OFFSET
1,1
COMMENTS
The maximum k for which T(n,k) is nonzero, denoted as Cl(n), is usually referred to as the Clar number of O(3,3,n); one has: Cl(1)=3, Cl(2)=5, Cl(3)=7, Cl(4)=8, and Cl(n)=9 for n>4.
T(n,k) denotes the number of Clar covers of order k in the hexagonal graphene flake O(3,3,n).
The Kekulé number of O(3,3,n) is given by T(n, 0).
ZZ polynomials of hexagonal graphene flakes O(3,3,n) with n=1..10 are listed in Eq.(36) of Chou, Li and Witek.
ZZ polynomials of hexagonal graphene flakes O(3,3,n) with any n can be obtained from Eq.(13) of Witek, Langner, Mos and Chou.
ZZ polynomials of hexagonal graphene flakes O(3,3,n) can be also computed using ZZDecomposer (see links below), a graphical program to compute ZZ polynomials of general benzenoids.
LINKS
C.-P. Chou, ZZDecomposer<\a>.
C.-P. Chou, Y. Li and H. A. Witek, Zhang-Zhang Polynomials of Various Classes of Benzenoid Systems, MATCH Commun. Math. Comput. Chem. 68 (2012), 31-64.
C.-P. Chou and H. A. Witek, ZZDecomposer: A Graphical Toolkit for Analyzing the Zhang-Zhang Polynomials of Benzenoid Structures, MATCH Commun. Math. Comput. Chem. 71 (2014), 741-764.
C.-P. Chou and H. A. Witek, Determination of Zhang-Zhang Polynomials for various Classes of Benzenoid Systems: Non-Heuristic Approach, MATCH Commun. Math. Comput. Chem. 72 (2014), 75-104.
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 105 for a graphical definition of O(3,3,n)).
H. A. Witek, J. Langner, G. Mos, and C.-P. Chou, Zhang-Zhang Polynomials of Regular 5-tier Benzeonid Strips, MATCH Commun. Math. Comput. Chem. 78 (2017), 487-504.
H. Zhang and F. Zhang, The Clar covering polynomial of hexagonal systems I, Discrete Appl. Math. 69 (1996), 147-167 (ZZ polynomial is defined by Eq.(2.1) and working formula is given by Eq.(2.2)).
FORMULA
T(n,k) = Sum_{l=0..9} (binomial(k+l,k)*(binomial(9,k+l)*binomial(n,k+l)+(10*binomial(7,k+l-2)-binomial(6,k+l-2))*binomial(n+1,k+l)+(20*binomial(5,k+l-4)+binomial(3,k+l-3)-binomial(3,k+l-5))*binomial(n+2,k+l)+(10*binomial(3,k+l-6)+binomial(2,k+l-5)+binomial(3,k+l-5))*binomial(n+3,k+l)+binomial(2,k+l-7)*binomial(n+4,k+l)).
This formula can be obtained by a double sum rotation from Eq.(13) of Witek, Langner, Mos and Chou. Eq.(13) was first discovered heuristically as Eq.(37) of Chou, Li and Witek; a formal proof was given in Eqs.(66-71) on pp. 100-102 of Chou and Witek.
EXAMPLE
Triangle begins:
k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9
n=1: 20 30 12 1
n=2: 175 450 425 180 33 2
n=3: 980 3308 4458 3065 1140 225 22 1
n=4: 4116 16468 27293 24262 12521 3796 653 58 2
n=5: 14112 63522 120848 126518 79506 30681 7132 933 58 1
n=6: 41580 204180 429030 503664 361690 163380 45885 7588 648 20
...
Row n=4 corresponds to the polynomial 4116 + 16468*x + 27293*x^2 + 24262*x^3 + 12521*x^4 + 3796*x^5 + 653*x^6 + 58*x^7 + 2*x^8.
MAPLE
(n, k)->add(binomial(i+k, k)*(binomial(9, i+k)*binomial(n, i+k)+(10*binomial(7, i+k-2)-binomial(6, i+k-2))*binomial(n+1, i+k)+(20*binomial(5, i+k-4)+binomial(3, i+k-3)-binomial(3, i+k-5))*binomial(n+2, i+k)+(10*binomial(3, i+k-6)+binomial(2, i+k-5)+binomial(3, i+k-5))*binomial(n+3, i+k)+binomial(2, i+k-7)*binomial(n+4, i+k)), i = 0..9)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Henryk A. Witek, Oct 14 2020
STATUS
approved