Numbers of Monosubstituted Alkanes as Stereoisomers
Shinsaku FUJITA
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1 Introduction
Combinatorial enumeration of rooted trees as models of monosubstituted alkanes has been initiated by a mathematician Cayley [1, 2]. Henze and Blair [3] obtained the numbers of aliphatic alcohols of a given carbon content by means of recursive equations. Later, a more systematic method based on Polya's theorem [4, 5] was applied to the evaluation of the numbers of rooted trees, which was differently formulated as planted trees.
Although Polya's theorem [4, 5] has been widely applied to chemical combinatorics [6  11], enumerated objects in most cases are "graphs", not as "threedimensional objects". Chemically speaking, they are constitutional (structural) isomers, but not stereoisomers. As pointed out in our recent reports [12  14], Polya's theorem lacks the concept of sphericities [15, 16] so as to be incapable of solving stereochemical problems. Although Robinson et al.[17] have reported the enumeration of monosubstituted alkanes by modifying Polya's cycle indices (CIs), their treatment also lacks the concept of sphericities.
Without the sphericity concept, in general, any methods are incapable of treating fundamental stereochemical problems such as mesocompounds and pseudoasymmetry. Because the stereochemical problems have been already solved in the last quarter of the 19th century in a descriptive or nonmathematical fashion [18  21], the present status of chemical combinatorics based on Polya's theorem does not properly reflect the advance of stereochemistry before and during the 20th century.
On the basis of the sphericity concept, we have developed a general method named the proligand method for enumerating stereoisomers [12  14]. To show the versatility of the proligand method, we have recently applied it to the enumeration of monosubstituted alkanes (planted 3Dtrees) as 3Dobjects [22].
The sphericity concept was originally proposed in the form of "the sphericities of orbits", which provides us with various tools for combinatorics in connection with Fujita's USCI (unitsubducedcycleindex) approach [16]. Then, it was modified into the concept "the sphericities of orbits based on cyclic groups", which gives a basis to other types of combinatorial tools [23  25]. Further, it was transformed into the concept "the sphericities of cycles", which gives the proligand method [12  14].
As found by the short history described in the preceding paragraph, the original formulation [12  14] was too mathematical to assure chemists of the traceability of the proligand method. However, there is a short cut to become practised in the usage of the proligand method without acquainting such mathematical formulation. The short cut is to study suitable examples which disclose the details of calculation procedures. We can safely say that understanding mathematics is usually different from implementing mathematics as a program for concrete calculation. In other words, "The proof of the pudding is in the eating. You can't eat mathematics, but you can digest it.", as Polya phrased it [10](page 68). Note that the word "eat" corresponds to "understand" and the word "digest" corresponds to "implement".
As clarified in the preceding paragraphs, the aim of the present paper is to report the details of calculation procedures for obtaining the numbers of monosubstituted alkanes as 3Dobjects (stereoisomers), but not as graphs, where the procedures are implemented by the Maple programming language [26].
2 Enumeration of Monosubstituted Alkanes as Stereoisomers
2. 1 Planted Promolecules and Their Nesting
A monosubstituted alkane is regarded as a planted 3Dtree of degree 4, where a principal vertex (a central carbon atom) is connected with a root (the monosubstituent) and three substituents (alkyl ligands or hydrogen atoms). Each one of the three substituents is called a proligand, which is regarded as being a hypothetical structureless object with chirality/achirality. By considering such proligands, the original monosubstituted alkane (or the planted 3Dtree) is regarded as a planted promolecule, which is a 3Dskeleton (a tetrahedral skeleton) having the root () and the tree proligands on its vertices.
Figure 1. Monosubstituted alkane as a planted 3Dtree and a planted promolecule
For example, let us examine a 3,4dimethyl3Xhexane (1), which is selected from a set of four diastereomers. The monoXsubstituted alkane (1) is regarded as a planted 3Dtree with a root X. When the three substituents are replaced by proligands, i.e., A = CH_{3}, B = CH_{2}CH_{3}, and p = RCH(CH_{3})CH_{2}CH_{3}, and when the X is replaced by a solid circle, the planted 3Dtree (1) is converted into a planted promolecule (2) with a root (). The proligands A and B are achiral, while the proligand p is chiral. Note that the Sconfiguration is tentatively assigned by considering a vacant bond (incident to the X originally) as the lowest priority (p>B>A>).
