Received May.12, 2002.

Recently, Enumeration of isomers was done on the more and more small scope of compounds,such as, each symmetry group or a special kind of compounds.
    Besides hydrocarbons, acyclic saturated compounds consisting of O, C and H are the most important one class of all organic compounds. Alcohols, ethers and hydroxyl ethers are the most valuable and the most useful kind in all acyclic saturated organic compounds and used in many aspects.
    The enumerations of constitutions, chiral and configurations of alkanes have been reported^[1-4]. The enumerations of isomers of alcohols, ethers and some other compounds also have been reported^[5-10], but the enumeration of isomers of hydroxyl ethers has not been seen. Three generating functions were presented for counting constitutions, and configurations of acyclic saturated compounds consisting of O, C and H, and the numbers of constitutions and configurations of hydroxyl ethers were obtained with the graph-theoretical method.

1.DEFINITIONS
1.1 Stable C and O trees (alcohols, ether and hydroxyl ethers)
Consider a tree T that consists of a finite set of two different kinds of points. A kind of points has degree 1-4, and is colored into black, representing carbon atom; another kind of points has only degree 1 and 2, and is colored into red. The number of each kind of points can be an arbitrary natural number. The tree is called a C and O tree.
    Stable C and O trees represent constitutional formulae for the acyclic saturated hydrocarbons, which some oxygen atoms insert in. Acyclic saturated compounds consisting of O, C and H can be alcohols, ethers or hydroxyl ethers, we simply call them "C and O compounds". The "C and O compounds" have a characteristic, that is to say, in which each carbon atom only can adjoin one oxygen atom, such as,
CH₂(OH)CH(OH)CH₂OH, CH₃OCH₂CH₂OCH₂CH₃
and CH₃OCH₂CH₂OCH₂CH₂OH.
    But CH₃-O-CH₂OCH₃ and (CH₃)₂C(OCH₃)₂ are not considered, because hydrolysis of them gives aldehyde or ketone, being considered as derivatives of unsaturated compounds. Similarly, orthocarbonates and orthoformates are not considered.
    In this paper, each point of degree 1-4 represents a carbon atom. Each point of degree 1 or 2 represents an oxygen atom, hydrogen atoms being omitted. Each line joins two points.

1.2 Planted C and O tree
Carefully inspect the acyclic saturated C and O compounds C_iH_2i+2O_j , It can be found that they consist of alkyls (R) and alkoxyls (RO). The substituted alkyls can be classified into two kinds: carbon-carbon dissever alkyl--R( I ) and carbon-oxygen dissever alkyl--R( II).
    For R( I ), the root carbon atom can or cannot adjoin an oxygen atom . For R(Ⅱ), the root carbon atom can not adjoin any more oxygen atom.
    A planted C and O tree represents an alkyl, has a distinguishable point, which has degree 1-3, this point is called a root.
    A planted C and O tree (I) represents a R(I).
    A planted C and O tree(II) represents a R(II).
1.3 Stereo C and O tree
A stereo C and O tree with two different kinds of points is a C and O tree, in which four neighbors (neighboring points) of every carbon point are given a tetrahedral configuration. A stereo C and O tree with two different kinds of points represents a configuration formula of an acyclic saturated compound consisting of O, C and H.
1.4 planted stereo C and O tree
Similarly, a planted stereo C and O tree with two different kinds of points is a stereo C and O tree, which also contains distinguishable root point besides the above-mentioned two kinds of points.
2.COUNTING
2.1 Constitution of alkyls R(1) and alkyls RO(II)
Let A(x,y) be the generating function of constitutions of R(I) and B(x,y) be the generating function of constitutions of R(II). Then they obey the following alternate recurrence relations:
B(x,y)= b_ij xⁱy^j
=1 + y + 1/6x[A³(x,y)+3A(x,y)A(x²,y²)+2A(x³,y³)],       <1>
where the b₀₀ = 1 and b_ij is the number of constitutions of R(Ⅱ) containing i carbon atoms and j oxygen atoms;
A(x,y) = a_ijxⁱy^j
=B(x,y)+1/2 x[yB(x,y)A²(x,y)+yB(x,y)A(x²,y²)],             <2>
where a_ij is the number of constitutions of R(Ⅰ) containing i carbon atoms and j oxygen atoms.
2.2 Constitutions of C and O compounds and hydroxyl ethers
Let C(x,y) be the generating function of counting the Stable C and O trees with two different kinds of points, in which the coefficient c_ij of the term xⁱy^j is the number of constitutions of acyclic saturated compounds consisting of O, C and H containing i carbon atoms and j oxygen atoms. Using the method of Harary et al. (see Refs. 3,4.), we can obtain the function
C(x,y)= c_ij xⁱy^j
=1/24x[A⁴(x,y)+4yB(x,y)A³(x,y)+6A²(x,y)A(x²,y²)
+12yB(x,y)U(x,y)A(x²,y²)+3A²(x²,y²)+8A(x,y)A(x³,y³)
+8yB(x,y)A(x³,y³)+6A(x⁴,y⁴)]
-1/2[A(x,y)-1]²+1/2[A(x²,y²)-1]
-1/2y[B(x,y)-1]²+1/2y[B(x²,y²)-1],                                   <3>
    The numerical results are given in Table 1.

