%I #36 Jun 17 2019 02:01:39

%S 0,0,0,3,15,105,525,3255,17703,112455,669735,4485195,29023995,

%T 205768563,1432735395,10728177915,79665069435,627587657595,

%U 4933313794683,40724759240235,336819780949995,2902978545030795,25135723046974155,225455477000793963

%N Number of symmetric difference-closed 4-sets consisting of the empty set and sets consisting of pairwise disjoint 2-subsets of {1,2,...,n}.

%C Suppose that n people arrive at a meeting, and suppose that the n people arrange themselves into couples, except for a loner if n is odd. The couples then form various organizations. Then a(n) is the number of possible collections of three distinct organizations such that given two organizations in this collection, if a couple belongs to only one of these two organizations, then this couple must be a member of the remaining (third) organization.

%C A set of this form forms a group (isomorphic to the Klein four-group) under the symmetric difference operation.

%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-i-j))/(k!)/(i-k)!/(j-k)!/(n-2i-2j+2k)!/(delta(i, j)+delta(i, 2k)+1)!, k=max(ceiling(i/2), i+j-[n/2])..min(j, [1/4*(2i+2j-1)])), j=1..i), i=1..[n/2]), where delta denotes the Kronecker delta function.

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

%F Recurrence: (n-4)*(n-2)*a(n) = 3*(n^2 - 5*n + 5)*a(n-1) + (n-1)*(4*n^2 - 27*n + 41)*a(n-2) - (n-2)*(n-1)*(8*n - 29)*a(n-3) - (n-3)*(n-2)*(n-1)*(3*n - 16)*a(n-4) + 3*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).

%F E.g.f.: exp(x)/3 - exp(x*(x+2)/2)/2 + exp(x*(3*x+2)/2)/6.

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

%F (End)

%F a(n) = 1/3 - A000085(n)/2 + A115327(n)/6. - _Vaclav Kotesovec_, May 28 2016

%e There are a(n) = 15 sets of this form in the case whereby n=5:

%e {{}, {{1, 2}}, {{3, 4}}, {{1, 2}, {3, 4}}}

%e {{}, {{1, 2}}, {{3, 5}}, {{1, 2}, {3, 5}}}

%e {{}, {{1, 2}}, {{4, 5}}, {{1, 2}, {4, 5}}}

%e {{}, {{1, 3}}, {{2, 4}}, {{1, 3}, {2, 4}}}

%e {{}, {{1, 3}}, {{2, 5}}, {{1, 3}, {2, 5}}}

%e {{}, {{1, 3}}, {{4, 5}}, {{1, 3}, {4, 5}}}

%e {{}, {{1, 4}}, {{2, 3}}, {{1, 4}, {2, 3}}}

%e {{}, {{1, 4}}, {{2, 5}}, {{1, 4}, {2, 5}}}

%e {{}, {{1, 4}}, {{3, 5}}, {{1, 4}, {3, 5}}}

%e {{}, {{1, 5}}, {{2, 3}}, {{1, 5}, {2, 3}}}

%e {{}, {{1, 5}}, {{2, 4}}, {{1, 5}, {2, 4}}}

%e {{}, {{1, 5}}, {{3, 4}}, {{1, 5}, {3, 4}}}

%e {{}, {{2, 3}}, {{4, 5}}, {{2, 3}, {4, 5}}}

%e {{}, {{2, 4}}, {{3, 5}}, {{2, 4}, {3, 5}}}

%e {{}, {{2, 5}}, {{3, 4}}, {{2, 5}, {3, 4}}}

%t Table[n!*Sum[Sum[Sum[(2^(k-i-j))/(k!)/(i-k)!/(j-k)!/(n-2i-2j+2k)!/(KroneckerDelta[i, j]+KroneckerDelta[i, 2 k]+1)!, {k, Max[Ceiling[i/2], i+j-Floor[n/2]], Min[j, Floor[1/4*(2i+2j-1)]]}], {j, 1, i}], {i, 1, Floor[n/2]}], {n, 1, 26}]

%t Rest[CoefficientList[Series[E^x/3 - E^(x*(x+2)/2)/2 + E^(x*(3*x+2)/2)/6, {x, 0, 20}], x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Apr 10 2016 *)

%Y Cf. A266503.

%K nonn,easy

%O 1,4

%A _John M. Campbell_, Jan 22 2016