J. Chem. Inf. Comput. Sci. 1998, 38, 1145-1150
1145
Algorithm for the Direct Enumeration of Chiral and Achiral Skeletons of a Homosubstituted Derivative of a Monocyclic Cycloalkane with a Large and Factorizable Ring Size n Robert M. Nemba*,† and Alexandru T. Balaban‡ Faculty of Science, Laboratory of Physical Chemistry, Section of Molecular Topology, University of Yaounde I, P.O. Box 812 YDE, Cameroon, and Polytechnic University, Department of Organic Chemistry, Splaiul Independentei 313, 77207 Bucharest, Romania Received April 22, 1998
An algorithm with combinatorial formulas is derived for the direct enumeration of chiral and achiral skeletons of stereo and position isomers of homopolysubstituted monocyclic cycloalkanes (CnH2n-mXm) with any large ring size n. Some developments are given when n is factorizable into the forms n ) Rj or n ) 2pRj where j, p, and R are integer numbers while R is odd prime. Some illustrations are given for n ) 27 and 36 with 0 e m e 9. 1. INTRODUCTION
In some previous papers,1-3 it was shown that numbers of stereo and position isomers of cycloalkanes where one or more hydrogen atoms have been substituted can be calculated by means of Po´lya’s counting theorem.4-9 Such calculations have been limited to systems with small ring size because of the unwieldiness of the expressions involved and the difficulty of expanding those expressions. Our purpose is to circumvent this limitation and present an algorithm for the direct determination of the numbers of chiral and achiral skeletons of stereo and position isomers of homopolysubstituted cycloalkanes (CnH2n-mXm) with a higher ring size n factorizable into the two forms previously indicated. We assume ring flip to be fast enough to equilibrate conformers. 2. MATHEMATICAL FORMULATION
Let us consider the system CnH2n-mXm ) (n,2n,m). The three non-negative integers n, 2n, and m denote, respectively, the ring size, the number of available substitution sites, and the degree of substitution. The tridimensional graph or stereograph G of its parent monocyclic cycloalkane CnH2n shown in Figure 1 belongs to the symmetry point group Dnh which includes 4n symmetry operations. The n labeled vertices symbolize the carbon atoms, and the unlabeled vertices indicate the locations of the 2n hydrogen atoms which constitute the substitution sites. These latter collected in two subsets E ) {1,2,3,...,n} and E′ ) {1′,2′, 3′,...,n′} are permutable under the action of the 4n symmetry operations of Dnh. The elements of E and E′ joined as indicated in Figure 2 constitute the vertices of an orthogonal prism which contains two regular n-gons as parallel faces. Some considerations and rules can be made from the spatial distribution of the substitution sites and the symmetry properties of the point group Dnh acting on the molecular † ‡
University of Yaounde I. Polytechnic, University of Bucharest.
Figure 1. Molecular stereograph G of a monocyclic cycloalkane.
Figure 2. Orthogonal prism symbolizing the locations of the 2n hydrogen atoms (substitution sites) at the vertices of the two parallel n-gons.
stereograph G. If n is factorizable, each divisor d of n gives rise to a d-fold rotation axis which generates proper rotations Crd (r e d - 1) and improper rotations Sr′d (r′ e d - 1 if d odd or r′ e 2d - 1 if d even). According to the parity of n one may notice the following. Rule 1: For d odd or eVen, each proper rotation Crd of Dnh permutes some elements of the subset E and produces n/ d-cycles of permutations. This operation is simultaneously d undertaken on the subset E′ and giVes the same results. Therefore each proper rotation Crd permuting in parallel some elements of the subsets E and E′ produces 2n/d d-cycles of permutations noted f2n/d d . To exemplify this situation we consider the stereograph G of cyclohexane which belongs to the symmetry point group D6h. If we set up d ) 3 and r ) 1, the proper rotation C3 produces on E ) {1,2,3,4,5,6} two three-cycles of permutations noted (135) (246) and on E′ ) {1′,2′,3′,4′,5′,6′} the other two three-cycles (1′3′5′) (2′4′6′). This symmetry operation gives rise to the term (135) (246) (1′3′5′) (2′4′6′) also noted as f43 which is a product of four disjoint permutation cycles of length 3.
