J Math Chem (2016) 54:310–330 DOI 10.1007/s10910-015-0562-y ORIGINAL PAPER

Type-itemized enumeration of quadruplets of RS-stereoisomers: II—cycle indices with chirality fittingness modulated by type-V quadruplets Shinsaku Fujita1

Received: 14 May 2015 / Accepted: 17 September 2015 / Published online: 25 September 2015 © Springer International Publishing Switzerland 2015

Abstract Fujita’s proligand method developed originally for combinatorial enumeration under point groups (Fujita in Theor Chem Acc 113:73–79, 2005) has been extended to meet combinatorial enumeration under Fujita’s stereoisogram approach (Fujita in J Org Chem 69:3158–3165, 2004). A new succinct method for type-itemized enumeration of quadruplets of RS-stereoisomers has been developed on the basis of cycle indices with chirality fittingness (CI–CF) modulated by type-V quadruplets, where CI–CFs for the subgroups of the corresponding RS-stereoisomeric group are used to calculate CI–CFs of the respective types (type I to type V). By introducing ligand-inventory functions to these CI–CFs, generating functions have been obtained to give the numbers of inequivalent quadruplets, which are itemized with respect to five types. The method using a modulated CI–CF is applied to enumerations based on an oxirane skeleton, an allene skeleton, and a tetrahedral skeleton. The relationship of RS-stereoisomers versus stereoisomers is discussed by referring to the enumeration results of oxirane derivatives. Keywords RS-stereoisomer · RS-stereoisomeric group · Stereoisogram · Type-itemized enumeration

1 Introduction Enumeration of chemical compounds has been traditionally conducted by considering permutation groups, as found in Pólya’s accomplishments on the enumeration of chemical compounds as graphs [1–3]. The usefulness of Pólya’s method in chemistry

B 1

Shinsaku Fujita [email protected] Shonan Institute of Chemoinformatics and Mathematical Chemistry, Kaneko 479-7 Ooimachi, Ashigara-Kami-Gun, Kanagawa-Ken 258-0019, Japan

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has been summarized in reviews [4–6] and books [7–9]. These traditional approaches have made light of the difference between permutation groups and point groups, so that the effect of chirality due to point groups has not been fully investigated. On the other hand, enumeration by considering point groups has been more recent topics. For example, Fujita’s unit-subduced-cycle-index (USCI) approach [10,11] and Fujita’s proligand method [12] have properly treated the effect of chirality on enumeration of chemical compounds. To clarify the difference between permutation groups and point groups, however, a more comprehensive approach should be developed. For this purpose, Fujita’s stereoisogram approach [13–15] has integrated permutation groups and point groups by developing RS-stereoisomeric groups. Stereoisograms have been proposed as diagrammatic representations of RS-stereoisomeric groups for integrating point groups and RS-permutation groups [13–15]. Their potentialities for reorganizing the theoretical foundations of stereochemistry and stereoisomerism have recently been discussed in detail [16–19]. Each quadruplet of RS-stereoisomers contained in a stereoisogram is regarded as an equivalence class under the action of the corresponding RS-stereoisomeric group. Such quadruplets of RS-stereoisomers have been proven to be categorized into five types [20]. Type-itemized enumerations of quadruplets have been conducted according to three kinds of relationships contained in stereoisograms [21,22]. More recently, symmetry-itemized enumerations of quadruplets have been conducted by extending Fujita’s USCI approach [10], where tetrahedral derivatives [23,24], allene derivatives [25,26], and three-membered heterocycles [27–29] are combinatorially enumerated after the preparation of necessary data (e.g., mark tables, inverse mark tables, subductions of coset representations). Because such necessary data to symmetry-itemized enumerations are not so easily obtained in general, type-itemized enumerations are still important as succinct methods. In Part I, the method of type-itemized enumeration reported in [22] has been substantially simplified, where cycle indices with chirality fittingness (CI–CFs) have been modulated by type-IV quadruplets after the extension of Fujita’s proligand method to meet Fujita’s stereoisogram approch [30]. As a continuation of Part I, the present paper as Part II deals with another simplified method using CI–CFs modulated by type-V quadruplets.

2 Type-itemized enumeration 2.1 Derivation of CI–CFs for type-itemized enumeration Suppose that a stereoskeleton belongs to a point group GCσ . Then, an RSstereoisomeric group G is constructed according to Fujita’s stereoisogram approach [14,20]. Then, the point group GCσ is interpreted as the maximum point group of G, where 2|GCσ | = |G|. The RS-stereoisomeric group G contains the maximum RS-permutation group GC σ (2|GC σ | = |G|), which is isomorphic to the point group GCσ ; the maximum ligand-reflection group GC  I (2|GC  I | = |G|); and the maximum normal subgroup GC (4|GC  I | = |G|) as a common subgroup. Promolecules derived from the stereoskeleton of G are categorized into five types, which are correlated to

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the five subgroups, i.e., type-III promolecules belonging to subgroups of GC , typeV promolecules belonging to subgroups of GCσ (minus GC ), type-II promolecules belonging to subgroups of GC σ (minus GC ), type-I promolecules belonging to sub(minus G ), type-IV promolecules belonging to subgroups of G (minus groups of GC  C I GC , covering reflections, RS-permutations, and ligand-reflections) under the action of the RS-stereoisomeric group G. Under the action of RS-stereoisomeric group G, a quadruplet of promolecules contained in a stereoisogram is counting just once. As a result, the number of inequivalent quadruplets under G is evaluated by using the cycle index with chirality fittingness denoted as CI–CF(G; $d , bd ), where the symbol $d denotes a sphericity index ad or cd . Let the symbols CI–CF[K] (G) (K = I to V) be the CI–CFs for evaluating the numbers of inequivalent quadruplets of RS-stereoisomeric promolecules of respective types. Thereby, the CI–CF(G; $d , bd ) is represented by the sum of them: CI–CF(G; $d , bd ) = CI–CF[I] (G) + CI–CF[II] (G) + CI–CF[III] (G) + CI–CF[IV] (G) + CI–CF[V] (G),

(1)

where the symbols CI–CF[K] (G) (K = I to V) in the right-hand sides are used in place of CI–CF[K] (G; $d , bd ) (K = I to V) for the sake of simplicity. The stereoskeleton belonging to G can be alternatively considered to belong to the respective subgroups described above, where equivalence classes are changed according to the subgroups. For example, suppose that the stereoskeleton belongs to the maximum point group GCσ . Then each quadruplet is divided into pairs of (self-)enantiomers, where a type-I quadruplet gives one pair of enantiomers, a type-II quadruplet gives one pair of enantiomers, a type-III quadruplet gives two pairs of enantiomers, a type-IV quadruplet gives one achiral promolecule (one pair of selfenantiomers), and a type-V quadruplet gives two achiral promolecules (two pairs of self-enantiomers). Because each pair of (self-)enantiomers is counted once as an equivalence class under GCσ , the CI–CF(GCσ ; $d , bd ) is represented as follows: CI–CF(GCσ ; $d , bd ) = CI–CF[I] (G) + CI–CF[II] (G) + 2CI–CF[III] (G) + CI–CF[IV] (G) + 2CI–CF[V] (G).

