Enumeration of Primary, Secondary, and Tertiary Monosubstituted Alkanes as Stereoisomers
Shinsaku FUJITA
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1 Introduction
Henze and Blair [1] have obtained the numbers of aliphatic alcohols of given carbon contents, where the numbers were itemized with respect to their degrees of branching, i.e., primary, secondary, and tertiary alcohols. Their enumeration based on recursive equations was concerned with constitutional isomers (or mathematically graphs), but not with stereoisomers as threedimensional (3D) objects.
The situation has not been restricted to such specific enumeration by Henze and Blair [1]. From the beginning when Cayley [2, 3] started the combinatorial enumeration of monosubstituted alkanes (or mathematically rooted or planted trees), calculated numbers have been concerned with constitutional isomers (graphs), but not with stereoisomers (3Dobjects) in most cases with or without such itemization due to the degrees of branching.
These situations have not been so dramatically changed even after the development of a more systematic method based on Polya's theorem [4, 5], because stereoisomers were not treated properly. Thus, by using symmetric groups, enumeration results based on Polya's theorem [4, 5] were concerned with graphs. By using alternating groups, on the other hand, achiral isomers, chiral isomers, and their enantiomers were counted independently, where pairwise recognition between a chiral isomer and its enantiomer was not been accomplished by Polya's theorem. Moreover, Polya's theorem did not take account of stereochemical problems on pseudoasymmetry and mesocompounds.
Among many contributions based on Polya's theorem [4, 5] to chemistry [6  11], only Robinson et al. [12] to our best knowledge reported the enumeration of monosubstituted alkanes as stereoisomers by modifying Polya's cycle indices (CIs). Even their treatment, however, did not take account of the classical stereochemical problems described above.
The disregard for pseudoasymmetry and mesocompounds has turned out to stem from the fact that both Polya's theorem itself and its modification by Robinson et al. lack the concept of sphericity, which has been recently formulated by us [13, 14] in solving stereochemical problems and has been more recently modified so as to provide Fujita's proligand method [15  17]. Because the sphericity concept is a key to differentiate Fujita's proligand method from Polya's theorem, it is important to exemplify how the sphericity concept works in various enumeration problems.
The sphericity concept has been developed as three versions, i.e., the concept "the sphericities of orbits" [13, 14], the concept "the sphericities of orbits based on cyclic groups" [18  20], and the concept "the sphericities of cycles" [15  17]. Among them, the concept "the sphericities of cycles" gave a basis of Fujita's proligand method [15  17], which is here adopted as a principal tool for solving the enumeration problems described in the first paragraph of the Introduction.
Because the enumeration of monosubstituted alkanes (rooted or planted 3Dtrees) as 3Dobjects has been recently accomplished by using the proligand method [21], remaining problems are to obtain the numbers of primary, secondary, and tertiary aliphatic alcohols which are regarded as 3Dobjects. The present paper is devoted to solving these problems, where their calculation procedures implemented by Maple programming language [22] are discussed in detail.
2 Itemized Enumeration of Monosubstituted Alkanes
2. 1 Recursive Functional Equations
Let a(x) be a generating function in which the coefficient (a_{k}) of the term x^{k} represents the number of achiral monosubstituted alkanes (or planted 3Dtrees) of carbon content k. Let c(x^{2}) be a generating function in which the coefficient (g_{2k}) of the term x^{2k} represents the number of achiral and chiral monosubstituted alkanes (or planted 3Dtrees) of carbon content 2k, where a diploid of enantiomers is counted just once [21]. Let b(x) be a generating function in which the coefficient (b_{k}) of the term x^{k} represents the number of achiral and chiral monosubstituted alkanes (or planted 3Dtrees) of carbon content k, where two enantiomers of each pair are separately counted. Thus, we can put the following ligand inventories:
where a_{0} = 1, g_{0} = 1, and b_{0} = 1 to treat trivial cases of terminal vertices (or hydrogen atoms).
A C_{3v}skeleton for enumerating such monosubstituted alkanes is characterized by the right coset representation (C_{s}\)C_{3v}, i.e., (1)(2)(3) for I, (1 2 3) for C_{3}, (1 3 2) for C_{3}^{2}, (1)(2 3) for s_{v(1)}, (1 3)(2) for s_{v(2)}, and (1 2)(3) for s_{v(3)}, so that the resulting cycles are transformed into the corresponding products of sphericity indices, i.e., b_{1}^{3} for I, b_{3} for two threefold rotations, and a_{1}c_{2} for three reflections. The proligand method [15  17] is applied to this case, where relevant cycle indices with chirality fittingness (CICFs) are obtained and transformed into eqs. 46 by replacing a_{d} by a(x^{d}), c_{d} by c(x^{d}), and b_{d} by b(x^{d}).