Each of the proligands can be further regarded as a planted promolecule, if a root is attached to its central vertex (as a new principal vertex). For example, the proligand p (i.e., a 2butyl ligand) can be regarded as a planted promolecule, if a root is attached to the p through the vacant bond and the three achiral substituents (H, A = CH_{3}, and B = CH_{2}CH_{3}) are replaced by the corresponding achiral proligands. The resulting planted promolecule is represented by CHAB, which has the same skeleton as the starting planted promolecule (2). The proligand B (i.e., an ethyl ligand) can be regarded as a planted promolecule, if a root is attached to the B through the vacant bond and the three achiral substituents (two H's and A = CH_{3}) are replaced as above. The resulting planted promolecule is represented by CH_{2}A, which has the same skeleton as 2.
This procedure is repeated to reach terminal vertices of the original monosubstituted alkane (or the planted 3Dtree). Thereby, the original monosubstituted alkane (or the planted 3Dtree) is recognized to have a nested structure of such planted promolecules.
2. 2 Sphericities of Cycles
It is to be noted that each planted promolecule contained in a nested fashion is a 3Dobject (not a graph) which has a common skeleton belonging to C_{3v}. Hence, such planted promolecules can be enumerated by means of Fujita's proligand method [12  14].
Figure 2. Symmetry operations for a C_{3v}skeleton (3). The identity operation (I) converts 3 into itself, where the resulting skeleton is denoted as 3a.
For this purpose, we examine the three positions of the C_{3v}skeleton under all operations of C_{3v}. In terms of the USCI approach [16], the three positions other than the root construct an orbit governed by a right coset representation (RCR) represented as (C_{s}\)C_{3v}, whose concrete form is shown as a product of cycles as follows:
The numbering used in the RCR (Cs\)C_{3v} corresponds to the numbering shown in the skeleton (3), where the product of cycles corresponding to each improper rotation (s_{v(1)}, s_{v(2)}, or s_{v(3)}) is attached by an overbar, which shows the inverse of chirality.
The sphericity of each cycle is determined according to its shape, so that a sphericity index (SI) is assigned to the cycle. The chirality fittingness of each orbit is determined by the corresponding SI.

A cycle belonging to an improper rotation is determined to be homospheric, if the length d is odd; and a sphericity index a_{d} is assigned to it. The chirality fittingness of a homospheric cycle permits the transitivity of achiral (pro)ligands only.

If the length d is even, on the other hand, the cycle in an improper rotation is determined to be enantiospheric; and a sphericity index c_{d} is assigned to it. The chirality fittingness of an enantiospheric cycle permits the transitivity of achiral (pro)ligands or the transitivity of pairs of enantiomeric ligands (i.e., d/2 of chiral proligands of the same kind and d/2 of their enantiomeric proligands).

Any cycle belonging to a proper rotation is determined to be hemispheric and a sphericity index b_{d} is assigned to it, where the subscript d is the length of the cycle. The chirality fittingness of a hemispheric cycle permits the transitivity of achiral and chiral (pro)ligands.
Thereby, each symmetry operation acting on the skeleton is characterized by a product of such sphericity indices, as collected along with the corresponding product of cycles in Figure 2.
To count achiral planted promolecules and enantiomeric pairs of chiral planted promolecules, we apply Theorem 2 of [14] to the enumeration of planted promolecules without any modification. Hence, the cycle index with chirality fittingness (CICF) for this case is calculated by using the product of SIs collected in Figure 2:
To count achiral and chiral planted promolecules, Theorem 3 of [14] for the enumeration of ligands under the action of the maximum chiral subgroup is applied to this case so as to derive the following CICF:
To count achiral planted promolecules only, the first proposition of Theorem 4 for the enumeration of achiral ligands [14] is applied to this case so as to derive the following CICF_{A}:
To count chiral planted promolecules only, the second proposition of Theorem 4 for the enumeration of chiral ligands [14] is applied to obtain the following CICF_{C}:
2. 3 Recursive Procedures
By using eq. 3, let us consider the enumeration of achiral and chiral planted promolecules, where two enantiomers of each pair are counted separately. For this purpose, the SI appearing in the righthand side of eq. 3 is replaced by the following ligand inventory:
where achiral and chiral proligands for substitution are itemized by their carbon contents k (k = 1 to ¥) and we put b_{0} = 1. If the ligand inventory is introduced into eq. 3, the resulting equation is a generating function for counting achiral and chiral planted promolecules. The generating function can be regarded as a new ligand inventory for the hemispheric cycle. This means that this process is recursive, so that we can obtain the following functional equation:
where the principal node of the parent planted promolecule is taken into consideration by multiplying by x and the initial (trivial) planted promolecule is considered by adding 1. This type of equation was first noted by Polya [4, 5], although the sphericity concept was not taken into consideration.