Table 1 The numbers (c_ij) of constitutions of acyclic saturated compounds consisting of C, O and H ,C_iH_2i+2O_j.

2.3 Configurations of alkyls R(1) and alkyls RO(II)
Let D(x,y) be the generating function of configurations of R(I) and E(x,y) be the generating function of configurations of RO(II). Then they obey the following alternate recurrence relations:
E (x, y) = e_ij xⁱy^j
=1+1/3x[D³ (x, y)+2D(x³, y³)],                     <6>
where the d₀₀=1 and e_ijis the number of configurations of RO(II) containing i carbon atoms and j oxygen atoms;
D (x, y) = d_ij xⁱy^j
=E (x, y)+ x y E(x, y)D²(x, y),                   <7>
where d_ij is the number of configurations of R(I) containing i carbon atoms and j oxygen atoms.
2.4. Configurations of C and O compounds and hydroxyl ethers
Similarly, let F(x,y) be the generating function of configurations of acyclic saturated compounds consisting of O, C and H . We can get
F(x,y)= f_ijxⁱy^j
=1/12x[D⁴(x,y)+4yE(x,y)D³(x,y)+3D²(x²,y²)+8D(x,y)D(x³,y³)
+8yE(x,y)V(x³,y³)]-1/2[D(x,y)-1]²+1/2[D(x²,y²)-1]
-1/2y[E(x,y)-1]²+1/2y[E(x²,y²)-1]                   <8>
where f_ij is the number of numbers of configurations of acyclic saturated compounds
consisting of C, O and H ,C_iH_2i+2O_j containing i carbon atoms and j oxygen atoms. Some results are given in Table 3.

Table3 The numbers (f_ij) of configurations of acyclic saturated compounds consisting of C, O and H ,C_iH_2i+2O_j

Let F1(x,y) be the generating function of counting constitutions of the acyclic saturated alcohols(For the numerical results ,see cf.6 Table2).
Let F2(x,y) be the generating function of counting constitutions of the acyclic saturated ethers(For the numerical results ,see cf.7 Table2).
Let F3(x,y) be the generating function of counting constitutions of the acyclic saturated hydroxyl ethers. Then
F3(x,y)= F(x,y)- F1(x,y)- F2(x,y).                   <9>
Some results are given in Table 4.

2.5 Achiral configurations of C and O compounds and hydroxyl ethers
Let G(x,y) be the generating function of achiral configurations of R(I) and H(x,y) be the generating function of achiral configurations of R(II). Then they obey the following alternate recurrence relations:
H(x,y)= g_ij xⁱy^j = 1+xG(x,y)D(x²,y²),                   <10>
where the g₀₀ = 1 and h_ij is the number of achiral configurations of R(II) containing i carbon atoms and j oxygen atoms;
G(x,y)= g_ij xⁱy^j = H(x,y)+xy H(x,y)D(x²,y²),                   <11>
where g_ij is the number of achiral configurations of R(Ⅰ) containing i carbon atoms and j oxygen atoms.Let F(x,y) be the generating function of configurations of acyclic saturated compounds consisting of O, C and H . We can get
L(x,y)= l_ijxⁱy^j
=1/2xD(x⁴,y⁴)+1/2xG²(x,y)D(x²,y²+xyH(x,y)G(x,y)D(x²,y²)-1/2[G(x,y)-1]²
+1/2[D(x²,y²)-1]-1/2y[H(x,y)-1]²+1/2y[E(x²,y²)-1],                   <12>
where l_ij is the number of numbers of achiral configurations of acyclic saturated compounds consisting of C, O and H ,C_iH_2i+2O_j containing i carbon atoms and j oxygen atoms.
    Let L1(x,y) be the generating function of counting constitutions of the acyclic saturated alcohols(For the numerical results ,see cf.6 Table3).
    Let L2(x,y) be the generating function of counting constitutions of the acyclic saturated ethers(For the numerical results ,see cf.7 Table3).
    Let L3(x,y) be the generating function of counting constitutions of the acyclic saturated hydroxyl ethers. Then
L3(x,y)= L(x,y)- L1(x,y)- L2(x,y).                      <13>
Some numerical results are given in Table 5-6.

Table6 The numbers of achiral configurations of hydroxyl ethers

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