It follows that the three ligand inventories regarded as generating functions (eqs. 1 to 3) satisfy the following recursive functional equations:
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 to treat a trivial case of carbon content 0. Because we can place B_{k} = a_{k} + C_{k} and b_{k} = a_{k} + 2C_{k}, eq. 4 (for a(x)) and eq. 6 (for b(x)) give the following equations:
2. 2 Primary Monosubstituted Alkanes
Primary monosubstituted alkanes can be derived by placing an appropriate ligand on the vertex 1 of the C_{s}skeleton (1). The action of symmetry operations is shown in Figure 1. The transitivity of the vertex 1 is characterized by a trivial right coset representation (C_{s}\)C_{s} = {(1), (1)}, where the cycle (1) is hemispheric and the cycle (1)} is homospheric.
Figure 1. Symmetry operations of the C_{s}skeleton for counting primary monosubstituted alkanes (1).
We can apply Theorem 2 of [17] to the enumeration of primary monosubstituted alkanes without any modification. Hence, by using the product of SIs collected in Figure 1, the cycle index with chirality fittingness (CICF) for this case is calculated as follows:
which counts achiral primary monosubstituted alkanes and enantiomeric pairs of chiral ones. Theorem 3 of [17] 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:
which counts achiral primary monosubstituted alkanes and chiral ones, where two enantiomers of each pair are counted separately. We apply the first proposition of Theorem 4 for the enumeration of achiral ligands [17] to this case so as to derive the following CICF_{A}:
which counts achiral primary monosubstituted alkanes only. We use the second proposition of Theorem 4 for the enumeration of chiral ligands [17] to obtain the following CICF_{C}:
which counts enantiomeric pairs of chiral primary monosubstituted alkanes only, where each pair of enantiomers is counted just once.
Let P_{k}, P_{k}^{(A)}, and P_{k}^{(C)} be the number of achiral primary monosubstituted alkanes plus enantiomeric pairs, the number of achiral ones, and the number of enantiomeric pairs. They are represented as the following generating functions:
where x^{k} corresponds to the carbon content k >= 1.
By starting from eq. 11, the SIs b_{1} and a_{1} are replaced by b(x) and a(x) respectively. Thereby, we obtain the following functional equation by multiplying by x for the principal vertex:
It should be noted that the first term 1 in b(x) (eq. 6) or a(x) (eq. 4) is not omitted because a monosubstituted methane (CH_{3}) should be counted as a trivial case in addition to usual primary monosubstituted alkanes. On a similar line, eq. 13 gives the following functional equation:
after multiplied by x to take account of the principal vertex. Further, eq. 14 gives the following functional equation:
after multiplied by x.
Because a(x) (by eq. 4), c(x^{2}) (by eq. 5), and b(x) (by eq. 6) have been evaluated recursively, we are able to obtain the coefficients P_{k} (eq. 15), P_{k}^{(A)} (eq. 16), and P_{k}^{(C)} (eq. 17) by expanding eqs. 18, 19, and 20. Procedures for calculating them were programmed by using Maple programming language as follows.
Maple program for enumeration primary monosubstituted alkanes, "P1100.mpl":
Table 1. Numbers of Primary Monosubstituted Alkanes as Stereoisomers^{a}
^{a}In each value of P_{k}^{(C)} (Chiral) or P_{k} (Chiral + Achiral), each enantiomeric pair is counted just once for chiral primary monosubstituted alkanes.
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}), and so on. The first paragraph ("Definition of Functional Equations") declares three functional equations (eqs. 46). In the 2nd paragraph ("Initial Values"), the initial values for the initial (trivial) planted promolecule are set as a_{0} = 1, g_{0} = 1, and b_{0} = 1 by encoding
a1 := 1; c2 := 1;
and so on. The 3rd paragraph ("Do Loop for Calculation") shows a
do
loop for calculating a_{k}, g_{k}, and b_{k} recursively by using a Maple command
coeff
After escaping from the
do
loop, the 4th paragraph ("Primary Monosubstituted Alkanes") declares the calculation of P(x) (eq. 18), P(x)^{(A)} (eq. 19), and P(x)^{(C)} (eq. 20). The 5th paragraph (the final
do
loop named "PrintOut of Results") shows the printout of the calculation results.
The code was stored in a file named "P1100.mpl"and then executed by inputting the following command on the Maple inputting window:
read "P1100.mpl";
Thereby, we obtained the coefficients P_{k} (eq. 15) for P(x) (eq. 18), P_{k}^{(A)} (eq. 16) for P(x)^{(A)} (eq. 19), and P_{k}^{(C)} (eq. 17) for P(x)^{(C)} (eq. 20), as collected in Table 1. It should be noted that each enantiomeric pair is counted just once throughout the present enumeration. Hence, the value of P_{k}^{(C)} is the number concerned with pairs of enantiomeric primary monosubstituted alkanes. On the same line, the value of P_{k} is the number concerned with achiral primary monosubstituted alkanes plus pairs of enantiomeric primary monosubstituted alkanes.