(2)

This process is repeated to cover all the five subgroups described above. The results are summarized as a matrix representation: ⎞⎛ ⎞ ⎞ ⎛ ⎛ CI–CF[I] (G) 22412 CI–CF(GC ; bd ) ⎟⎜ ⎟ ⎜ CI–CF(G ; $ , b ) ⎟ ⎜ ⎜ 1 1 2 1 2 ⎟ ⎜ CI–CF[II] (G) ⎟ Cσ d d ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ CI–CF[III] (G) ⎟ , 1 2 2 1 1⎟ (3) ⎜ CI–CF(GC ; bd ) ⎟ = ⎜ σ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟⎜ ⎟ [IV] ⎝ CI–CF(G ; $d , bd ) ⎠ ⎜ 2 1 2 1 1 CI–CF (G) ⎝ ⎠ ⎝ ⎠ CI CI–CF(G; $d , bd ) 11111 CI–CF[V] (G) where the result of Eq. 2 appears in the second row. This equation has been once noted in Part I of this series. Because the central 5 × 5 matrix is singular, its inverse does not exist. This means that Eq. 3 cannot be solved to give CI–CF[I] (G) and so on.

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To convert the central 5×5 matrix of Eq. 3 into a non-singular matrix, the CI–CF for the point group, i.e., CI–CF(GCσ ; $d , bd ), is modulated by subtracting CI–CF[V] (G) from Eq. 2 as follows: Definition 1 (A Modulated CI–CF) CI–CF (GCσ ; $d , bd ) = CI–CF(GCσ ; $d , bd ) − CI–CF[V] (G) = CI–CF[I] (G)+CI–CF[II] (G)+2CI–CF[III] (G)+CI–CF[IV] (G)+CI–CF[V] (G), (4) which is called a modulated CI–CF by type-V quadruplets. Then, Eq. 3 is converted into the following equation: ⎞⎛ ⎛ ⎞ ⎛ ⎞ CI–CF[I] (G) 22412 CI–CF(GC ; bd ) ⎟⎜ ⎜ ⎟ ⎜ ⎟ ⎜ CI–CF (GCσ ; $d , bd ) ⎟ ⎜ 1 1 2 1 1 ⎟ ⎜ CI–CF[II] (G) ⎟ ⎟⎜ ⎜ ⎟ ⎜ ⎟ [III] ⎟⎜ ⎜ CI–CF(GC ⎟ ⎜ (G) ⎟ σ ; bd ) ⎟ = ⎜ 1 2 2 1 1 ⎟ ⎜ CI–CF ⎜ ⎟, ⎟⎜ ⎜ ⎟ ⎜ ⎟ [IV] ⎝ 2 1 2 1 1 ⎠ ⎝ CI–CF (G) ⎠ ⎝ CI–CF(GC  I ; $d , bd ) ⎠ 11111 CI–CF(G; $d , bd ) CI–CF[V] (G)

(5)

where the value 2 surrounded by a frame in the central 5 × 5 matrix of Eq. 3 is changed into 1 surrounded by a frame in the corresponding matrix of Eq. 5. Thus, Eq. 4 appears in the second row of the central 5×5 matrix of Eq. 5. Because the 5×5 matrix of Eq. 5 is non-singular, its inverse can be calculated so as to give the following equation: ⎞⎛ ⎛ ⎞ ⎛ ⎞ CI–CF(GC ; bd ) CI–CF[I] (G) 0 −1 0 1 0 ⎟⎜ ⎜ ⎟ ⎜ ⎟ ⎜ CI–CF[II] (G) ⎟ ⎜ 0 −1 1 0 0 ⎟ ⎜ CI–CF (GCσ ; $d , bd ) ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ CI–CF[III] (G) ⎟ = ⎜ 0 1 0 0 −1 ⎟ ⎜ CI–CF(GC ⎟ (6) σ ; bd ) ⎟ . ⎟⎜ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ [IV] ⎝ CI–CF (G) ⎠ ⎝ −1 2 0 0 0 ⎠ ⎝ CI–CF(GC  I ; $d , bd ) ⎠ 1 −1 −1 −1 2 CI–CF[V] (G) CI–CF(G; $d , bd ) If the modulated CI–CF (GCσ ; $d , bd ) is evaluated, Eq. 6 means that CI–CFs for type-itemized enumeration can be obtained as follows: CI–CF[I] (G) = −CI–CF (GCσ ; $d , bd ) + CI–CF(GC  I ; $d , bd ) [II]  CI–CF (G) = −CI–CF (GCσ ; $d , bd ) + CI–CF(GC σ ; bd )

CI–CF[III] (G) = CI–CF (GCσ ; $d , bd ) − CI–CF(G; $d , bd ) CI–CF[IV] (G) = −CI–CF(GC ; bd ) + 2CI–CF (GCσ ; $d , bd )

(7) (8) (9) (10)

CI–CF[V] (G) = CI–CF(GC ; bd ) − CI–CF (GCσ ; $d , bd ) − CI–CF(GC σ ; bd ) − CI–CF(GC  (11) I ; $d , bd ) + 2CI–CF(G; $d , bd ). The next task is the evaluation of the modulated CI–CF (GCσ ; $d , bd ) (Eq. 4), which requires the evaluation of CI–CF[V] (G).

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2.2 Evaluation of the CI–CF for enumerating type-V quadruplets The number of achiral promolecules (type IV and type V) is calculated by using the first and second rows of Eq. 3 as follows: 2CI–CF(GCσ ; $d , bd ) − CI–CF(GC ; bd ) = CI–CF[IV] (G) + 2CI–CF[V] (G). (12) On the other hand, Eqs. 12 and 13 of Part I of this series gives the following result: 2CI–CF(GCσ ; $d , bd ) − CI–CF(GC ; bd ) =

ν (P) ν (P) 2 $11 $22 · · · $νnn (P) |GCσ | P∈GC σ

(13) Thereby, Eqs. 12 and 13 of the present article (Part II) gives the following CI–CF for enumerating type-V quadruplets: CI–CF[V] (G) =

ν (P) ν (P) 1 1 $11 $22 · · · $νnn (P) − CI–CF[IV] (G), |GCσ | 2

(14)

P∈GC σ

where the symbol $d represents ad or cd . ν (P) ν (P) ν (P) Each product of sphericity indices (PSI) $11 $22 · · · $nn in Eq. 14 can contribute to the evaluated value of type-V quadruplets if it contains a sphericity index cd for an enantiospheric orbit. However, the evaluated value due to the PSI ν (P) ν (P) ν (P) $11 $22 · · · $nn is contaminated by the contribution of the type-IV quadruplets. ν1 (P) ν2 (P) ν (P) $2 · · · $nn should be corrected by subtracting PSIs for typeHence, the PSI $1 IV quadruplets. This process of correction can be done in a trial-and-error fashion by introducing the corresponding ligand-inventory functions to a tentative corrected PSI. Once the the CI–CF for enumerating type-V quadruplets (Eq. 14) is evaluated, the modulated CI–CF is calculated according to Definition 1 (Eq. 4). Thereby, the CI–CFs for the respective types, i.e., CI–CF[K] (G; $d , bd ) (K = I to V), are calculated by means of Eqs. 7–11.