10.1021/ci980079f CCC: $15.00 © 1998 American Chemical Society Published on Web 11/06/1998
1146 J. Chem. Inf. Comput. Sci., Vol. 38, No. 6, 1998
NEMBA
Table 1. Permutations Generated by Binary Axes and Reflections of Dnh n (odd) σh f f
σh f f
nσv f nf 21 f n-1 2
n 2;
[
n n σ f fn 2 d 2 2
1 4n
[
∑
n 2 n-1 φ(d)(f 2n/d d ) + (n + 1)f 2 + n(f 1 f 2 )]
(d*2)|2n
ZE(n-) )
(1) 1 2n
[
∑
(d*2)|n
n φ′(d)(f 2n/d d ) + nf 2]
∑
ZE(n+) )
Rule 2: For d odd, each improper rotation Sr′d of Dnh merges together into a single 2d-cycle, one d-cycle of the elements of E and another d-cycle of the elements of E′. Such permutations of length 2d applied to the sequence of 2n elements generate a product of n/d permutations cycles of length 2d which produce the term f n/d 2d . For instance the action of the improper rotation S3 on the subsets E and E′ with the same cardinality n ) 6 as in the preceding example gives the cycle structure (13′51′35′)(24′62′46′) which is the product of two disjoint permutation cycles of length 6 noted f 26. Rule 3: For d eVen, each improper rotation Sr′d of Dnh permuting the elements of the subsets E and E′ of G generates a product of 2n/d d-cycles of disjoint permutations noted f 2n/d d . For n ) 6 and d ) 2, the preceding rule applied to S2 ≡ i generates the term f 62 which corresponds to the product of six disjoint transpositions (or two-cycle permutations) explicitly noted as follows: (14′) (25′) (36′) (1′4) (2′5) (3′6). The permutations generated by the other symmetry elements of Dnh including the horizontal binary axes C′2, C′′2 and the reflections σh, σv, and σd are reported in Table 1 according to the parity of n. The application of Po´lya’s theorem on the molecular stereograph G which belongs to the symmetry point group Dnh requires all the 4n symmetry operations to be expressed according to the parity of n, as the cycle indices ZT(n-) and ZT(n+) given in eqs 1 and 3 usually used for the topological enumeration which gives the figure inventory of geometrical isomers. The cycle indices ZE(n-) and ZE(n+) given in eqs 2 and 4 are obtained after eliminating with respect to the principle of Parks and Hendrickson10 all the terms generated by reflections and improper rotations in eqs 1 and 3, respectively. Equations 2 and 4 apply to the remaining 2n symmetry operations and hold for enantiomeric enumeration in order to count all stereoisomers (including chiral pairs and achiral forms) of substituted derivatives of monocyclic cycloalkanes. Similarly, the cycle indices obtained from the 2n symmetry operations of Dnh including only reflections and improper rotations lead to the enumeration of achiral skeletons. It is to be noticed throughout this paper that the notation (n-), (n+), and (n() is assigned to the integer numbers n odd, n even, and n odd or even, respectively. If n odd, then
ZT(n-) )
]
3 n 4 n-2 n φ(d)(f 2n/d d ) + (n + 2)f 2 + (f 1 f 2 ) (3) 4n (d*2)|n 2 2
n n σ f f 4f n-2 2 v 2 1 2 n n n n C′ f f n; C′′ f f n 2 2 2 2 2 2 2 2
nC′2 f nf n2
BALABAN
and if n even, then
ZT(n+) ) 1
n (even)
n 2
AND
(2)
1
[
2n
∑
n φ′(d)(f 2n/d d ) + (n + 1)f 2]
(4)
(d*2)|n
where the summation is over all the integers d * 2 that divide 2n in eq 1 or n in eqs 2-4. The number φ(d) of symmetry operations (proper or improper rotations) that induce permutations of length d is obtained from eqs 5-7:
φ(d) ) φ′(d)pr if d odd
(5)
φ(d) ) φ′(d)pr + φ′(d)ir + φ′(Pi)ir if d(even) ) 2Pi and Pi (odd prime integer) (6) φ(d) ) φ′(d)pr + φ′(d)ir if d(even) * 2Pi
(7)
while φ′(d) the number of proper rotations that generate permutations of length d is given by (8)
φ′(d) ) φ′(d)pr
(8)