2. 3 Secondary Monosubstituted Alkanes
Secondary monosubstituted alkanes are derived by placing two appropriate ligands on the vertices 2 and 3 of the C_{s}skeleton (2). The action of symmetry operations is shown in Figure 2. The transitivity of the vertices 2 and 3 is characterized by a right coset representation (C_{1}\)C_{s} = {(2)(3), (2 3)}, where the onecycle (2) or (3) is hemispheric so as to take an SI b_{1}, while the twocycle (2 3) is enantiospheric so as to take an SI c_{2}.
Figure 2. Symmetry operations of the C_{s}skeleton for counting secondary monosubstituted alkanes (2).
We apply Theorem 2 of [17] to the enumeration of secondary monosubstituted alkanes, so that we calculate the CICF for this case (Figure 2) as follows:
which counts achiral secondary monosubstituted alkanes and enantiomeric pairs of chiral ones. According to Theorem 3 of [17] for the enumeration of ligands under the action of the maximum chiral subgroup, we derive the following CICF in order to count achiral secondary monosubstituted alkanes and chiral ones:
where two enantiomers of each pair are counted separately. We apply the first proposition of Theorem 4 for the enumeration of achiral ligands [17] to this case so as to derive the following CICF_{A}:
which counts achiral secondary monosubstituted alkanes only. We adopt the second proposition of Theorem 4 for the enumeration of chiral ligands [17] to obtain the following CICF_{C}:
which counts enantiomeric pairs of chiral secondary monosubstituted alkanes only, where each pair of enantiomers is counted just once.
Let S_{k}, S_{k}^{(A)}, and S_{k}^{(C)} be the number of achiral secondary monosubstituted alkanes plus enantiomeric pairs, the number of achiral ones, and the number of enantiomeric pairs. Then, we obtain the corresponding generating functions as follows:
where x^{k} corresponds to the carbon content k >= 1.
To evaluate the generating function S(x) (eq. 25), we replace the SIs b_{1} and c_{2} in eq. 21 by b(x)  1 and c(x^{2})  1, respectively. The resulting equation is multiplied by x for the principal vertex to give the following functional equation:
It should be noted that the first term 1 in b(x) or c(x^{2}) is omitted; otherwise, a monosubstituted methane (CH_{3}) and primary monosubstituted alkanes (e.g., CH_{2}R) would be erroneously counted in.
Table 2. Numbers of Secondary Monosubstituted Alkanes as Stereoisomers^{a}
^{a}In each value of S_{k}^{(C)} (Chiral) or S_{k} (Chiral + Achiral), each enantiomeric pair is counted just once for chiral secondary monosubstituted alkanes.
On a similar line, eq. 23 when modified and multiplied by x gives the following functional equation:
Further, the modification of eq. 24 and the subsequent multiplication by x give the following functional equation:
Procedures for calculating S_{k} (eq. 25) by S(x) (eq. 28), S_{k}^{(A)} (eq. 26) by S(x)^{(A)} (eq. 29), and S_{k}^{(C)} (eq. 27) by S(x)^{(C)} (eq. 30) were programmed by using Maple programming language as follows (the omitted paragraphs are the same as those of "P1100.mpl").
Maple program for enumeration primary monosubstituted alkanes: "S1100.mpl":
The execution of the program gave the results collected in Table 2.
2. 4 Tertiary Monosubstituted Alkanes
Tertiary monosubstituted alkanes are derived from the same skeleton (3) as discussed for monosubstituted alkanes without itemization due to branching (Figure 3). A different point is that all of the three substituents in a tertiary monosubstituted alkane must be alkyl ligands, but not hydrogens.
We apply Theorem 2, Theorem 3, and the two propositions of Theorem 4 of [17] to the enumeration of tertiary monosubstituted alkanes. To count achiral planted promolecules and enantiomeric pairs of chiral planted promolecules, we employ the following CICF:
We use the following CICF:
in order to count achiral and chiral planted promolecules. For the purpose of counting achiral planted promolecules only, we start from the following CICF_{A}:
To count chiral planted promolecules only, we use the following CICF_{C}:
We have already used these equations to obtain the functional equations for recursive calculations, i.e., a(x) (eq. 4), c(x^{2}) (eq. 5), and b(x) (eq. 6). But, we here use them in a slightly different manner in order to treat tertiary monosubstituted alkanes.
Figure 3. Symmetry operations for a C_{3v}skeleton (3). The identity operation (I) converts 3 into itself, where the resulting skeleton is denoted as 3a.
Table 3. Numbers of Tertiary Monosubstituted Alkanes as Stereoisomers^{a}
^{a}In each value of T_{k}^{(C)} (Chiral) or T_{k} (Chiral + Achiral), each enantiomeric pair is counted just once for chiral tertiary monosubstituted alkanes.
Let T_{k}, T_{k}^{(A)}, and T_{k}^{(C)} be the number of achiral tertiary monosubstituted alkanes plus enantiomeric pairs, the number of achiral ones, and the number of enantiomeric pairs. They satisfy the following generating functions:
where x^{k} corresponds to the carbon content k >= 1.