2.3 Type-itemized enumeration using CI–CFs derived from the modulated CI–CF Type-itemized enumeration by starting from the CI–CFs for the respective types (Eqs. 7–11) can be accomplished in a similar way to the procedure described in Part I of this series. To maintain the self-contained nature of the present article (Part II), a minimum set of information is described here. A set of n proligands selected from the ligand inventory represented by X = {X1 , X2 , . . . , Xm ; p1 , p2 , . . . , pm  ; p1 , p2 , . . . pm  }

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(15)

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is applied to the n substitution positions of a given skeleton belonging to an RSstereoisomeric group G. Each of the resulting quadruplets of promolecules (as RSstereoisomers) has the following composition (a molecular formula): θ θ

θ

θ  θ 

θ  

Wθ = Xθ11 Xθ22 · · · Xθmm p11 p22 · · · pmm p11 p22 · · · pmm , 

(16)

where X1 , X2 , etc. represent achiral proligands; p1 , p2 , etc. represent chiral proligands; p1 , p2 , etc. represent chiral proligands of opposite chirality; and m and m  represent non-negative integers. Each exponent of the composition Wθ (Eq. 16) indicates the number of proligands of the same kind, i.e., θ1 of X1 , θ2 of X2 , · · · θm of Xm ; θ1 of p1 , θ2 of p2 , · · · θm  of pm  ; θ1 of p1 , θ2 of p2 , · · · , or θm  of pm  , which satisfies the following partition: [θ ] = θ1 + θ2 + · · · + θm + θ1 + θ2 + · · · + θm  + θ1 + θ2 + · · · + θm  = n.

(17)

The following theorem using the CI–CFs (Eqs. 7–11) is equivalent to Theorem 1 of Part I of this series, although the modulated CI–CF (Eq. 4) of Definition 1 is used in place of another modulated CI–CF adopted in Part I. Theorem 1 Let the symbol NθK denote the type-itemized number of such quadruplets as having the composition Wθ (Eq. 16) and type K (K = I to V), where each quadruplet of RS-stereoisomers is counted just once. A generating function for evaluating the number NθK is obtained to be:

  NθK Wθ = CI–CF[K] (G; $d , bd )

[θ]

(18)

$d ,bd = X

for K = I to V, where the more definite symbol CI–CF[K] (G; $d , bd ) is used in place of the simplified form CI–CF[K] (G) (Eqs. 7–11) and the symbol $d , bd = X denotes the introduction of a ligand-inventory function to the sphericity index $d (or bd ): ad = Xd1 + Xd2 + · · · + Xdm cd = bd =

Xd1 Xd1

+ Xd2 + Xd2

+ · · · + Xdm + · · · + Xdm

(19) d/2 d/2 d/2 d/2 d/2 d/2 + 2p1 p1 + 2p2 p2 + · · · + 2pm  pm  + pd1 + pd2 + · · · + pdm  + pd1 + pd2 + · · · + pdm  .

(20) (21)

The composition Wθ (Eq. 16) is conjugate to the following composition: θ  θ 

θ   θ  θ 

θ

W θ = Xθ11 Xθ22 · · · Xθmm p11 p22 · · · pmm p11 p22 · · · pmm . 

(22)

Thereby, the term 21 (Wθ + W θ ) corresponds to one quadruplet in each generating function represented by Eq. 18.

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3 Examples 3.1 Type-itemized enumeration of oxirane derivatives Let us first examine oxirane derivatives on the basis of an oxirane skeleton 1, which is one of 4-ligancy stereoskeletons exemplified in Fig. 1. The point group of the oxirane skeleton 1 is determined to be C2v of order 4, where the maximum chiral subgroup is the point group C2 of order 2. The corresponding RSstereoisomeric group C2v σ I is constructed on the basis of the point group C2v , as shown in Table 1. The four positions of the oxirane skeleton 1 belong to a C2v (/C1 )-orbit under the point group C2v , while they are regarded as belonging to a C2v I )σ I (/C [27–29]. Each operation of C orbit under the RS-stereoisomeric group C2v   σI 2v σ I is represented by a product of cycles, which is characterized by a product of sphericity indices (PSI) as listed in the PSI-column of Table 1. By referring to the data of the PSI-column of Table 1, the CI–CF for C2v (A and B) and the CI–CF for C2 (A) are calculated as follows: 1 4 (b + b22 + 2c22 ) 4 1 1 CI–CF(C2 ; bd ) = (b14 + b22 ), 2

CI–CF(C2v ; $d , bd ) =

(23) (24)

where the PSIs for C2v or C2 are summed up and divided by the order |C2v | = 4 or |C2 | = 2. By referring to Eq. 23, the CI–CF for the corresponding RS-permutation group C2 σ (A and C in Table 1) and the CI–CF for the corresponding ligand-reflection group C2 I (A and D in Table 1) are calculated as follows:

3 1

H

4 C

C

2

H

4

2

O

1

2

1

3

H

H1

H

≡

C 4

C

H

H3

H

2

3

Fig. 1 Oxirane, allene, and tetrahedral skeletons of ligancy 4 Table 1 Operations of C2v σ I and coset representation of C2v σ I (/C I)

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1

C ⇐= (View) 2

3 4

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1 4 (b + 3b22 ) 4 1 1 4 2 4 2 CI–CF(C2 I ; $d , bd ) = (b1 + b2 + a1 + c2 ). 4 CI–CF(C2 σ ; bd ) =

(25) (26)

Finally, Eqs. 23, 25, and 26 are combined to give the CI–CF for the RS-stereoisomeric group C2v σ I (A, B, C, and D in Table 1) as follows: CI–CF(C2v σ I ; $d , bd ) =

1 4 (b + 3b22 + a14 + 3c22 ). 8 1

(27)

According to Eq. 14, the CI–CF for type V can be evaluated by the following equation: CI–CF[V] (C2v σ I) =

1 2 1 c − CI–CF[IV] (C2v σ I ). 2 2 2

(28)

The PSI c22 in Eq. 28 contains a PSI a22 for type IV as a contaminant. Moreover, The PSI c22 contains a PSI c4 as a contaminant, because such a dually generated term as p2 p2 can be ascribed to c4 . Then, a PSI a4 is added to correct overestimation: CI–CF[V] (C2v σ I) =

1 2 (c − a22 − c4 + a4 ). 2 2

(29)