The terms φ′(d)pr and φ′(d)ir are the Euler totient function11 for the integer d (odd or even), and the indices (pr) and (ir) refer to proper and improper rotations induced by the d-fold rotation axis. The substitution in eqs 1-4 of the terms f ba by the corresponding figure counting series (1 + xa)b and the expansion of the resulting algebraic expression then lead to the polynomials 9 and 10: 2n
PT(n(,m() )
∑ AT(n(,m()(xm)
(9)
m)0 2n
PE(n(,m() )
∑ AE(n(,m()(xm)
(10)
m)0
the coefficients of which satisfy the following relations
AT(n(,m() ) Ac(n(,m() + Aac(n(,m()
(11)
AE(n(,m() ) 2Ac(n(,m() + Aac(n(,m()
(12)
where Ac(n(,m() and Aac(n(,m() denote respectively the numbers of chiral and achiral skeletons of stereo and position isomers of a homopolysubstituted monocyclic cycloalkane CnH2n-mXm. It should be observed that the coefficient of the term xm and that of the term x2n-m in the generating functions 9 and 10 obey the following conditions:
AT(n(,m() ) AT(n(,2n( - m() and AE(n(,m() ) AE(n(,2n - m()
ENUMERATION
OF
CHIRAL/ACHIRAL SKELETONS
OF A
CYCLOALKANE
These equivalences are due to the complementarity of the degrees of substitution m and 2n - m. These conditions induce also the equivalences:
Ac(n(,m() ) Ac(n(,2n( - m() and Aac(n(,m() ) Aac(n(,2n( - m() The derivation of the numerical sequences Ac(n(,m() and Aac(n(,m() through the determination of AT(n(,m() and AE(n(,m() by the classical Po´lya method is laborious for large ring size n and higher values of m. A direct and convenient way to circumvent this difficulty and compute these coefficients for singular values (n, m) is an everlasting problem for chemists. The emphasis in this work has been to set up a general pattern inventory of stereo and position isomers for a single system CnH2n-mXm. A long time ago, related results for monocyclic aromatics had been published in Roumanian language. Obviously, no chirality was present.12 The algorithm begins as follows: consider the system CnH2n-mXm and its triade of numbers (n, 2n, m). Let us define for n odd, and m odd or even, the sets F2n- ) {1,2,...,d,...,n,2n} and Fm( ) {1,...,d′,...,m,} containing the divisors of 2n and m. We extract from these sets the subsets F2n- ∩ Fm-, F2n- ∩ Fm+ which contain the common divisors {d} of the integer numbers (2n, m). Then the enantiomerpairs and achiral skeletons inventories are given by (13)(16). If (d * 2) ∈ F2n- ∩ Fm- and µ′ ) (m - 1)/2, then
Ac(n-,m-) ) 1
[
( ) ( )] (13)
2n/d n-1 (2φ′(d) - φ(d)) - 2n ∑ m/d µ′ 4n (d*2)
Aac(n-,m-) ) 1
[
2n
(2n/d ) + 2n(nµ′- 1 )] (14)
∑ (φ(d) - φ′(d)) m/d
(d*2)
If (d * 2) ∈ F2n- ∩ Fm+, then
Ac(n-,m+) ) 1 4n
[
∑ (2φ′(d) - φ(d))
(d*2)
( ) ( )] 2n/d n m/d m/2
( )
If (d * 2) ∈ Fn+ ∩ Fm- and µ′ ) (m - 1)/2, then
Ac(n+,m-) ) 1 4n
[
(2n/d )- 2n(nµ′- 1 )] (17)
∑ (2φ′(d) - φ(d)) m/d
(d*2)
Aac(n+,m-) ) 1 2n
[∑
(φ(d) - φ′(d))
(d*2)
n-1 + 2n( (2n/d )] (18) m/d ) µ′
If (d * 2) ∈ Fn+ ∩ Fm+, then
Ac(n+,m+) )
1 4n
[∑
+ (2n/d m/d ) 1 m n - n(m + 1) + 1)( (19) ( m/2 )] n-1 2 (2φ′(d) - φ(d))
(d*2)
[∑
2
+ (2n/d m/d ) m 1 n - 2)( n + n(m + 1) (20) ( m/2 )] n-1 2
Aac(n+,m+) )
1
2n
(φ(d) - φ′(d))
(d*2)
2
2
In eqs 13-20 φ(d) and φ′(d) have the same definitions as previously given and the notation of type (r/s) refers to binomial coefficients. We illustrate the preceding developments by the following applications. 3. APPLICATIONS
Case 1. Enantiomer-pairs and achiral skeletons of CnH2n-mXm where n (odd) ) Rj (R and j are integer numbers and R is prime). Let F2Rj ) {1,2,...,Rk,...Rj,2R,...2Rk,...,2Rj} be the set of divisors of 2Rj and 1 e k e j. The coefficient φ′(R)ir ) φ′(R)pr ) (Rk - Rk-1). According to eqs 1 and 2 the cycle indices ZT(Rj) and ZE(Rj) are given by (21) and (22).