According to the chirality fittingness of an enantiospheric cycle, diploids (ordered pairs) of achiral (pro)ligands or enantiomeric ligands satisfy the following ligand inventory:
where achiral and chiral proligands for substitution are itemized by their carbon contents k (k = 1 to ¥) and we put g_{0} = 1. The concept of diploid is here used without definition, which will be reported elsewhere [22]. Because the diploids for this case are characterized by C_{3}, we modify eq. 7 into the following functional equation:
where the principal nodes of the parent planted promolecules contained in a diploid are characterized by multiplying by x^{2} and the initial (trivial) planted promolecule is taken into consideration by adding 1.
By using eq. 4, we are able to enumerate achiral planted promolecules only. Thus, the SI appearing in the righthand side of eq. 4 is replaced by the following ligand inventory:
where achiral and chiral proligands for substitution are itemized by their carbon contents k (k = 1 to ¥) and we put a_{0} = 1. If the ligand inventories (eq. 10) along with eq. 6 are introduced into eq. 4, the resulting equation is a generating function for counting achiral planted promolecules only. Because the generating function can be regarded as a new ligand inventory for the homospheric cycle, this process is recognized to be recursive, so that we obtain the following functional equation:
where the principal node of the parent planted promolecule is taken into consideration by multiplying by x and the initial (trivial) planted promolecule is considered by adding 1. Although this type of equation was first noted by Robinson et al. [17], their treatment did not take the sphericity concept into consideration.
Let B(x) be a generating function for counting achiral and chiral planted promolecules, where each pair of enantiomers is counted just once. Let C(x) be a generating function for counting chiral planted promolecules only, where each pair of enantiomers is counted just once. They are represented by the following equations:
where we put B_{0} = 1 and C_{0} = 0. Because we can place B_{k} = a_{k} + C_{k} and b_{k} = a_{k} + 2C_{k}, eq. 7 (for b(x)) and eq. 11 (for a(x)) give the following equations:
It should be noted that a(x) (eq. 10), c(x^{2}) (eq. 9), and b(x) (eq. 7) have recursive nature, while B(x) (eq. 14 or eq. 15) and C(x) (eq. 16 or eq. 17) have no recursive nature.
2. 4 Preliminary Calculations for Implementation
To collect data for implementing the process of recursive calculation, suppose that the coefficients of x^{k} (k<=5) have been already calculated to give intermediate generating functions as follows:
where we add the coefficients of the terms x^{6}, x^{7}, and x^{8} for calculation and omit the terms for k>=9 tentatively. The intermediate generating functions (eqs. 1820) are introduced into the respective righthand sides of eqs. 11, 9, and 7. The resulting equations are expanded to give the following equations:
which are compared with eqs. 1820. Thereby, the coefficients of the terms x^{6} (for a(x) and b(x)) or x^{12} (for c(x)) give the following relationships:
which show that they are determined as distinct numbers, if the generating functions up to x^{5} (for a(x) and b(x)) or to x^{10} (for c(x)) are given.
Rule 1: In general, if such generating functions up to x^{n} (for a(x) and b(x)) or to x^{2n} (for c(x)) are given, the coefficients (a_{n+1}, g_{n+1}, and b_{n+1}) are recursively generated as distinct numbers.
Comparison of the coefficients of the terms x^{7} (for a(x) and b(x)) or x^{14} (for c(x)) gives the following relationships:
These equations (eqs. 2729) mean that the coefficients (a_{7}, g_{7}, and b_{7}) can be calculated by using the coefficients (a_{6}, g_{6}, and b_{6}), which have been already obtained by eqs. 2426.
Rule 2: In general, if such generating functions up to x^{n} (for a(x) and b(x)) or to x^{2n} (for c(x)) are given, the coefficients a_{n+2}, g_{n+2}, and b_{n+2}) are recursively calculated by using the generated coefficients (a_{n+1}, g_{n+1}, and b_{n+1}).
Comparison of the coefficients of the terms x^{8} (for a(x) and b(x)) or x^{8} (for c(x)) gives the following relationships:
These equations (eqs. 3032) mean that the coefficients (a_{8}, g_{8}, and b_{8}) can be calculated by using the coefficients (g_{6} and b_{6}) and the coefficients (a_{7}, g_{7}, and b_{7}), which have been already obtained by eqs. 25 and 26 as well as by eqs. 2729.
Rule 3: In general, if such generating functions up to x^{n} (for a(x) and b(x)) or to x^{2n} (for c(x)) are given, the coefficients (a_{n+3}, g_{n+3}, and b_{n+3}) are recursively calculated by using the generated coefficients (g_{n+1} and b_{n+1}) and the calculated coefficients (a_{n+2}, g_{n+2}, and b_{n+2}).
2. 5 Implementation and Results
In the preceding subsection, we have obtained three rules, each of which is capable of working as a basis of recursive calculation.