By applying Definition 1 (Eq. 4) to this case, the modulated CI–CF is calculated from Eq. 23 and Eq. 29: CI–CF (C2v ; $d , bd ) =

1 4 1 (b1 + b22 + 2c22 ) − (c22 − a22 − c4 + a4 ). 4 2

(30)

The set of CI–CFs (Eqs. 24–27) and the modulated CI–CF (Eq. 30) are applied to Eqs. 7–11. Thereby, the CI–CFs for type-itemized enumeration of oxirane derivatives are obtained as follows:  CI–CF[I] (C2v σ I ) = −CI–CF (C2v ; $d , bd ) + CI–CF(C2 I ; $d , bd ) 1 4 = (a1 + c22 − 2a22 − 2c4 + 2a4 ) 4  CI–CF[II] (C2v ) = −CI–CF (C2v ; $d , bd ) + CI–CF(C2  σ ; bd ) σI 1 2 = (b2 − a22 − c4 + a4) 2  CI–CF[III] (C2v ) = CI–CF (C2v ; $d , bd ) − CI–CF(C2v  σI σ I ; $d , bd ) 1 4 = (b1 − b22 + 4a22 + 4c4 − 4a4 − a14 − 3c22 ) 8  CI–CF[IV] (C2v ) = −CI–CF(C  2 ; bd ) + 2CI–CF (C2v ; $d , bd ) σI

= a22 + c4 − a4

(31)

(32)

(33)

(34)

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 CI–CF[V] (C2v σ ; bd ) σ I ) = CI–CF(C2 ; bd ) − CI–CF (C2v ; $d , bd ) − CI–CF(C2

− CI–CF(C2 I ; $d , bd ) + 2CI–CF(C2v σ I ; $d , bd ) 1 2 = (c2 − a22 − c4 + a4 ). 2

(35)

These CI–CFs are identical with the CI–CFs obtained via symmetry-itemized enumeration, which has been based on the extension of the partial-cycle-index (PCI) method of Fujita’s USCI approach (Eqs. 111–115 of [27]). Suppose that a set of four proligands for the oxirane skeleton 1 (Fig. 1) is selected from the following ligand inventory: X = {A, B, X, Y; p, q, r, s; p, q, r, s}.

(36)

According to Eqs. 19–21, the corresponding ligand-inventory functions for counting oxirane derivatives are obtained as follows: ad = Ad + Bd + Xd + Yd

(37)

cd = A + B + X + Y + 2p d

d

d

d

d/2 d/2

p

+ 2q

d/2 d/2

q

+ 2r

d/2 d/2

r

+ 2s

d/2 d/2

s

(38) bd = A + B + X + Y + p + q + r + s + p + q + r + s . d

d

d

d

d

d

d

d

d

d

d

d

(39)

where the letters A, B, X, and Y represent achiral proligands and the pairs of p/p, q/q, r/r, and s/s represent pairs of enantiomeric proligands. These ligand-inventory functions (Eqs. 37–39) are introduced into the CI–CFs for type-itemized enumeration (Eqs. 31–35). The resulting equations are expanded to give the following generating functions for type I to type V: f 1[I] = {(A3 B) + · · · } + {3(A2 BX) + · · · } + {6(ABXY) + · · · } f 1[II] f 1[III]

+{(A2 pp) + · · · } + {2(ppqq) + · · · } + {(A2 B2 ) + · · · }

 1 4 (p + p4 ) + · · · = {(A2 p2 + A2 p2 ) + · · · } + {(p2 q2 + p2 q2 ) + · · · } 2

(40)

(41)

 1 3 (A p + A3 p) + · · · = 2  

3 2 3 2 (A Bp + A2 Bp) + · · · + (A pq + A2 pq) + · · · + 2 2

 3 2 2 (ABp + ABp ) + · · · +{3(ABXp + ABXp) + · · · } + {3ABpp + · · · } + 2

 3 (Ap2 p + App2 ) + · · · +{3(ABpq + ABpq) + · · · } + 2

  3 1 2 2 3 3 + (Ap q + Ap q) + · · · + (Ap + Ap ) + · · · 2 2 +{3(Appq + Appq) + · · · } + {3(Apqr + Apqr) + · · · }

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  1 3 1 3 (p p + pp3 ) + · · · + (p q + p3 q) + · · · 2 2  

3 2 3 2 2 (p pq + pp q) + · · · + (p qq + p2 qq) + · · · + 2 2

 3 2 (p qr + p2 qr) + · · · + {3(ppqr + ppqr) + · · · } + {3(pqrs + pqrs) + · · · } + 2  

1 2 2 1 2 2 2 2 2 2 (A p + A p ) + · · · + (p q + p q ) + · · · (42) + 2 2 +

f 1[IV] = {2(A2 B2 ) + · · · } + {2(p2 p2 ) + · · · } + {(A4 ) + · · · }

(43)

f 1[V]

(44)

= {2(A pp) + · · · } + {4(ppqq) + · · · } + {(p p ) + · · · } 2

2 2

These generating functions are equivalent to those reported in a previous paper (Eqs. 116–120 of [27]). Note that a pair of parentheses containing a conjugate pair of compositions indicates that its half, e.g., 21 (A2 p2 + A2 p2 ), represents one quadruplet. A pair of parentheses containing a single composition indicates that it represents one quadruplet; e.g., the term (A3 B) represents the presence of one quadruplet with the composition A3 B.

3.2 Type-itemized enumeration of allene derivatives Let us second examine the type-itemized enumeration of allene derivatives on the basis of an allene skeleton 2 (or 3) as another 4-ligancy stereoskeleton (Fig. 1). Note that the allene skeleton 2 (or 3) belongs to the point group D2d of order 8, the maximum chiral subgroup D2 of order 4, the RS-permutation group D2 σ of order 8, the ligandof order 8, and the RS-stereoisomeric group D2d reflection group D2 I σ I of order 16. The CI–CFs necessary to the type-itemized enumeration are cited from Part I of this series and listed as follows: CI–CF(D2d ; $d , bd ) = CI–CF(D2 ; bd ) = CI–CF(D2 σ ; bd ) = CI–CF(D2 I ; $d , bd ) = CI–CF(D2d σ I ; $d , bd ) =

1 4 (b + 3b22 + 2a12 c2 + 2c4 ) (45) 8 1 1 4 (b + 3b22 ) (46) 4 1 1 4 (b + 3b22 + 2b12 b2 + 2b4 ) (47) 8 1 1 4 (b + 3b22 + a14 + 3c22 ) (48) 8 1 1 4 (b + 3b22 + 2a12 c2 + 2c4 + 2b12 b2 + 2b4 + a14 + 3c22 ). 16 1 (49)

According to Eq. 14, the CI–CF for type V can be evaluated by the following equation:

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CI–CF[V] (D2d σ I) =

1 2 1 (a1 c2 + c4 ) − CI–CF[IV] (D2d σ I ). 4 2

(50)