ZT(Rj) )
1
n 2 n-1 [f 2n + 1 + (n + 1)f 2 + nf 1 f 2
4n
j
∑ (Rk - Rk-1)(f 2R R
j-k
(15)
Aac(n-,m+) ) 1 2n/d n (φ(d) - φ′(d)) + (n + 1) ∑ m/d m/2 2n (d*2)
[
J. Chem. Inf. Comput. Sci., Vol. 38, No. 6, 1998 1147
( )] (16)
But if n is even, and m odd or even, we define this time the sets Fn+ ) {1,2,...,d,...,n} and Fm( ) {1,...,d′,...,m} containing the divisors of the integer numbers n and m and extract from these sets and according to the parities of both n and m one of the following subsets Fn+ ∩ Fm- or Fn+ ∩ Fm+ which contain the common divisors {d} of the integer numbers (n, m). Then the enantiomer-pairs and achiral skeletons inventories are given by eqs 17-20, respectively.
k
k)1
ZE(Rj) )
1
R + f 2R k )] (21) j-k
j
2n
n k k-1 [f 2n )(f 2R 1 + nf 2 + ∑ (R - R Rk )] (22) j-k
k)1
The generating functions for topological and enantiomeric enumerations can be obtained by replacing f ba by (1 + xa)b and the expansion of the algebraic expressions depends upon the magnitude of the integers a and b. To avoid this difficult task and compute directly the numbers Ac(Rj,m() and Aac(Rj,m() we shall deduce some recurrence formulas and generalizations depending on the factorization and parity of m. Let m- ) Rk and F2Rj ∩ FRk ) 1,R,...,Ri,...,Rk. Thus eqs 13 and 14 become
1148 J. Chem. Inf. Comput. Sci., Vol. 38, No. 6, 1998
Ac(Rj,Rk) ) 1
[(
)
( ) (
i)1
n-1 - 2n (m - 1)/2
j-i
2n 2R + ∑(Ri - Ri-1) k-i R 4n m k Aac(Rj,Rk) )
[(
n-1 1 2n (m - 1)/2 2n
)]
)]
NEMBA
Ac(27,4) ) (23)
Aac(27,4) ) (24)
Example 1. Enantiomer-pairs and achiral skeletons of C27H45X9. Let n ) 33 and m ) 32 and F54 ∩ F9 ) {1,3,32}, then
( ) [( ) ( ) ( )]
2‚33-1 1 54 + (31 - 30) 2-1 + 108 9 3
Ac(27,9) )
(32 - 31)
(27 - 1) 2‚33-2 - 2‚27 2-2 (9 -1)/2 3
Aac(27,9) )
[ (
(27 - 1) 1 54‚ (9 - 1)/2 54
)]
[( ) ( )] [ ( )]
27 1 54 2 108 4 27 1 (28) 2 54
AND
BALABAN
) 2925
) 182
Case 2. Enumeration of enantiomer-pairs and achiral skeletons when n ) 2pRj. The divisors d of n ) 2pRj are deduced from the relation d ) 2p-qRk under the conditions 0 e q e p, 0 e k e j. The couples of integer values (q,k) needed for the computation of d and the determination of the cycle indices ZT(2pRj) and ZE(2pRj) are developed as shown in the following grid:
) 49 232 691
) 14 950
If m+ ) 2Rk, then
Ac(R ,2R ) ) j
[(
k
)
[( ) ( )]
k
2n 2Rj-i Rj-i + ∑(Ri - Ri-1) 2Rk-i Rk-i 4n m i)1 1
Aac(R ,2R ) ) j
k
[∑
1 2n
k
(R - R i
-
(nm/2 )]
(25)
Therefore for the even integer n ) 2pRj, eqs 3 and 4 become, respectively