First, we implement Rule 1 as a Maple program for calculating the numbers of planted 3Dtrees (monosubstituted alkanes) which are itemized with respect to carbon content (1 to 100). The upper limit of the carbon content (
ccntt
) is tentatively fixed to be 100, but it is freely selected.
Maple program for Rule 1, "EnumAlkyl1100RR.mpl":
This code is stored in a file named "EnumAlkyl1100RR.mpl" tentatively. In this code, the abbreviated symbols for functional equations are used as follows:
a1 for a(x), c2 for c(x^{2}), c6 for c(x^{6}), b1 for b(x) and b3 for b(x^{3}),
Such symbols as
a1 and c2
are used to remember that such functions as a(x) and c(x^{2}) correspond to the SIs a_{1}, c_{2} and so on. The first paragraph of this code declares three functional equations (eqs. 11, 9, and 7). In the 2nd paragraph, the initial values a(x) = 1, c(x^{2}) = 1, and b(x) = 1 are set by encoding
a1 := 1;c2 := 1; ... c6 := 1;
, which means that we put a_{0} = 1, g_{0} = 1, and b_{0} = 1 for the initial (trivial) planted promolecule. The 3rd paragraph of the code shows a
do
loop, in which the next coefficients are calculated by using a Maple command
coeff
and added to the end of respective functional equations so as to generate intermediate generating functions. After escaping from the
do
loop, the 4th paragraph declares the calculation of B(x) (eq. 14) and C(x) (eq. 16). The 5th paragraph of the code (the final
do
loop) shows the printout of the calculation results.
We execute the code by inputting the following command on the Maple inputting window:
read "EnumAlkyl100RR.mpl"
Thereby, we obtain the coefficients a_{k} (eq. 10) for a(x) (eq. 11), B_{k} (eq. 12) for B(x) (eq. 14), and C_{k} (eq. 13) for C(x) (eq. 16), as collected in Table 1.
Table 1. Numbers of Monosubstituted Alkanes as Stereoisomers^{a}
^{a}In each value of C_{k} (Chiral) or B_{k} (Chiral + Achiral), each enantiomeric pair is counted just once for chiral monosubstituted alkanes.
Rule 2 is implemented as a Maple program for calculating the numbers of planted 3Dtrees (monosubstituted alkanes), which are itemized with respect to carbon content (1 to 100).
Maple program for Rule 2, "Enum2by1100RR.mpl":
Thereby, we alternatively obtain the coefficients a_{k} (eq. 10) for a(x) (eq. 11), B_{k} (eq. 12) for B(x) (eq. 14), and C_{k} (eq. 13) for C(x) (eq. 16), which turn out to be identical to those collected in Table 1. It should be noted that a pair of enantiomers is counted just once for the values of B_{k} and C_{k}.
Rule 3 is further implemented as a Maple program for calculating the numbers of planted 3Dtrees (monosubstituted alkanes), which are itemized with respect to carbon content (1 to 100).
Maple program for Rule 3, "Enum3by1100RR.mpl":
We execute this code to obtain the coefficients a_{k} (eq. 10) for a(x) (eq. 11), B_{k} (eq. 12) for B(x) (eq. 14), and C_{k} (eq. 13) for C(x) (eq. 16), which are identical with those collected in Table 1.
3 Enumeration of Monosubstituted Alkanes as Graphs
It is worthwhile to mention the difference between the present work and previous results based on Polya's theorem. For this purpose, we implement the Polya's treatment [4, 5](Section 40) by using the Maple programming language on the same line as the present treatment based on Fujita's proligand method.
3. 1 Implementation of Polya's Treatment
Polya's treatment has been implicitly based on the condition that substituents for a given skeleton are limited to atoms (or graphs), because it uses the following functional equation:
in place of eq. 15. Obviously, eq. 33 is a special case of eq. 15, where we put r(x) = B(x) in the lefthand side and r(x) = b(x) = a(x) = c(x) in the righthand side. As a result, eq. 33 turns out to have recursive nature. The functional equation eq. 33 was first noted by Polya [4] by using the symmetric group of degree 3
, where an equation equivalent to eq. 7 was also derived by using the alternating group of degree 3
We implement the functional equation (eq. 33) as a Maple program for calculating the numbers of planted trees (monosubstituted alkanes as graphs), which are itemized with respect to carbon content (1 to 100).
