Vaclav Kotesovec_, Apr 10 2016: (Start)

%F 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).

%F 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.

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

%F (End)

%F a(n) = 1/3 + (-1)^n*A001464(n)/8 - A000085(n)/4 + A115327(n)/24. - _Vaclav Kotesovec_, May 28 2016

%e For example, there are a(n) = 15 sets of this form in the case whereby n=6:

%e {{{12}, {34}}, {{34}, {56}}, {{12}, {56}}, {}}

%e {{{12}, {35}}, {{35}, {46}}, {{12}, {46}}, {}}

%e {{{12}, {36}}, {{36}, {45}}, {{12}, {45}}, {}}

%e {{{13}, {24}}, {{24}, {56}}, {{13}, {56}}, {}}

%e {{{13}, {25}}, {{25}, {46}}, {{13}, {46}}, {}}

%e {{{13}, {26}}, {{26}, {45}}, {{13}, {45}}, {}}

%e {{{14}, {23}}, {{23}, {56}}, {{14}, {56}}, {}}

%e {{{14}, {25}}, {{25}, {36}}, {{14}, {36}}, {}}

%e {{{14}, {26}}, {{26}, {35}}, {{14}, {35}}, {}}

%e {{{15}, {23}}, {{23}, {46}}, {{15}, {46}}, {}}

%e {{{15}, {24}}, {{24}, {36}}, {{15}, {36}}, {}}

%e {{{15}, {26}}, {{26}, {34}}, {{15}, {34}}, {}}

%e {{{16}, {23}}, {{23}, {45}}, {{16}, {45}}, {}}

%e {{{16}, {24}}, {{24}, {35}}, {{16}, {35}}, {}}

%e {{{16}, {25}}, {{25}, {34}}, {{16}, {34}}, {}}

%t 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}]] ;

%t 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 *)

%Y Cf. A267840.

%K nonn,easy

%O 1,6

%A _John M. Campbell_, Jan 24 2016