The PSI a12 c2 in Eq. 50 is a candidate to represent type V, where it contains PSIs a12 a2 and a2 c2 for enumeration of type IV. These are subtracted as contaminants, but an PSI a22 should be added as a duplicated component. The other PSI c4 in Eq. 50 is not related to type V. Hence, Eq. 50 is converted into the following CI–CF for type V: CI–CF[V] (D2d σ I) =

1 2 (a c2 − a12 a2 − a2 c2 + a22 ). 4 1

(51)

By applying Definition 1 (Eq. 4) to the allene case, the modulated CI–CF is calculated from Eq. 45 and Eq. 51: CI–CF (D2d ; $d , bd ) = CI–CF(D2d ; $d , bd ) − CI–CF[V] (D2d σ I) 1 4 1 = (b1 + 3b22 + 2a12 c2 + 2c4 ) − (a12 c2 − a12 a2 − a2 c2 + a22 ). 8 4

(52)

The set of CI–CFs (Eqs. 46–49) and the modulated CI–CF (Eq. 52) are applied to Eqs. 7–11. Thereby, the CI–CFs for type-itemized enumeration of allene derivatives are obtained as follows:  CI–CF[I] (D2d σ I ) = −CI–CF (D2d ; $d , bd ) + CI–CF(D2 I ; $d , bd ) 1 = (a14 + 3c22 − 2a12 a2 − 2a2 c2 + 2a22 − 2c4 ) 8  CI–CF[II] (D2d σ ; bd ) σ I ) = −CI–CF (D2d ; $d , bd ) + CI–CF(D2 1 = (b12 b2 + b4 − a12 a2 − a2 c2 + a22 − c4 ) 4  CI–CF[III] (D2d σ I ) = CI–CF (D2d ; $d , bd ) − CI–CF(D2d σ I ; $d , bd ) 1 4 (b + 3b22 − 2b12 b2 − 2b4 + 4a12 a2 + 4a2 c2 − 4a22 = 16 1 + 2c4 − 2a12 c2 − a14 − 3c22 )

(53)

(54)

(55)

 CI–CF[IV] (D2d σ I ) = −CI–CF(D2 ; bd ) + 2CI–CF (D2d ; $d , bd ) 1 = (a12 a2 + a2 c2 − a22 + c4) (56) 2  CI–CF[V] (D2d σ ; bd ) σ I ) = CI–CF(D2 ; bd ) − CI–CF (D2d ; $d , bd ) − CI–CF(D2 − CI–CF(D2 I ; $d , bd ) + 2CI–CF(D2d σ I ; $d , bd ) 1 2 = (a1 c2 − a12 a2 − a2 c2 + a22 ). (57) 4

These CI–CFs are identical with those obtained by an alternative method described in Part I of this series. Moreover, they are identical with the CI–CFs obtained via

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symmetry-itemized enumeration, which has been based on the extension of the partialcycle-index (PCI) method of Fujita’s USCI approach (Eqs. 65–69 of [26]). The ligandinventory functions (Eqs. 37–39) are introduced into the CI–CFs for type-itemized enumeration (Eqs. 53–57) to give the corresponding generating functions, which are omitted here (cf. Part I).

3.3 Type-itemized enumeration of tetrahedral derivatives Let us third examine the type-itemized enumeration of tetrahedral derivatives on the basis of a tetrahedral skeleton 4 as a further 4-ligancy stereoskeleton (Fig. 1). Note that the tetrahedral skeleton 4 belongs to the point group Td of order 24, the maximum chiral subgroup T of order 12, the RS-permutation group T σ of order 24, the ligandof order 24, and the RS-stereoisomeric group Td reflection group T I σ I of order 48. The CI–CFs necessary to the type-itemized enumeration of tetrahedral derivatives are cited from Part I of this series and listed as follows: CI–CF(Td ; $d , bd ) = CI–CF(T; bd ) = CI–CF(T σ ; bd ) = CI–CF(T I ; $d , bd ) = CI–CF(Td σ I ; $d , bd ) =

1 4 (b + 3b22 + 8b1 b3 + 6a12 c2 + 6c4 ) 24 1 1 4 (b + 3b22 + 8b1 b3 ) 12 1 1 4 (b + 3b22 + 8b1 b3 + 6b12 b2 + 6b4 ) 24 1 1 4 (b + 3b22 + 8b1 b3 + a14 + 3c22 + 8a1 a3 ) 24 1 1 4 (b + 3b22 + 8b1 b3 + 6a12 c2 + 6c4 48 1 + 6b12 b2 + 6b4 + a14 + 3c22 + 8a1 a3 ).

(58) (59) (60) (61)

(62)

According to Eq. 14, the CI–CF for type V can be evaluated by the following equation: CI–CF[V] (Td σ I) =

1 2 1 (a1 c2 + c4 ) − CI–CF[IV] (Td σ I ). 4 2

(63)

Because the terms to be evaluated are equal to those of Eq. 50, a similar process of evaluation can be applied to Eq. 63. Hence, Eq. 63 is converted into the following CI–CF for type V: CI–CF[V] (Td σ I) =

1 2 (a c2 − a12 a2 − a2 c2 + a22 ), 4 1

(64)

which is equal to Eq. 51. By applying Definition 1 (Eq. 4) to the tetrahedral case, the modulated CI–CF is calculated from Eq. 58 and Eq. 64:

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CI–CF (Td ; $d , bd ) = CI–CF(Td ; $d , bd ) − CI–CF[V] (Td σ I) 1 4 1 (b1 + 3b22 + 8b1 b3 + 6a12 c2 + 6c4 ) − (a12 c2 − a12 a2 − a2 c2 + a22 ). = 24 4 (65) The set of CI–CFs (Eqs. 59–62) and the modulated CI–CF (Eq. 65) are applied to Eqs. 7–11. Thereby, the CI–CFs for type-itemized enumeration of tetrahedral derivatives are obtained as follows:  CI–CF[I] (Td I ; $d , bd ) σ I ) = −CI–CF (Td ; $d , bd ) + CI–CF(T 1 4 (a + 3c22 + 8a1 a3 − 6a12 a2 − 6a2 c2 + 6a22 − 6c4 ) (66) = 24 1  CI–CF[II] (Td σ ; bd ) σ I ) = −CI–CF (Td ; $d , bd ) + CI–CF(T 1 2 = (b1 b2 + b4 − a12 a2 − a2 c2 + a22 − c4 ) (67) 4  CI–CF[III] (Td σ I ) = CI–CF (Td ; $d , bd ) − CI–CF(Td σ I ; $d , bd ) 1 4 (b + 3b22 + 8b1 b3 + 12a12 a2 + 12a2 c2 − 12a22 = 48 1 + 6c4 − 6a12 c2 − 6b12 b2 − 6b4 − a14 − 3c22 − 8a1 a3 ) (68)  CI–CF[IV] (Td σ I ) = −CI–CF(T; bd ) + 2CI–CF (Td ; $d , bd ) 1 = (a12 a2 + a2 c2 − a22 + c4) (69) 2  CI–CF[V] (Td σ ; bd ) σ I ) = CI–CF(T; bd ) − CI–CF (Td ; $d , bd ) − CI–CF(T

− CI–CF(T σ I ; $d , bd ) I ; $d , bd ) + 2CI–CF(Td 1 2 = (a1 c2 − a12 a2 − a2 c2 + a22 ). 4

(70)

These CI–CFs are identical with those obtained by an alternative method described in Part I of this series. Moreover, they are equal to the CI–CFs obtained via symmetryitemized enumeration, which has been based on the extension of the partial-cycle-index (PCI) method of Fujita’s USCI approach (Eqs. 69–73 of [24]). The ligand-inventory functions (Eqs. 37–39) are introduced into the CI–CFs for type-itemized enumeration (Eqs. 66–70) to give the corresponding generating functions, which are omitted here (cf. Part I).