Maple program for counting planted trees, "EnumPTr1100RR.mpl":
We execute this code to obtain the coefficients b_{k} for b(x) (eq. 7) and R_{k} for r(x) (eq. 33), as collected in Table 2. It should be noted that the two enantiomers of each pair are counted separately in the evaluation of b_{k}, while structural isomers (constitutional isomers) are counted as graphs in the evaluation of R_{k}. The results of R_{k} up to 20 are identical to those of Henze and Blair [3], the results from carbon content 21 to 30 are identical to those of Perry [27], and the results from carbon up to 50 are identical to those collected in [28](pages 153154 of Vol. II).
Table 2. Numbers of Monosubstituted Alkanes as Graphs
4 Discussions
4. 1 Stereoisomers vs. Graphs
Let us exemplify the difference between the enumeration results based on Fujita's proligand method (Table 1) and those based on Polya's theorem (Table 2).
Figure 3. Planted 3Dtrees or monosubstituted alkanes of carbon content 6. A planted 3Dtree with an asterisk is chiral, where the chirality is shown by a wedged bond and/or a hashed dash bond.
The k = 6 row of Table 1 indicates the presence of 8 achiral monosubstituted alkanes and of 10 enantiomeric pairs of chiral monosubstituted alkanes. The structural formulas of monosubstituted alkanes of carbon content 6 are shown in Figure 3, where either one selected from each pair of enantiomers is depicted as a representative, which is designated by an asterisk. In agreement with the calculation result shown in Table 1, there exist 8 achiral monosubstituted alkanes (4, 8, 11, 12, 13, 17, 19, and 21) and 10 enantiomeric pairs of chiral monosubstituted alkanes (5, 6, 7, 9, 10, 14, 15, 16, 18, and 20).
On the other hand, Table 2 (R_{6} = 17) indicates the presence of 17 monosubstituted alkanes as graphs (constitutional isomers). The difference between the value and the value of Table 1 (B_{6} = 18) comes from the the presence of diastereoisomers, 14 and 15, which are equalized in the enumeration based on Polya's theorem so as to be counted once as one graph. Chemically speaking, 14 and 15 are so congruent as to be regarded as one constitutional (structural) isomer. The value b_{6} = 28 in the k = 6 row of Table 2 can be calculated from the data collected in the k = 6 row of Table 1, i.e., 8 + 10 × 2 = 28, which are the number of stereoisomers, where enantiomers are counted separately.
The k = 7 row of Table 1 indicates the presence of 14 achiral monosubstituted alkanes and of 30 enantiomeric pairs of chiral monosubstituted alkanes. On the other hand, Table 2 indicates the presence of 39 monosubstituted alkanes as graphs. Manual enumeration shows that there are 14 achiral stereoisomers, 20 pairs of enantiomers with one asymmetric carbon, and 5 quartets of diastereomers with two asymmetric carbons. Thereby, we obtain the relationships 14 + 20 × 2 + 5 × 4 = 74 ( = b_{7}) and 14 + 20 + 5 = 39 ( = R_{7}). Note that b_{7} can be alternatively calculated by the data of Table 1, i.e., 14 + 30 × 2 = 74 ( = b_{7}).
The evaluation of R_{7} from B_{7} and C_{7} would require the evaluation of the number of diastereomeric quartets. In general, the evaluation of R_{k} from B_{k} and C_{k} requires the evaluation of the number of diastereomeric quartets and higher stereoisomeric sets. The latter evaluation is open to further investigation.
4. 2 Comments on Earlier Works
Robinson et al. [17] reported the enumeration of monosubstituted alkanes by modifying Polya's cycle indices (CIs) as follows:
where the r(x) in eq. 33 is replaced by p(x) for achiral ligands. However, the transitivity between a chiral ligand and its enantiomeric ligand is specified by the term r(x^{2}), which is the same as the term for two chiral ligands of the same chirality. By comparison between eq. 15 and eq. 34, the latter presumes p(x) = a(x) and r(x) = b(x) = c(x) in the present context, where the transitivity for p/p (or p/p) is mixed up with the transitivity for p/p (or p/p) if p and p represent a pair of enantiomeric proligands. In other words, their treatment also lacks the concept of sphericities.
Figure 4. Erroneous interpretation of transitivity which comes from the permutation (1)(2 3) assigned erroneously to s_{v(1)}.
An example of erroneous interpretation of transitivity by using eq. 34 is illustrated in Figure 4, where the term p(x)r(x^{2}) in eq. 34 corresponds to the action of the permutation (1)(2 3). When we put p = RCH(CH_{3})CH_{2}CH_{3} and p = SCH(CH_{3})CH_{2}CH_{3}, the formula 22 represents pseudoasymmetry for carbon content 9. The action of the permutation (1)(2 3) converts 22 into its diastereomer 23. This means that 22 and 23 are equalized so as to be regarded as one isomer, which is counted just once.