The SIs a_{d}, c_{d}, and b_{d} in eq. 31 are replaced by a(x^{d})  1, c(x^{d})  1 and b(x^{d})  1 respectively. The resulting equation is multiplied by x to take the principal vertex into consideration. Thereby, we obtain the following functional equation:
It should be noted that the first term 1 in b(x), a(x), or c(x^{2}) is omitted in order to exclude monosubstituted methane (CH_{3}), primary (e.g., CH_{2}R), and secondary monosubstituted alkanes (e.g., CHRR'). On a similar line, eq. 33 gives the following functional equation:
Further, eq. 34 gives the following functional equation:
Procedures for calculating T_{k} (eq. 35) by T(x) (eq. 38), T_{k}^{(A)} (eq. 36) by T(x)^{(A)} (eq. 39), and T_{k}^{(C)} (eq. 37) by T(x)^{(C)} (eq. 40) were programmed by using the Maple programming language as follows (the omitted paragraphs are the same as those of "P1100.mpl").
Maple program for enumeration primary monosubstituted alkanes: "T1100.mpl":
The execution of the program gave the results collected in Table 3.
2. 5 Total Numbers
The total numbers of achiral stereoisomers of primary, secondary, and tertiary monosubstituted alkanes are calculated by adding the coefficients of the term x^{k} appearing in eqs. 19, 29, and 39. The results should be identical with the values a_{k} alternatively calculated by using eq. 4. In fact, we can obtain the following relationship:
where the coefficient of each term is identical with that of eq. 4 except x^{0}.
The total numbers of achiral enantiomeric pairs of primary, secondary, and tertiary monosubstituted alkanes are calculated by adding the coefficients of the term x^{k} appearing in eqs. 20, 30, and 40. The results should be identical with the values C_{k} alternatively calculated by using C(x) (eq. 10). In fact, we can obtain the following relationship:
which is identical with the expanded equation derived from C(x) (eq. 10).
The total numbers of achiral plus enantiomeric pairs of primary, secondary, and tertiary monosubstituted alkanes are calculated by adding the coefficients of the term x^{k} appearing in eqs. 18, 28, and 38. The results should be identical with the values B_{k} alternatively obtained by B(x) (eq. 9). In fact, we can obtain
which is identical with the expanded equation derived from B(x) (eq. 9) except the term x^{0}.
We are able to verify these facts numerically by the following Maple program "Test1100.mpl", where we put:
AA for eq. 41, CC for eq. 42, BB for eq. 43, BBx for eq. 9, CCx for eq. 10.
Maple program for enumeration monosubstituted alkanes: "Test1100.mpl":
Table 4. Total Numbers of Monosubstituted Alkanes as Stereoisomers^{a}
^{a}In each value of B_{k}^{(C)}', C_{k}, B_{k}', and C_{k}, each enantiomeric pair is counted just once for chiral monosubstituted alkanes.
The results are summarized in Table 4, where the coefficients appearing in
AA, CC, and BB
are listed in the B_{k}^{(A)}', B_{k}^{(C)}', and B_{k}'columns, respectively. For comparison, Table 4 also collects the values of a_{k} (eq. 4), C_{k} (eq. 10), and B_{k} (eq. 9). Obviously, we can find equality between B_{k}^{(A)}' and a_{k}, between B_{k}^{(C)}' and C_{k}, as well as between B_{k}' and B_{k}.
By comparing eq. 9 and eq. 18, we obtain the following relationship:
This equation results in the equality between the coefficients of eq. 9 (P_{k+1}) and those of eq. 18 (B_{k}), i.e.,
In fact, this relationship can be verified by comparing the P_{k}column of Table 1 with the B_{k}column of Table 4.
Because eq. 19 for achiral primary monosubstituted alkanes is akin to eq. 44, we are able to obtain the following equation:
This relationship can be verified by comparing the P_{k}^{(A)}column of Table 1 with the a_{k}column of Table 4.
On the same line as eq. 44, we obtain the following relationship by comparing eq. 10 and eq. 20:
This equation results in the equality between the coefficients of eq. 9 (P_{k+1}) and those of eq. 20 (C_{k}), i.e.,
This relationship can be verified by comparing the P_{k}^{(C)}column of Table 1 with the C_{k}column of Table 4.
Chemically speaking, eq. 46 reflects the exclusive derivation of achiral primary monosubstituted alkanes from lower achiral alkyl ligands, e.g.,
On the other hand, eq. 47 reflects the exclusive derivation of chiral primary monosubstituted alkanes from lower chiral alkyl ligands, e.g.,
where each enantiomeric pair is counted just once. It follows that eq. 45 reflects the total derivation of primary monosubstituted alkanes from lower alkyl ligands.