4 RS-stereoisomers versus stereoisomers A quadruplet of RS-stereoisomers is an intermediate concept which mediates between a pair of enantiomers and a set of stereoisomers. The type-itemized enumeration of quadruplets under an RS-stereoisomeric group provides useful pieces of information on RS-stereoisomers vs. stereoisomers.

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4.1 Quadruplets with the compositions A2 pp and A2 p2 Let us examine quadruplets of RS-stereoisomers with the composition A2 pp. The term A2 pp in f 1[I] (Eq. 40) indicates the presence of one quadruplet of type I, which is depicted in the first stereoisogram containing the representative promolecule 5 (Fig. 2). On the other hand, the term 2A2 pp in f 1[V] (Eq. 44) indicates the presence of two typeV quadruplets, which are depicted in the second and third stereoisograms containing the representative promolecules 7 and 9 (Fig. 2). Type I

Type V S

p A

3

4

1

A p 2

A p

1

p

3

4

O

p A

A

3

4

p

1

2

O

A p

A

1

2

3

A p 4

O

O

5

6 (= 5)

7

8

(1)(2)(3)(4)

(1 3)(2 4)

(1)(2)(3)(4)

(1 3)(2 4)

A p

1

2

p

p A

3

4

O

C

2

S

3

4

A 1

A p

A p

1

2

A p

3

2

p

3

4

p A

1

2

O

O

4

A O

5

6 (= 5)

7 (= 7)

8 (= 8)

(1 3)(2 4)

(1)(2)(3)(4)

(1 3)(2 4)

(1)(2)(3)(4)

C

Type V S A A

3

4

p

1

2

p

A A

1

2

p

3

4

O

O

9

10

(1)(2)(3)(4)

(1 3)(2 4)

A A

1

2

p p

3

4

A A

3

4

p

p p

1

2

O

O

9 (= 9)

10 (= 10)

(1 3)(2 4)

(1)(2)(3)(4)

C

Fig. 2 Stereoisograms for characterizing three quadruplets of RS-stereoisomers with the composition A2 pp. The uppercase letter A represents an achiral proligand in isolation and the lowercase letters p and p¯ represent chiral proligands with opposite chirality senses. The priority sequence is presumed to be A > p >p

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The promolecules 5 and 5 are enantiomeric to each other and they are contained as one pair of enantiomers of This profile of the stereoisogram is   the stereoisogram. characterized by the symbol [5 5] , where a pair of square brackets represents a pair of enantiomers, while a pair of parentheses represents a quadruplet of RS-stereoisomers. Note that the pair of enantiomers [5 5] is a pair of RS-diastereomers at the same time. The promolecule 7 (or 8) is achiral so that it is represented by the symbol [7] (or [8]). This symbol means that the achirality is regarded as being ascribed to a selfenantiomeric relationship or to a one-membered orbit under the point group C2v . Because 7 and 8 are contained in the same stereoisogram, the profile of the stereoisogram is characterized by the symbol ([7] [8]). The promolecules 5 and 7 (or 8) exhibit cis/trans-isomerism (Z /E-isomerism), so that they are stereoisomeric to each other. If the priority sequence is presumed to be A > p > p, 5 is characterized by the label E, while 7 (or 8) is characterized by the label Z . On the other hand, the achiral promolecules 9 and 10 construct a quadruplet of type V, where they do not exhibit cis/trans-isomerism (Z /E-isomerism). From a viewpoint of Fujita’s stereoisogram approach [27–29], the stereoisomerism at issue is represented by the following scheme:     [5 5] ([7] [8]) ([9] [10]) ,

(71)

where a pair of angle brackets represents a set of and  stereoisomers   a pair of braces represents a set of isoskeletomers. The symbol [5 5] ([7] [8]) surrounded by a   pair of angle brackets indicates that the quadruplet [5 5] is stereoisomeric to the quadruplet ([7] [8]). On the other hand, the symbol ([9] [10]) surrounded by a pair of angle brackets indicates that the quadruplet ([9] [10]) is self-stereoisomeric not to exhibit cis/trans-isomerism (Z /E-isomerism). The list of Fig. 2 is essentially equivalent to to the list of Fig. 26 of [28]. However, it should be noted that the numbering of four positions of the oxirane skeletons in Fig. 2 is based on the action of the RS-stereoisomeric group C2v σ I , while the numbering in Fig. 26 of [28] is based on the action of the corresponding isoskeletal (isoskeletomeric) group (cf. Fig. 7 of [28]). Let us next examine quadruplets of RS-stereoisomers with the composition A2 p2 (or A2 p2 ). The term 21 (A2 p2 + A2 p2 ) in f 1[III] (Eq. 42) indicates the presence of one quadruplet of type III, which is depicted in the first stereoisogram containing the representative promolecule 11 (Fig. 3). On the other hand, the term (A2 p2 + A2 p2 ) in f 1[II] (Eq. 41) is interpreted to be 2 × 21 (A2 p2 + A2 p2 ), which indicates the presence of two quadruplets of type II. They are depicted in the second and third stereoisograms containing the representative promolecules 13 and 15 (Fig. 3). The promolecules 11 and 13 exhibit cis/trans-isomerism (Z /E-isomerism), so that they are stereoisomeric to each other. If the priority sequence is presumed to be A > p > p, 11 is characterized by the label E, while 13 is characterized by the label Z . On the other hand, the chiral promolecule 15 constructs a quadruplet of type II so as not to exhibit cis/trans-isomerism (Z /E-isomerism). From a viewpoint of Fujita’s stereoisogram approach [27–29], the stereoisomerism at issue is represented by the following scheme:

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325

Type III

Type II S

p A

3

4

1

A p 2

A p

1

S

p

3

4

O

p A

A

3

4

p

1

2

O

A p

A

1

2

3

A p 4

O

O

11

12

13

14 (= 13)

(1)(2)(3)(4)

(1 3)(2 4)

(1)(2)(3)(4)

(1 3)(2 4)

A p

1

2

p

p A

3

C

2

4

3

4

A 1

A p

A p

2

3

1

2

A p

p

3

4

p

A

4

A 1

2

O

O

O

O

11

12

13

14 (= 13)

(1 3)(2 4)

(1)(2)(3)(4)

(1 3)(2 4)

(1)(2)(3)(4)

C

Type II S A A

3

4

p

1

2

p

A A

1

2

p

3

4

O

O

15

16 (= 15)

(1)(2)(3)(4)

(1 3)(2 4)

A A

1

2

p p

3

4

A A

3

4

p

p p

1

2

O

O

15

16 (= 15)

(1 3)(2 4)

(1)(2)(3)(4)

C

Fig. 3 Stereoisograms for characterizing three quadruplets of RS-stereoisomers with the composition A2 p2 (or A2 p2 ). The uppercase letter A represents an achiral proligand in isolation and the lowercase letters p and p¯ represent chiral proligands with opposite chirality senses. The priority sequence is presumed to be A>p>p

      [11 11] [12 12] [13 13] [15 15] .

