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REFERENCES
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CRC Handbook of Combinatorial Designs, 1996, p. 651.
R. L. Davis, The numbers of structures of finite relations, Proc. Amer. Math. Soc., 4 (1953), 486-494.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 522.
F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 225.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 124, Table 5.1.2 and p. 241, Table A4.
M. D. McIlroy, Calculation of numbers of structures of relations on finite sets, Massachusetts Institute of Technology, Research Laboratory of Electronics, Quarterly Progress Reports, No. 17, Sept. 15, 1955, pp. 14-22.
W. Oberschelp, Kombinatorische Anzahlbestimmungen in Relationen, Math. Ann., 174 (1967), 53-78.
J. Qian, Enumeration of unlabeled directed hypergraphs, Electronic Journal of Combinatorics, 20(1) (2013), #P46. - From N. J. A. Sloane, Mar 01 2013
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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MAPLE
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(Maple code from Vladeta Jovovic, Jun 06 2006) with(combinat):with(numtheory):
for n from 1 to 20 do p:=partition(n):
s:=0:for k from 1 to nops(p) do
q:=convert(p[k], multiset):
for i from 1 to n do a(i):=0:od:for i from 1 to nops(q) do a(q[i][1]):=q[i][2]:od:
c:=1:ord:=1:for i from 1 to n do c:=c*a(i)!*i^a(i): if a(i)<>0 then ord:=lcm(ord, i):fi:od:
g:=0:for d from 1 to ord do if ord mod d=0 then g1:=0:for del from 1 to d do if del<=n and d mod del=0 then g1:=g1+del*a(del):fi:od:g:=g+phi(ord/d)*g1*(g1-1):fi:od:
s:=s+2^(g/ord)/c:
od:
print(n, s):
od:
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MATHEMATICA
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Table[CycleIndex[PairGroup[SymmetricGroup[n], Ordered], t]/.Table[t[i]->1+x^i, {i, 1, n^2}]/.{x->1}, {n, 1, 7}] (* or *)
Table[GraphPolynomial[n, t, Directed]/.{t->1}, {n, 1, 20}]
(* Geoffrey Critzer, Mar 08 2011 *)
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