ZT(2pRj) )
i)1
( )]
( )
4n
f 2n 1 +
p-1 p
[(
)
( ) ( ) ( )]
2‚33-1 1 54 + (31 - 30) 108 6 2‚31-1 (31 - 30)
Aac(27,6) )
[
33-1 27 1-1 3 3
( )
) 239 116
( )]
q+1)(Rj) p-q)
+
q)0 p-1 j
∑ ∑2(p-q)(Rk - Rk-1)(f (2(2
q+1)(Rj-k) p-q)(Rk)
]
) (29)
q)0k)1
ZE(2pRj) )
1
n [f 2n 1 + nf 2 +
2n
j
∑(Rk - Rk-1)(f (R(2 k)
p+1)(Rj-k)
) 1517
k)1
p-1
)+
∑ 2(p-q-1)(f (2(2
q+1)(Rj) p-q)
)+
q)0
p-1 j
If there is no common factor between 2Rj and m except 1 or 2, i.e., F2Rj ∩ Fm ) {1,2} or F2Rj ∩ Fm ) {1}, the eqs 25 and 26 become
[( ) ( )] [ ( )]
∑ 2(p-q)f (2(2
and
3 -1 27 1 1 (3 - 30) 1 + (27 + 1) 3 54 3 -1 3
j-k
(2 )(R ) )+ f (2R k)
Example 2. Enantiomer-pairs and achiral skeletons of C27H48X6. Let n ) 33, m ) 2R ) 6, R ) 3, j ) 3, k ) 1 and F54 ∩ F6 ) {1,2,3,6}
Ac(27,6) )
3 n + 1 f n2 2
j n (2p+1)(Rj-k) + f 41 f n-2 + (Rk - Rk-1)(f (R + k) ∑ 2 2 k)1
Rj-i n ) k-i + (n + 1) m/2 R (26)
i-1
[ ( )
1
Ac(Rj,m) )
n 1 2n m/2 4n m
(27)
Aac(Rj,m) )
n 1 (n + 1) m/2 2n
(28)
Example 3. Enantiomer-pairs and achiral skeletons of C27H50X4. Let n ) 33, m ) 22, F54 ∩ F4 ) {1,2}.
∑ ∑2(p-q-1)(Rk - Rk-1)(f (2(2
q+1)(Rj-k) p-q)(Rk)
)] (30)
q)0k)1
To obtain the global solution of our enumeration problem, the classical method requires transforming of these cycle indices into Po´lya counting polynomials of order 2n. To circumvent such a procedure which cannot be done by hand and needs computer assistance for higher values of the ring size n and the degree of substitution m we shall use eqs 1720. Let us illustrate this case by enumerating the stereoisomers of a system CnH2n-mXm where n+ ) 2pRj and m+ ) 2p-qRk.
ENUMERATION
CHIRAL/ACHIRAL SKELETONS
OF
OF A
CYCLOALKANE
J. Chem. Inf. Comput. Sci., Vol. 38, No. 6, 1998 1149
Table 2. Numbers of Enantiomer-Pairs and Achiral Skeletons of Homopolysubstituted Cycloalkanes (CnH2n-mXm) Where the Degree of Substitution 0 e m e 9 and the Ring Size n ) 6, 7, 8, 9 , 12 , 27, and 36 n 6
7
8
9
12
27
m Ac(m,n) Aac(m,n) Ac(m,n) Aac(m,n) Ac(m,n) Aac(m,n) Ac(m,n) Aac(m,n) Ac(m,n) Aac(m,n) 0 1 2 3 4 5 6 7 8 9
0 0 2 7 18 28 35
1 1 5 5 14 10 20
0 0 3 10 35 64 106 113
1 1 4 6 12 15 20 20
0 0 3 14 53 126 241 340 390
1 1 6 7 24 21 50 35 65
0 0 4 19 84 226 514 856 1212 1317
The subset of common divisors is Fn+ ∩ Fm+ ) 1,...{2p-q},...{Rk}...,{2p-qRk} and the transformation of eqs 17 and 18 gives the following recurrence formulas 31 and 32 for the determination of enantiomeric pairs and achiral skeletons, respectively.