On the other hand, 24 is converted into the same 24 by the action of the permutation (1)(2 3). This means that 24 is counted just once. In a similar way, its enantiomer 24 is converted into the same 24 by the action of the permutation (1)(2 3). Thereby, 24 is counted just once, independent of 24.
Consequently, in the enumeration based on eq. 34, individual isomers to be counted are 22 (or 23), 24, and 24, which contribute by three to the coefficient of the term x^{9}. Obviously, the pseudoasymmetry between 22 and 23 is not treated properly.
Moreover, Robinson et al. [17] used the following functional equation:
to enumerate achiral planted 3Dtrees. Although this equation is akin to eq. 11, its connotation is entirely different from that of eq. 11. The misunderstanding connotation of eq. 35 stems from an unconscious assumption that the permutation (1)(2 3) can be assigned to the reflection operation s_{v(1)}. The term r(x^{2}) in eq. 35 means that 24 is "achiral" and independently 24 is also "achiral" because the permutation (1)(2 3) is erroneously assigned to the reflection operation s_{v(1)}, as found in Figure 4. Note that an object superposable to its mirror image is achiral and that the permutation (1)(2 3) (~s_{v(1)} erroneously) converts 24 (or 24}) into itself, which is inevitably but erroneously regarded as a mirror image (Figure 4). Hence, in the enumeration based on eq. 35, individual isomers to be counted as "achiral" are 24 and 24, which contribute by two to the coefficient of the term x^{9}.
By the same reason, we are forced to say that 22 and 23 are "chiral" so as to be "enantiomeric" to each other under the action of the permutation (1)(2 3) (~s_{v(1)} erroneously). This means that the set of 22 and 23 is regarded as one isomer and counted just once as an "enantiomeric" pair of "chiral" stereoisomers. Note that the term pseudoasymmetry has been coined to explain this type of phenomenon (22 vs. 23). It can be said that chemical combinatorics based on eq. 35 using the criteria of Figure 4 still remains before the coinage of the term pseudoasymmetry.
In consequence, so long as we rely on eqs. 34 and 35, "achiral" and "chiral" are reversely recognized in cases of pseudoasymmetry, as shown in Figure 4.
In contrast to eqs. 34 and 35, the present enumeration based on Fujita's proligand method uses eqs. 15 and 11. Because the permutation (1)(2 3) is assigned to the reflection operation s_{v(1)}, the case of pseudoasymmetry described above is interpreted in agreement with geometry and pointgroup theory, as shown in Figure 5.
The action of the permutation (1)(2 3) converts 22 into itself as shown in Figure 5, because the onecycle (1) is homospheric and the twocycle (2 3) is enantiospheric. This means that 22 is achiral so as to be regarded as one stereoisomer, which is counted just once. On the same line, 23 is converted into itself by the action of the permutation (1)(2 3), so that it is achiral so as to be regarded as one stereoisomer, which is counted just once.
On the other hand, 24 is converted into its enantiomer 24 by the action of the permutation (1)(2 3). This means that 24 and 24 are regarded as one pair of enantiomers, which is counted just once.
Consequently, in the enumeration based on eq. 15, individual isomers to be counted are 22, 23, 24 (as one enantiomeric pair with 24), which contribute by three to the coefficient of the term x_{9}. Thus the pseudoasymmetry between 22 and 23 is treated properly.
In Fujita's proligand method, the term c(x^{2}) in eq. 11 corresponds to the enantiosphericity, so that 22 is achiral and independently 23 is also achiral, because the permutation (1)(2 3) is correctly assigned to the reflection operation s_{v(1)}, as found in Figure 5. Hence, in the enumeration based on eq. 11, individual isomers to be counted as being achiral are 22 and 23, which contribute by two to the coefficient of the term x^{9}.
Figure 5. Correct interpretation of transitivity which comes from the permutation (1)(2 3) assigned properly to s_{v(1)}.
In the enumeration based on eq. 17, a set of 24 and 24 is counted just once as one pair of enantiomers, which contribute by one to the coefficient of the term x^{9}.
The enumeration based on the modified Polya's CIs (eq. 34 etc.) fortunately gives the same isomer numbers as obtained by means of Fujita's CICFs (eq. 15 etc.). Enumeration of the two types provides the apparently same contribution (e.g., three to the coefficient of the term x^{9}) because such a relationship between Figure 4 and Figure 5 holds true for any case of pseudoasymmetry. It should be noted, however, that the content of the former enumeration (cf. Figure 4) is quite different from the content of the latter enumeration (cf. Figure 5). Thus, the former enumeration erroneously shows the presence of one "chiral" isomer 22 (or 23) and two "achiral" isomers 24 and 24, which contribute by three to the coefficient of the term x^{9} (Figure 4). The latter enumeration correctly shows the presence of two achiral isomers 22 and 23 and one chiral isomer 24 (as one enantiomeric pair with 24), which contribute by three to the coefficient of the term x^{9} (Figure 5).