3 Enumeration of Monosubstituted Alkanes as Graphs
To emphasize the difference between stereoisomers and graphs (constitutional isomers), let us count monosubstituted alkanes as graphs by using Polya's theorem. Note that any CIs based on Polya's theorem [4, 5] can be derived from the CICFs of Fujita's proligand method [15  17] by omitting sphericities. In other words, Polya's theorem lacks the sphericity concept. Although Polya put it, "The cycle index knows many things." [10](page 67), the cycle index (CI) does not know the sphericities of cycles. In connection with the present viewpoint of stereoisomer enumeration, it is important to know how Polya's theorem lacks the sphericity concept.
Suppose that the number (R_{k}) of monosubstituted alkanes of carbon content k is itemized into those of primary (P_{k}^{(G)}), secondary (S_{k}^{(G)}), and tertiary ones (T_{k}^{(G)}), as shown in the following generating functions:
where x^{k} corresponds to the carbon content k and we put R_{0} = 1, P_{(G)} = 1, S_{0}^{(G)} = 0 (omitted), and T_{0}^{(G)} = 0 (omitted).
Because any Polya's CI can be regarded as a special case of the CICF of Fujita's proligand method with putting r_{d} = a_{d} = c_{d} = b_{d}, eq. 43 can be converted to the following functional equation by putting r(x) = P(x) + S(x) + T(x) in the lefthand side and r(x) = b(x) = a(x) = c(x) in the righthand side and by adding 1 to assure recursive character:
Note that this equation gains recursive nature at the expense of the sphericity concept in contrast to eq. 43.
Enumeration of primary monosubstituted alkanes as graphs is regarded as a special case of eq. 18. Hence, by putting r(x) = a(x) = b(x), we obtain:
Similarly, enumeration of secondary monosubstituted alkanes as graphs can be conducted as a special case of eq. 28, where we put r(x) = b(x) = c(x) as follows:
On a similar line, enumeration of tertiary monosubstituted alkanes as graphs is regarded as a special case of eq. 38. Hence, we obtain the following functional equation by putting r(x) = a(x) = b(x) = c(x):
These generating functions were programmed by means of Maple programming language and the code stores in a file named "Graph1100.mpl" tentatively.
Table 5. Numbers of Monosubstituted Alkanes as Graphs
Maple program for enumeration monosubstituted alkanes, "Graph1100.mpl":
In this code, we use the following symbols:
rx for r(x) (eq. 53), Pg for P(x)^{(G)} (eq. 54), Sg for S(x)^{(G)} (eq. 55), and Tg for T(x)^{(G)} (eq. 56); as well as r1 for r(x), r2 for r(x^{2}), and r3 for r(x^{2}).
The paragraph "Validity Test" confirms that the values by the recursive calculation (
rx
for r(x)) are equal to the sum of the itemized values (
RR
or P(x)^{(G)}+S(x)^{(G)}+T(x)^{(G)}). This is directly confirmed by summing up the functional equations:
which is identical with r(x) (eq. 53) except for the initial value 1 added for recursion.
The results are summarized in Table 5, where the coefficients appear in the P_{k}^{(G)} (primary), the S_{k}^{(G)} (secondary), the T_{k}^{(G)} (tertiary), and the R_{k}columns, respectively. The last column V_{k} contains zero values for indicating the validity of the present calculation.
The results up to 20 shown in Table 5 are identical with those of Henze and Blair [1], the results from carbon content 21 to 30 are identical with those of Perry [23], and the results from carbon content up to 50 are identical with those collected in Trinajstic's book [24](pages 153154 of Vol. II).
According to eq. 54, we obtain the following relationship:
The relationship can be verified by comparing the P_{k}^{(G)}column of Table 5 with the R_{k}column of the same table. Obviously, this is a graphcounting counterpart of eq. 45. Chemically speaking, eq. 58 reflects the exclusive derivation of primary monosubstituted alkanes (as graphs) from lower alkyl ligands (as graphs).
4 Discussions
4. 1 Pseudoasymmetric Cases
Among the enumeration results collected in Tables 1,  3, the S_{k}^{(A)}column in Table 2 exhibits a special feature to be pointed out. Thus, we can find that each value of S_{k}^{(A)} (Table 2) vanishes to zero when the carbon content k is even. This feature holds true in general. In order to be achiral, a secondary monosubstituted alkane should have a formula represented by XCHR_{2} or XCHpp where the R is an achiral ligand or the pair p/p represents a pair of enantiomeric ligands. As a result, the total carbon content (k) is calculated to be odd because the number represented by 2
l
+1 ( = k) means an odd number whether the carbon content (
l
) of the R (or p) is odd or even. Hence, there exists no achiral secondary monosubstituted alkane with an even k (i.e., S_{k}^{(A)} = 0 if k is even). In other words, all of the secondary monosubstituted alkanes characterized by even carbon contents (k) are chiral and contribute to S_{k}^{(C)} (k: even).
Achiral secondary monosubstituted alkanes having the general formula XCHpp are recognized as pseudoasymmetric cases. The two diastereomers corresponding to each formula XCHpp with carbon content k contribute by two to the value of S_{k}^{(A)} (Table 2).