[(
)
k
( )
2n 2n/Ri + ∑(Ri - Ri-1) + m/Ri 4n m i)1 1 m2 n - n(m + 1) + 1 m/2 n-1 2 1
Ac(2pRj,2p-qRk) )
(
)(
k
1
Aac(2pRj,2p-qRk) )
2n
p-2 k
[∑
)
( )] ( ) ( ) )( )]
∑(Ri - Ri-1) m/2Ri i)1
p-2
2n/2R
2p-q-1
q)0
p-q
2n/2 m/2p-q
2n/2p-qRi
∑ ∑2p-q-1(Ri - Ri-1) m/2p-qRi
q)0i)1
1 n-1
(
n2 + n(m + 1) -
m
2
2
-2
[( )
ac
2-0-1
2
1
0
Aac(m,n)
Ac(m,n)
Aac(m,n)
0 0 13 217 2925 29120 239116 1638520 9633780 49232691
1 1 14 26 182 325 1517 2600 9100 14950
0 0 17 397 7123 96866 1084666 10226656 83114768 591036384
1 1 20 35 367 595 4330 6545 37077 52360
If m+ ) 2p-q, then
Ac(2pRj,2p-q) )
[( )
(
)( )]
(33)
)( )]
(34)
n 1 m2 1 2n - n(m + 1) + 1 + m/2 4n m n-1 2 Aac(2pRj,2p-q) )
1
[
p-2
(
2n/2p-q
2p-q-1 ∑ m/2p-q 2n
1 n-1
(
q)0
n + n(m + 1) 2
)
m2 2
+ -2
n m/2
+
n m/2
Ac(36,4) ) (32)
( )] 72/6 1 + 2(3 - 3 )( + A (36,12) ) [2 (72/4 12/4 ) 12/6 ) 72 72/12 (3 - 3 )( + 2 12/12 ) 36 12 1 36 + 36(12 + 1) - 2)( )] ) 1 306 284 6 36 - 1( 2 1
Ac(m,n)
Example 5. Enantiomer-pairs and achiral skeletons of C36H68X4. Let n ) 36 ) 2232, m ) 4 ) 22, p ) 2, q ) 0, F36 ∩ F4 ) (1,2,4), φ(1) ) 1, φ′(1) ) 1; φ(4) ) 4, φ′(4) ) 2.
+
( ) )( )
2-0-1
1 1 8 11 49 55 174 165 410 330
(31)
72/3 1 72 + (31 - 31-1) + 12/3 144 12 36 122 1 - 36(12 + 1) + 1 12/2 36 - 1 2 72/6 ) 106 689 321 808 (31 - 31-1) 12/6
(
0 0 5 37 215 858 2778 7128 15252 27077
i
Example 4. Enantiomer-pairs and achiral skeletons of homopolysubstituted cyclohexatriacontane C36H72-mXm. For C36H72-mXm, let n ) 36 ) 2232, m ) 12 ) 223, p ) q ) 2, k ) 1, F36 ∩ F12 ) (1,2,3,4,6,12), φ(1) ) 1, φ′(1) ) 1; φ(3) ) 2, φ′(3) ) 2; φ(4) ) 4, φ′(4) ) 2; φ(6) ) 6 , φ′(6) ) 2; φ(12) ) 8, φ′(12) ) 4.