The discussions described in this section have clearly shown that Fujita's proligand method [12  14] is concerned with the enumeration of stereoisomers (3Dobjects), while Polya's theorem [4, 5] is concerned with the enumeration of constitutional isomers (graphs). The difference between the two methodologies stems from the presence/absence of the sphericity concept so that Fujita's proligand method involves Polya's theorem as a special case without sphericity. Moreover, it has been shown that the modification of Polya's theorem by Robinson et al. [17] has failed in the proper treatment of pseudoasymmetric cases because of the lack of the sphericity concept.
4. 3 Comments on a Stereochemical Convention
It is worthwhile to mention a stereochemical convention in comparison with the methodology adopted in the present approach. The IUPAC nomenclature (1996) [29] defines terms on stereoisomerism as follows:

(Def. 1) The term stereoisomerism is defined as "Isomerism due to differences in the spatial arrangement of atoms without any differences in connectivity or bond multiplicity between the isomers" [29](page 2219). The related term stereoisomers is defined as "Isomers that possess identical constitution, but which differ in the arrangement of their atoms in space. See enantiomer, diastereoisomer, cistrans isomers".

(Def. 2) The term enantiomerism is defined as "The isomerism of enantiomers" [29](page 2207). The related term enantiomer is defined as "One of a pair of molecular entities which are mirror images of each other and nonsuperposable. See also "enantiomorphism" [29](page 2207).
Because the term "stereoisomeric" (Def. 1) is concerned with a relationship between two compounds (as 3Dobjects), the relevant term "stereoisomer" is incapable of specifying a compound (a 3Dobject) without another 3Dobject stereoisomeric to it. For example, 22 is not a "stereoisomer" by itself; 22 can be called a "stereoisomer" just when 23 is taken into consideration. Thereby, we are able to say that the compounds 22 and 23 are "stereoisomers". In a similar way, 24 cannot be a "stereoisomer" nor an "enantiomer" by itself; the 24 can be called a "stereoisomer" or an "enantiomer" just when 22 (or 23) and/or 24 is taken into consideration. In the literal sense of the word "stereoisomer" defined in Def. 1, therefore, the expression "the number of stereoisomers" is permitted if a reference compound (e.g., 22) is given or if a common skeleton for generating stereoisomers is specified. When there exist two or more skeletons to be considered, therefore, the consideration of "the number of stereoisomers" is unsuitable for a systematic approach so as to require complicated treatments of various cases. In particular, the present enumeration problem for counting monosubstituted alkanes must consider an infinite number of skeletons which grow in their varieties in accord with the increase of carbon contents from zero to infinite. Then, what is aimed at by the expression "Numbers of Monosubstituted Alkanes as Stereoisomers"?
Obviously, Def. 1 contains two viewpoints, i.e., one viewpoint to recognize differences which are referred to by the expression "the differences in the spatial arrangement of atoms" and the other viewpoint to recognize equivalences which are referred to by the expression "without any differences in connectivity or bond multiplicity". However, a stereochemical convention tends to put emphasis on the viewpoint of "differences". On the same line, Def. 2 is mostly used in a context that enantiomers are nonsuperposable (i.e., different) instead of referring to their mirrorimage equivalence. So long as we obey the tendency of the stereochemical convention, it is difficult to count "different" objects, as found easily. In contrast, the present approach puts emphasis on equivalence classes (e.g., sets of compounds equivalent under Def. 1 or Def. 2) so that such equivalence classes can be counted by employing group theory.
From a viewpoint of the group theory which the present approach employs, the emphasis of "differences" in the stereochemical convention implies that the alternating group of degree 3 (
) is used to recognize stereochemical "equivalence". Under the action of
two enantiomers of each pair are different from each other, so that they are counted separately, as found in the b_{k}column of Table 2. Although the values collected in the b_{k}column of Table 2 are adopted as "the numbers of stereoisomers" in terms of the stereochemical convention or as "the numbers of steric trees" in terms of the combinatorial enumeration by Polya [4] and by Robinson et al. [17], we should mention several comments:

The use of is incapable of recognizing the chiral or achiral nature of each counted isomer. For example, the achirality of 22 (or 23) and the chirality of 24 are not recognized properly under the action of . The recognition of achirality/chirality necessitates subsequent manual examination by stereochemists.

The use of is incapable of recognizing enantiomeric relationships. For example, the enantiomeric relationship between 24 and 24 is not recognized properly under the action of . The recognition of enantiomeric relationships again necessitates subsequent manual examination by stereochimists.