Figure 4. Achiral secondary monosubstituted alkanes of carbon content 9. If necessary, configurations are shown by wedged bonds and/or hashed dash bonds.
For example, let us examine the case of S_{9}^{(A)} = 5 in Table 2. The five achiral monosubstituted alkanes are listed in Figure 4. Among them, 5 and 6 are diastereomeric and show pseudoasymmetric nature. The remaining monosubstituted alkanes 4, 7, and 8 are obviously represented by the general formula XCHR_{2}, where the R represents an achiral butyl ligand. It should be noted that Figure 4 contains all types of butyl substituents, i.e., nbutyl (for 4), secbutyl (for 5 and 6), isobutyl (for 7), and tbutyl (for 8). These butyls in turn are enumerated as P_{4}^{(A)} = 2 (Table 1), S_{4}^{(C)} = 1 (Table 2), and T_{4}^{(A)} = 1 (Table 3). Note that the numbers of such alkyl ligands are equal to the numbers of the corresponding monosubstituted alkanes.
The last finding in the preceding paragraph can be extended into general cases. When achiral secondary monosubstituted alkanes of carbon content k are represented by the formula XCHR_{2} or XCHpp, they are composed of two R's of carbon content (k  1)/2 or of a pair of p and p of carbon content (k  1)/2. The two R's and the pair of p and p cover all of alkyl ligands of carbon content (k  1)/2.
Figure 5. Achiral secondary monosubstituted alkanes of carbon content 11. If necessary, configurations are shown by wedged bonds and/or hashed dash bonds.
To exemplify a more complicated case, let us examine the case of S_{11}^{(A)} = 11 in Table 2. Because of the relationship (11  1)/2 = 5, we should take alkyl ligands of carbon content 5 into account. Thus, there are 8 alkyl ligands of carbon content 5, where we find P_{5}^{(A)} = 3 (for pentyl, 3methylbut1yl, and 2,2dimethylprop1yl) and P_{5}^{(C)} = 1 (for 2methylbut1yl) in Table 1; S_{5}^{(A)} = 1 (for pent3yl) and S_{5}^{(C)} = 2 (for 1methylbut1yl and 3methylbut2yl) in Table 2; as well as T_{5}^{(A)} = 1 (for tpentyl) and T_{5}^{(C)} = 0 in Table 3. Because each enantiomeric pair of chiral ligands (P_{5}^{(C)}, S_{5}^{(C)}, and T_{5}^{(C)}) generates two diastereomers of pseudoasymmetric nature, there should appear 11 achiral secondary monosubstituted alkanes, where we calculate P_{5}^{(A)} + 2P_{5}^{(C)} + S_{5}^{(A)} + 2S_{5}^{(C)} + T_{5}^{(A)} + 2T_{5}^{(C)} = 3 + 1 × 2 + 1 + 2 × 2 + 1 + 0 = 11.
The 11 achiral monosubstituted alkanes of carbon content 11 are listed in Figure 5. Among them, three pairs, i.e., 11/12 (from 2methylbut1yl), 13/14 (from 1methylbut1yl), and 15/16 (from 3methylbut2yl) are produced from pairs of chiral pentyl ligands according to the general formula XCHpp, where the two stereoisomers of each pair are diastereomeric and show pseudoasymmetric nature. The remaining 5 monosubstituted alkanes are obviously represented by the general formula XCHR_{2}, where the R represents an achiral pentyl ligand.
4. 2 Stereoisomers vs. Graphs
To show the effect of itemization due to the degree of branching, let us depict monosubstituted alkanes of carbon content 7.
The k = 7 row of Table 1 shows that there are 8 achiral primary monosubstituted alkanes and 10 enantiomeric pairs of chiral ones. They are listed in Figure 6, where either one enantiomer is depicted as a representative of each pair of enantiomers.
Because the 8 achiral primary monosubstituted alkanes (2027) are different from each other as graphs, they are constitutional isomers, which are counted so as to be 8 constitutional isomers (graphs).
As found easily, they are represented by XCH_{2}R, where the R covers all of the achiral hexyl ligands. In agreement with eq. 46, the presence of the 8 achiral primary monosubstituted alkanes of carbon content 7 (i.e., 2027) is related to the value a_{6} = 8 in Table 4.
In agreement with eq. 48, the presence of 10 chiral primary monosubstituted alkanes of carbon content 7 (P_{7}^{(C)} = 10 in Table 1) is related to the value C_{6} = 10 in Table 4. Note that they are represented by XCH_{2}p (or XCH_{2}p), where the p (or p) covers all the chiral hexyl ligands.
Among the 10 chiral primary monosubstituted alkanes (2837), there appears one diastereomeric case, i.e., 32 and 33, which are identical with each other as graphs. The set of 32 and 33 is regarded as a single constitutional isomer, which is counted just once, if we rely on Polya's theorem. It follows that the 10 primary monosubstituted alkanes (2837) are counted so as to be 9 constitutional isomers (graphs).
