OFFSET
1,2
COMMENTS
The maximum k for which T(n,k) is nonzero, denoted by 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 perfect matchings (i.e., Kekulé structures) with k proper sextets for the hexagonal graphene flake O(3,3,n).
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 computed using ZZDecomposer (see link below), a graphical program to compute ZZ polynomials of benzenoids, or using ZZCalculator (see link below).
LINKS
C.-P. Chou, ZZDecomposer executable.
C.-P. Chou, ZZCalculator source code.
C.-P. Chou and H. A. Witek, An Algorithm and FORTRAN Program for Automatic Computation of the Zhang-Zhang Polynomial of Benzenoids, MATCH Commun. Math. Comput. Chem. 68 (2012), 3-30.
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 III, Discrete Math. 212 (2000), 261-269 (proper sextet is defined in Fig.1 and ZZ polynomial in the basis of (1+x)^k monomials is defined by Theorem 2).
FORMULA
T(n,k) = binomial(9,k)*binomial(n,k) + (10*binomial(7,k-2) - binomial(6,k-2))*binomial(n+1,k) + (20*binomial(5,k-4) + binomial(3,k-3) - binomial(3,k-5))*binomial(n+2,k) + (10*binomial(3,k-6) + binomial(2,k-5) + binomial(3,k-5))*binomial(n+3,k) + binomial(2,k-7)*binomial(n+4,k).
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: 1 9 9 1
n=2: 1 18 63 68 23 2
n=3: 1 27 162 350 310 114 15 1
n=4: 1 36 306 996 1446 984 303 42 2
n=5: 1 45 495 2155 4360 4360 2141 505 49 1
n=6: 1 54 729 3976 10325 13650 9233 3124 468 20
n=7: 1 63 1008 6608 20958 34482 29750 13170 2685 175
n=8: 1 72 1332 10200 38220 75264 79002 43284 11190 980
...
Row n=4 corresponds to the polynomial 1 + 36*(1+x) + 306*(1+x)^2 + 996*(1+x)^3 + 1446*(1+x)^4 + 984*(1+x)^5 + 303*(1+x)^6 + 42*(1+x)^7 + 2*(1+x)^8.
MAPLE
(n, k)->binomial(9, k)*binomial(n, k)+(10*binomial(7, k-2)-binomial(6, k-2))*binomial(n+1, k)+(20*binomial(5, k-4)+binomial(3, k-3)-binomial(3, k-5))*binomial(n+2, k)+(10*binomial(3, k-6)+binomial(2, k-5)+binomial(3, k-5))*binomial(n+3, k)+binomial(2, k-7)*binomial(n+4, k)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Henryk A. Witek, Oct 17 2020
STATUS
approved