What is the subsequent manual examination described in the abovementioned comments? In the stereochemical convention, the manual examination for recognizing achirality (e.g., 22 or 23), chirality (e.g., 24 or 24}), and enantiomeric relationship (e.g., between 24 and 24}) is to consider the mirror image of each 3Dobject. However, concrete processes of considering mirror images have not been fully formulated from the viewpoint of group theory.

Even if we adopt the symmetric group of degree 3 () which is a supergroup of , the recognition of achirality/chirality and enantiomeric relationships has proved futile, as discussed in Section 3. The enumeration of this type has given the numbers of graphs, as shown in the R_{k}column of Table 2.

Even by the attempt in which the symmetric group of degree 3 was used after modification [17], insufficient interpretations on achirality/chirality and on enantiomeric relationships took place, as discussed in Subsection 4.2.
These comments reveal why the stereochemical convention has adopted the values b_{k} (Table 2) as the numbers of stereoisomers. That is to say, the stereochemical convention lacks a mathematical formulation for recognizing achirality/chirality and enantiomeric relationships so that the convention is forced to rely on the alternating group of degree 3 ().
In contrast, the present approach puts emphasis on "equivalences" or more precisely on "equivalence classes", which are frequently called orbits in a mathematical context. The crux of the present approach is to use RCRs (C_{s}\)C_{3v} and (C_{1}\)C_{3} (cf. Figure 2) in place of and . As shown in Figure 5, for example, the use of (1)(2 3) assigned to s_{v(1)} is capable of treating the achirality of 22 (or 23), which is superposable on its mirror image so as to be selfenantiomeric. The use of (1)(2 3) is also capable of treating the enantiomeric relationship between 24 and 24, which is characterized as being in a mirrorimage relationship. Thereby, the manual examinations described above are formulated by the RCR (C_{s}\)C_{3v}.
In terms of the present methodology, stereoisomer enumeration is concerned with the numbers of equivalence classes under the action of the RCR(C_{s}\)C_{3v}. Note that the achiral compound 22 (or 23) constructs an equivalence class under the action of (C_{s}\)C_{3v}, where the achiral compound is counted just once. On the same line, the pair of 24 and 24 constructs an equivalence class under the action of (C_{s}\)C_{3v}, where the pair is counted just once. It follows that each value in the C_{k}column and the B_{k}column of Table 1 represents the number in which each achiral compound is counted just once and each enantiomeric pair is counted just once.
Consequently, the expression "Numbers of Monosubstituted Alkanes as Stereoisomers" (i.e., the title of this article) is permissible if the phrase "as Stereoisomers" is used as a simplified expression of the phrase "under the criterion of stereoisomerism" (cf. Def. 1) or more precisely "under the criterion of enantiomerism" (cf. Def. 2). The phrase "under the criterion of enantiomerism" means the action of the RCR of a point group, e.g., (C_{s}\)C_{3v}. This viewpoint means that enumeration concerned with stereoisomers should be done on the basis of equivalence classes provided "under the criterion of stereoisomerism" (cf. Def. 1) or more precisely "under the criterion of enantiomerism" (cf. Def. 2). More mathematically speaking, the enumeration concerned with stereoisomers should be done on the basis of equivalence classes provided by point groups through such RCRs as (C_{s}\)C_{3v}.
When two enantiomers of each pair are counted separately according to the stereochemical convention, the value 2C_{k} is used for counting chiral compounds; and the value a_{k} + C_{k} ( = b_{k}) is used for counting achiral and chiral compounds. These values should be adopted under the implicit restrictions described above for the stereochemical convention.
5 Conclusion
Planted threedimensional (3D) trees or equivalently monosubstituted alkanes as stereoisomers are regarded as planted promolecules, which are enumerated by Fujita's proligand method [12  14]. Cycle indices with chirality fittingness (CICFs), which are composed of three kinds of sphericity indices (SIs), i.e., a_{d} for homospheric cycles, c_{d} for enantiospheric cycles, and b_{d} for hemispheric cycles, are used to obtain generating functions for calculating planted 3Dtrees. Functional equations a(x), c(x^{2}), and b(x) for recursive calculations are derived from the CICFs and programmed by means of the Maple programming language. Thereby, recursive calculations of the numbers of planted 3Dtrees (or equivalently those of monosubstituted alkanes as stereoisomers) are executed and collected up to 100 carbon content in a tabular form. The results are compared with the enumeration of planted trees (as graphs).
We gratefully acknowledge the financial support given to our recent project by the Japan Society for the Promotion of Science: GrantinAid for Scientific Research B (No. 18300033, 2006).
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