As a result, there are 17 constitutional isomers for primary monosubstituted alkanes (8 + 9 = 17). The number appears as P_{7}^{(G)} in Table 5, which is obtained by means of the enumeration based on Polya's theorem.
Because of eq. 58, the number of hexyl ligands as graphs is evaluated to be R_{6} = 17, as shown in Table 5. Hence we can confirm the relationship, P_{7}^{(G)} = R_{6} = 17, in agreement with eq. 58.
Figure 6. Primary monosubstituted alkanes of carbon content 7. Configurations are shown by wedged bonds and/or hashed dash bonds, if necessary. An arbitrary enantiomer is depicted for a representative of each pair of enantiomers.
The k = 7 row of Table 3 shows that there are 2 achiral secondary monosubstituted alkanes and 17 enantiomeric pairs of chiral ones. They are listed in Figure 7, where either one enantiomer is depicted as a representative of each pair of enantiomers.
The 2 achiral secondary monosubstituted alkanes (38 and 39) are different from each other as graphs so that there are 2 constitutional isomers.
Figure 7. Secondary monosubstituted alkanes of carbon content 7. Configurations are shown by wedged bonds and/or hashed dash bonds, if necessary. An arbitrary enantiomer is depicted for a representative of each pair of enantiomers.
Among the 17 chiral secondary monosubstituted alkanes (4056), there appear four sets of diastereomers, i.e., 45/46, 47/48, 49/50, and 54/55, each of which is regarded as a single constitutional isomer. Hence, the 17 chiral secondary monosubstituted alkanes (4056) are counted so as to be 13 constitutional isomers.
It follows that there are 15 constitutional isomers for secondary monosubstituted alkanes (2 + 13 = 15). The number appears as S_{7}^{(G)} in Table 5, which is obtained by means of the enumeration based on Polya's theorem.
Figure 8. Tertiary monosubstituted alkanes of carbon content 7. Configurations are shown by wedged bonds and/or hashed dash bonds, if necessary. An arbitrary enantiomer is depicted for a representative of each pair of enantiomers.
According to the k = 7 row of Table 3, there are 4 achiral tertiary monosubstituted alkanes and 3 enantiomeric pairs of chiral ones. They are listed in Figure 8, where either one enantiomer is depicted as a representative of each pair of enantiomers.
The 4 achiral tertiary monosubstituted alkanes (5760) are constitutional isomers and the 3 chiral tertiary monosubstituted alkanes (6163) are constitutional isomers. Hence, there are 7 constitutional isomers for tertiary monosubstituted alkanes. The number appears as T_{7}^{(G)} in Table 3, which is obtained by means of the enumeration based on Polya's theorem.
Consequently, we obtain 17 + 15 + 7 = 39 for the total number of constitutional isomers of monosubstituted alkanes, which appears as R_{7} in Table 5.
4. 3 Selection of Equivalence Classes
According to a stereochemical convention, two enantiomers of each pair are counted separately so as to contribute enumeration values by 2. In contrast, the present methodology adopts the criterion that each pair of enantiomers is counted just once so as to contribute enumeration values by 1. It is worthwhile to discuss the effect of the present criterion in comparison with the former criterion of the stereochemical convention.
Figure 9. Equivalence classes for secondary monosubstituted alkanes of carbon content 9. Pseudoasymmetric cases and a related pair of enantiomers.
The difference between the two criteria stems from the selection of equivalence classes, which depends on the selection of groups as bases of respective enumeration procedures. The difference is exemplified by using pseudoasymmetric and related cases, as shown in Figure 9.
The stereochemical convention counts 5, 6, 64, and 64 separately so as to give isomer number 4, as shown in Figure 9. This enumeration result implies the use of the alternating group of degree 1 (
), which is isomorphic to C_{1} in the present approach. To illustrate what happens under the stereochemical convention, let us adopt the CICF for C_{1} (eq. 22), where we use the following ligand inventory:
in accord with the substitution of enantiomeric secbutyl ligands p = x^{4} and p = x^{4} = x^{4}). Note that the symbol x represents the enantiomer of x, but is equal to x if carbon content is considered. By introducing eq. 59 into the righthand side of eq. 22 and by multiplying x for the central carbon atom, we obtain the following generating functions:
where the coefficient 4 of the term x^{9} indicates the presence of 4 isomers of carbon content 9 under the criterion provided by C_{1} (or equivalently
).
This result is in agreement with the stereochemical convention, where the four compounds 5, 6, 64, and 64 are counted separately. In other words, each of the compounds counted (i.e., 5, 6, 64, and 64) is independently regarded as an equivalence class due to C_{1} (
).
The enumeration result described in the preceding paragraph reveals implications concealed in the stereochemical convention:

The independent behavior under the action of C_{1}(
)
means that each compound is recognized to be "chiral" in spite of its realistic chirality/achirality, because the action of C_{1}(
)
contains no reflection operations. If the expression "chiral" used in characterizing the achiral compounds (5 and 6) seems to be rather strange, we can safely say that the compounds have no information on their chirality/achirality, as shown by the note "not specified" in Figure 9. Hence, the recognition of chirality/achirality is left afterward in the hands of stereochemists, who subsequently apply reflection operations to each compound for the purpose of symmetry characterization.