(72)

The first pair of angle brackets in Eq. 72 indicates that the quadruplet of type III   represented by the symbol [11 11] [12 12] is stereoisomeric to the quadruplet of   type II represented by the symbol [13 13] . The second pair of angle brackets in Eq. 72   indicates that the quadruplet of type II represented by [15 15] is self-stereoisomeric, so as not to exhibit cis/trans-isomerism (Z /E-isomerism).

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Because the proligand p and p are enantiomeric in isolation, a quadruplet with the composition A2 pp may be stereoisomeric with a quadruplet with the composition A2 p2 (or A2 p2 ). It follows that Eqs. 71 and 72 are combined to give the following scheme:       [13 13] [5 5] ([7] [8]) [11 11] [12 12]    ([9] [10]) [15 15] ,

(73)

where two pairs of angle brackets in Eqs. 71 and 72 are merged into each pair of angle brackets in Eq. 73 according to the coalescence of A2 pp and A2 p2 (or A2 p2). It should be noted that the symbols of type-I quadruplets, e.g., [5 5] , are not   differentiated from the symbols of type-II quadruplets, e.g., [13 13] , in Eq. 73. This means that the symbols adopted here lay stress on enantiomeric relationships among three relationships contained in an RS-stereoisomeric group (enantiomeric, RS-diastereomeric, and holantimeric relationships). In general, an enantiomeric relationship is coalescent with an RS-diastereomeric relationship in a type-I stereoisogram, while an enantiomeric relationship is coalescent with a holantimeric relationship in a type-II stereoisogram. If the differentiation between a type-I symbol and a type-II symbol is necessary in detaileddiscussions, of the label I or II as a  the attachment  subscript would be useful, e.g., [5 5] I and [13 13] II . 4.2 Quadruplets with the compositions ABpp and ABp2 Let us examine quadruplets of RS-stereoisomers with the composition ABpp. The term 3ABpp in f 1[III] (Eq. 42) indicates the presence of three quadruplets of type III, the stereoisograms of which are depicted in Fig. 4. Because the first type-III stereoisogram consists of two  pairs of enantiomers, the quadruplet is represented by the symbol  [17 17] [18 18] . In a similar way, the second and third type-III stereoisograms are     charachterized by the symbols, [19 19] [20 20] and [21 21] [22 22] , respec    tively. Among them, the quadruplets [17 17] [18 18] and [19 19] [20 20] are   stereoisomeric to each other. On the other hand, the quadruplet [21 21] [22 22] exhibits a self-stereoisomeric nature. Hence, these stereoisomeric relationships are schematically represented as follows:       [19 19] [20 20] [21 21] [22 22] , [17 17] [18 18]

(74)

where a stereoisomeric relationship is represented by a pair of angle brackets. The promolecules 17 and 19 exhibit cis/trans-isomerism (Z /E-isomerism), which is a kind of stereoisomerism. In other words, they are diastereomeric to each other from the viewpoint of modern stereochemistry. If the priority sequence is presumed to be A > p > p, 17 is characterized by the label E, while 19 is characterized by the label Z . Although 17 and 18 are diastereomeric to each other from the viewpoint of modern stereochemistry, they are characterized by the same label E. It follows that the concept

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327

Type III

Type III S

p A

3

4

1

B p 2

A p

1

S

p

3

4

O

p A

B

3

4

p

1

2

O

A p

B

1

2

3

B p 4

O

O

17

18

19

20

(1)(2)(3)(4)

(1 3)(2 4)

(1)(2)(3)(4)

(1 3)(2 4)

A p

1

2

p

p B

3

C

2

4

3

4

A 1

B p

A p

2

3

1

2

B p

p

3

4

p

A

4

B 1

2

O

O

O

O

17

18

19

20

(1 3)(2 4)

(1)(2)(3)(4)

(1 3)(2 4)

(1)(2)(3)(4)

C

Type III S B A

3

4

p

1

2

p

A B

1

2

p

3

4

O

21

22

(1)(2)(3)(4)

(1 3)(2 4)

A B

1

p

O

2

p

B p

3

4

3

4

p

A

p 1

2

O

O

21

22

(1 3)(2 4)

(1)(2)(3)(4)

C

Fig. 4 Stereoisograms for characterizing three quadruplets of RS-stereoisomers with the composition ABpp. The uppercase letters A and B represent achiral proligands in isolation and the lowercase letters p and p¯ represent chiral proligands with opposite chirality senses. The priority sequence is presumed to be A>B>p>p

of diastereomeric relationships of modern stereochemistry mixes up the two cases concerning the assignability of Z/E-descriptors. The assignment of the same label to 17 and 18, however, is consistent with the fact they are contained in a single stereoisogram, as shown in Fig. 3, where they are recognized to be RS-diastereomeric to each other from the viewpoint of Fujita’s stereoisogram approach. Hence, they should be differentiated by using a pair of Ra /S a descriptors defined in [28]. A similar situation holds true for a pair of RS-diastereomers 19 and 20.

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Type III

Type III S

p A

3

4

1

B p 2

A p

1

2

p

3

4

O

S p A

B

3

4

p

1

2

O

A p

B

1

2

3

B p 4

O

O

23

24

25

26

(1)(2)(3)(4)

(1 3)(2 4)

(1)(2)(3)(4)

(1 3)(2 4)

A p

1

2

p

p B

3

4

3

4

A 1

B p

A p

2

3

1

2

B p

p

3

4

p

A

4

B 1

2

O

O

O

O

23

24

25

26

(1 3)(2 4)

(1)(2)(3)(4)

(1 3)(2 4)

(1)(2)(3)(4)

C

C

Type III S B A

3

4

p

1

2

p

A B

1

2

p

3

4

O

27

28

(1)(2)(3)(4)

(1 3)(2 4)

A B

1

p

O

2

p

B p

3

4

3

4

p

A

p 1

2

O

O

27

28

(1 3)(2 4)

(1)(2)(3)(4)

C

Fig. 5 Stereoisograms for characterizing three quadruplets of RS-stereoisomers with the composition ABp2 (or ABp2 . The uppercase letters A and B represent achiral proligands in isolation and the lowercase letters p and p¯ represent chiral proligands with opposite chirality senses. The priority sequence is presumed to be A>B>p>p

The promolecules 21 and 21 are enantiomeric to each other. They are differentiated by Z/E-descriptors, because they are so-called ‘geometric enantiomers’. Note that a permutation (1 3)(2)(4) applied to 21 generates 21, even though the permutation (1 3)(2)(4) is not a (roto)reflection operation. If the priority sequence A > B  p > p is presumed, the pair of RS-diastereomers 21 and 22 is characterized by the same label Z, while the pair of RS-diastereomers 21 and 22 is characterized by the same label E. As a result, the pair of enantiomers 21 and 21 are labelled by a pair of Z/E-descriptors.