Ac(36,12) )
1 1 5 8 20 24 47 56 70 70
36
[( )
( [ ( ) (
)( )]
36 1 1 72 42 + - 36(4 + 1) + 1 4 2 144 36 - 1 2 Aac(36,4) )
1 2-0-1 72/4 2 + 4/4 72 42 36 1 362 + 36(4 + 1) 36 - 1 2 2
) 7123
)( )]
) 367
If the common factors between 2pRj and m+ are 1 or 2 i.e., F2pRj ∩ Fm+ ) {1} or {1,2} then
Ac(2pRj,m+) )
[( )
(
)( )]
(35)
)( )]
(36)
n 1 m2 1 2n - n(m + 1) + 1 + m/2 4n m n-1 2 Aac(2pRj,m+) )
[ (
n m2 1 1 -2 n2 + n(m + 1) m/2 2n n - 1 2
0
2
Example 6. Enantiomer-pairs and achiral skeletons of C36H62X10. Let n ) 36 ) 2232, m ) 10, F36 ∩ F10 ) (1,2). According to eqs 35 and 36
1150 J. Chem. Inf. Comput. Sci., Vol. 38, No. 6, 1998
Ac(36,10) )
[( )
(
2
)( )]
36 1 1 72 10 + ) - 36(10 + 1) + 1 10/2 144 10 36 - 1 2 3 723 668 168 Aac(36,10) )
[
)( )]
(
36 102 1 1 362 + 36(10 + 1) ) -2 10/2 72 36 - 1 2 245 344
NEMBA
REFERENCES AND NOTES (1) (2) (3) (4) (5)
(7)
4. CONCLUSION
As it is assumed to be known so far from the chemical literature13 the series of monocyclic cycloalkanes CnH2n have ring size 3 e n e 288. This wide range of ring sizes leads to distinct applications of Po´lya’s theorem. Po´lya’s classical procedure requires first to obtain the cycle index according to the parity and divisibility character of the ring size n and the transformation of the cycle index into a polynomial function of order 2n the coefficients of which are solutions of our enumeration problem. The emphasis in this paper has been to set up an algorithm that circumvents these difficulties and multiple calculations and allows a direct determination of enantiomer-pairs and achiral skeletons of
BALABAN
stereo and position isomers of any homopolysubstituted monocyclic cycloalkane CnH2n-mXm.
(6)
The numbers obtained from this counting procedure are given for illustration in Table 2 for n ) 27 and 36 and the degrees of substitution 0 e m e 9.
AND
(8) (9) (10) (11) (12) (13)
Balaban, A. T. Croat. Chem. Acta 1978, 51, 35-42. Nemba, R. M.; Ngouhouo, F. Tetrahedron 1994, 50, 6663-6670. Nemba, R. M.; Ngouhouo, F. New J. Chem. 1994, 18, 1175-1182. Po´lya, G. Acta Math. 1937, 69, 145-254. Po´lya, G.; Read, R. C. Combinatorial Enumeration of Graphs, Groups and Chemical Compounds; Springer Verlag: New York, 1987, pp 58-74. Po´lya, G; Tarjan, R. E.; Woods, D. R. Notes in introductory combinatorics; Birhauser: Boston, 1983; pp 55-85. (a) Harary, F.; Palmer, E.; Robinson, R. W.; Read, R. C. Polya’s contribution to chemical theory in Chemical Applications of Graph Theory; Balaban, A. T., Ed.; Academic Press: London, 1976; pp 1143. (b) Balaban, A. T.; Kennedy, J. W.; Quintas, L. V. The number of alkanes having n carbon and a longest chain of length d. An application of a theorem of Po´lya. J. Chem. Educ. 1988, 65, 304313. (c) Robinson, R. W., Harary, F.; Balaban, A. T. The numbers of chiral and achiral alkanes and monosubstituted alkanes. Tetrahedron 1976, 32, 355-361. Tucker, A. Po´lya’s Enumeration formula by example. Math. Magazine 1974, 47, 248-256. Rouvray, D. H. Isomers enumeration method. Chem. Soc. ReV. 1974, 3, 355-372. Parks, C. A.; Hendrickson, J. B. J. Chem. Inf. Comput. Sci. 1991, 31, 334-339. Kaufman, A. Introduction a` la combinatorique; Ed., Dunod, Paris, 1980; pp 84-86. Balaban, A. T. An attempt for the systematization of acyclic aromatic compounds. Studii Cercetari Chim., Acad. R. P. Romane 1959, 7, 257-295. Metanomski, W. V. Chem. Intl. 1987, 9, 125.
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