Moreover, the enumeration under the action of C_{1} (or equivalently
) specifies no relationships among the four compounds, 5, 6, 64, and 64. In particular, the enantiomeric relationship between 64 and 64 cannot be specified under the action of C_{1} (or equivalently
), because the C_{1} contains no reflection operations. Hence, the specification of enantiomeric relationships is regarded as a subsequent and distinct step using reflection operations. This step of symmetry characterization requires a manual examination by intelligence of stereochemists.
The implications itemized above reveal that the number 4 for counting 5, 6, 64, and 64 has been obtained at the expense of symmetry characterizations, i.e., the recognition of chirality/achirality and the specification of enantiomeric relationships. In other words, there is a logical gap between the first step using C_{1} or
(the enumeration step) and the second step using reflection operations (the step of symmetry characterization). The logical gap has been frequently overlooked by organic stereochemists, because they can easily and unconsciously skip the logical gap by their intuition.
In contrast, the present approach has attempted to fill the logical gap explicitly by using the point group C_{s}, as shown in Figure 2. It should be emphasized that an equivalence class under the action of C_{s} is composed of an achiral compound (as a selfenantiomeric pair) or of a pair of enantiomers. Thus, chiral compounds are counted once per pair of enantiomers.
By following Fujita's proligand method [15  17], we use the following ligand inventories:
where we select enantiomeric secbutyl ligands as substituents (p = x^{4} and p = x^{4}). These inventories are introduced into the CICF shown in eq. 21 and multiplied by x to give the following generating function:
The coefficient 3 of the term x^{9} indicates the presence of three compounds counted per pair of enantiomers, which are 5 (an achiral compound as a selfenantiomeric pair), 6 (an achiral compound as a selfenantiomeric pair), and 64/64 (an enantiomeric pair of two chiral compounds), as shown in Figure 6.
To count achiral compounds, the inventory (eq. 62) is introduced into the CICF_{A} shown in eq. 23 and multiplied by x to give the following generating function:
The coefficient 2 of the term x^{9} indicates the presence of two achiral compounds counted per pair of enantiomers. They are 5 and 6, each of which is an achiral compound regarded as being a selfenantiomeric pair (Figure 6).
After the inventories (eqs. 61 and 62) are introduced into the CICF_{C} shown in eq. 24 and x is multiplied, the resulting equation is expanded to give the following generating function:
The coefficient 1 of the term x^{9} indicates the presence of one pair of enantiomers, which is represented by the pair 64/64, as shown in Figure 6.
Moreover, eq. 22, which has been used to obtain eq. 60 is also involved in the present approach because the C_{1} is a subgroup of the C_{s}. Obviously, the combination of eqs. 6365 together with eq. 60 is more informative than the stereochemical convention using eq. 60 only.
From the viewpoint of the present approach, the values collected in the P_{k}^{(A)}, P_{k}^{(C)}, and P_{k}columns of Table 1 are concerned with equivalence classes due to C_{s}, because they are based on the C_{s}skeleton shown in Figure 1. The three columns contain consistent values under the action of C_{s}. This situation holds true for the columns contained in Tables 2, 3.
In place of the present column constitutions of Tables 1,  3, we can select, for example, a column listing the values of P_{k}^{(A)}, a column listing the values of 2P_{k}^{(C)}, and a column listing the values of P_{k}^{(A)} + 2P_{k}^{(C)} in agreement with the stereochemical convention. As a result of this type of column constitution, we find that the column listing the values of P_{k}^{(A)} is based on C_{s}, while the column listing the values of 2P_{k}^{(C)} and the column listing the values of P_{k}^{(A)} + 2P_{k}^{(C)} are based on C_{1}. Hence, this type of column constitution is inconsistent from the viewpoint of pointgroup theory.
5 Conclusions
Primary, secondary, and tertiary monosubstituted alkanes are counted as stereoisomers by means of Fujita's proligand method [15  17]. According to cycle indices with chirality fittingness (CICFs), 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 replaced by a(x^{d}), c(x^{d}), and b(x^{d}) so as to produce functional equations for recursive calculation. After the derivation of respective functional equations for counting primary, secondary, and tertiary monosubstituted alkanes, they are programmed by means of the Maple programming language and executed to obtain values up to carbon content 100. The respective results are collected in a tabular form, where the numbers of stereoisomers are further itemized into achiral and chiral ones. By omitting the sphericity concept, such CICFs are transformed into Polya's cycle indices (CIs), which are applied to the enumeration of primary, secondary, and tertiary monosubstituted alkanes as graphs (chemically, constitutional isomers). The results of Fujita's proligand method are compared with those based on Polya's theorem in connection with several cases of pseudoasymmetry.
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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