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If we obey the CIP system, the pair of enantiomers 21 and 21 is not directly differentiated by R/S-stereodescriptors. Instead, the two carbons of each oxirane ring are labelled by means of R/S-stereodescriptors. According to Fujita’s stereoisogram approach, the pair of RS-diastereomers 21 and 22 is characterized by a newly defined pair of Ra /S a -descriptors [28]. The assigned label Ra or S a may be interpreted to be assigned to the pair of enantiomers 21 and 21. Let us next examine quadruplets of RS-stereoisomers with the composition ABp2 or ABp2 . The term 23 (ABp2 + ABp2 ) in f 1[III] (Eq. 42) indicates the presence of three quadruplets of type III, because the term 21 (ABp2 + ABp2 ) corresponds to one quadruplet of RS-stereoisomers. Their stereoisograms are depicted in Fig. 5. Because the first type-III stereoisogram consists of two  pairs of enantiomers, the quadruplet is  represented by the symbol [23 23] [24 24] . The second and third type-III stereoiso    grams are charachterized by the symbols, [25 25] [26 26] and [27 27] [28 28] ,     respectively. Among them, the quadruplets [23 23] [24 24] and [25 25] [26 26]   are stereoisomeric to each other. On the other hand, the quadruplet [27 27] [28 28] exhibits a self-stereoisomeric nature. Hence, these stereoisomeric relationships are schematically represented as follows:       [25 25] [26 26] [27 27] [28 28] , [23 23] [24 24]

(75)

where a stereoisomeric relationship is represented by a pair of angle brackets. By putting p = p because of their enantiomeric relationship, the term ABpp becomes equal to the term 21 (ABp2 +ABp2 ). Thereby, a quadruplet with the composition ABpp may be stereoisomeric with a quadruplet with the composition ABp2 (or ABp2 ). It follows that Eqs. 74 and 75 are combined to give the following scheme:   [17 17] [18 18]   [23 23] [24 24]   [21 21] [22 22]

  [19 19] [20 20]   [25 25] [26 26]   [27 27] [28 28] ,

(76)

where a stereoisomeric relationship is represented by a pair of angle brackets.

5 Conclusion Type-itemized enumeration of quadruplets of RS-stereoisomers is conducted under the action of the corresponding RS-stereoisomeric group, where cycle indices with chirality fittingness (CI–CFs) are calculated for its subgroups, i.e., a maximum normal subgroup, a point group, an RS-permutation group, an ligand-reflection group, and the RS-stereoisomeric group itself. After the CI–CF for the point group is modulated by considering the participation of type-V quadruplets, these CI–CFs are used to calculate CI–CFs of the respective types (type I to type V). A set of ligand-inventory functions is introduced into the resulting CI–CFs so as to give generating functions, in which the coefficient of each term indicates the number of inequivalent quadruplets itemized with respect to five types. This method using a modulated CI–CF is applied

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to the enumeration based on an oxirane skeleton, an allene skeleton, and a tetrahedral skeleton. The relationship of RS-stereoisomers vs. stereoisomers is discussed by referring to the enumeration results of oxirane derivatives.

References 1. G. Pólya, Compt. Rend. 201, 1167–1169 (1935) 2. G. Pólya, Acta Math. 68, 145–254 (1937) 3. G. Pólya, R.C. Read, Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds (Springer, New York, 1987) 4. D.H. Rouvray, Chem. Soc. Rev. 3, 355–372 (1974) 5. O.E. Polansky, MATCH Commun. Math. Comput. Chem. 1, 11–31 (1975) 6. K. Balasubramanian, Chem. Rev. 85, 599–618 (1985) 7. A.T. Balaban (ed.), Chemical Applications of Graph Theory (Academic Press, London, 1976) 8. N. Trinajsti´c, Chemical Graph Theory, 2nd edn. (CRC Press, Boca Raton, 1992) 9. D. Bonchev, D.H. Rouvray (eds.), Chemical Graph Theory. Introduction and Fundamentals (Gordon & Breach, Yverdon, 1991) 10. S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry (Springer, Berlin-Heidelberg, 1991) 11. S. Fujita, Diagrammatical Approach to Molecular Symmetry and Enumeration of Stereoisomers (University of Kragujevac, Faculty of Science, Kragujevac, 2007) 12. S. Fujita, Combinatorial Enumeration of Graphs, Three-Dimensional Structures, and Chemical Compounds (University of Kragujevac, Faculty of Science, Kragujevac, 2013) 13. S. Fujita, J. Org. Chem. 69, 3158–3165 (2004) 14. S. Fujita, J. Math. Chem. 35, 265–287 (2004) 15. S. Fujita, Tetrahedron 60, 11629–11638 (2004) 16. S. Fujita, Tetrahedron: Asymmetry 25, 1153–1168 (2014) 17. S. Fujita, Tetrahedron: Asymmetry 25, 1169–1189 (2014) 18. S. Fujita, Tetrahedron: Asymmetry 25, 1190–1204 (2014) 19. S. Fujita, Tetrahedron: Asymmetry 25, 1612–1623 (2014) 20. S. Fujita, MATCH Commun. Math. Comput. Chem. 54, 39–52 (2005) 21. S. Fujita, MATCH Commun. Math. Comput. Chem. 58, 611–634 (2007) 22. S. Fujita, MATCH Commun. Math. Comput. Chem. 61, 71–115 (2009) 23. S. Fujita, J. Math. Chem. 52, 508–542 (2014) 24. S. Fujita, J. Math. Chem. 52, 543–574 (2014) 25. S. Fujita, J. Math. Chem. 52, 1717–1750 (2014) 26. S. Fujita, J. Math. Chem. 52, 1751–1793 (2014) 27. S. Fujita, J. Math. Chem. 53, 260–304 (2015) 28. S. Fujita, J. Math. Chem. 53, 305–352 (2015) 29. S. Fujita, J. Math. Chem. 53, 353–373 (2015) 30. S. Fujita, Theor. Chem. Acc. 113, 73–79 (2005)

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