TY - JOUR
T1 - State matrix recursion method and monomer–dimer problem
AU - Oh, Seungsang
N1 - Funding Information:
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. NRF-2017R1A2B2007216).
Publisher Copyright:
© 2019 Elsevier B.V.
PY - 2019/5
Y1 - 2019/5
N2 - The exact enumeration of pure dimer coverings on the square lattice was obtained by Kasteleyn, Temperley and Fisher in 1961. In this paper, we consider the monomer–dimer covering problem (allowing multiple monomers) which is an outstanding unsolved problem in lattice statistics. We have developed the state matrix recursion method that allows us to compute the number of monomer–dimer coverings and to know the partition function with monomer and dimer activities. This method proceeds with a recurrence relation of so-called state matrices of large size. The enumeration problem of pure dimer coverings and dimer coverings with single boundary monomer is revisited in partition function forms. We also provide the number of dimer coverings with multiple vacant sites. The related Hosoya index and the asymptotic behavior of its growth rate are considered. Lastly, we apply this method to the enumeration study of domino tilings of Aztec diamonds and more generalized regions, so-called Aztec octagons and multi-deficient Aztec octagons.
AB - The exact enumeration of pure dimer coverings on the square lattice was obtained by Kasteleyn, Temperley and Fisher in 1961. In this paper, we consider the monomer–dimer covering problem (allowing multiple monomers) which is an outstanding unsolved problem in lattice statistics. We have developed the state matrix recursion method that allows us to compute the number of monomer–dimer coverings and to know the partition function with monomer and dimer activities. This method proceeds with a recurrence relation of so-called state matrices of large size. The enumeration problem of pure dimer coverings and dimer coverings with single boundary monomer is revisited in partition function forms. We also provide the number of dimer coverings with multiple vacant sites. The related Hosoya index and the asymptotic behavior of its growth rate are considered. Lastly, we apply this method to the enumeration study of domino tilings of Aztec diamonds and more generalized regions, so-called Aztec octagons and multi-deficient Aztec octagons.
KW - Aztec diamond
KW - Dimer
KW - Domino tiling
KW - Hosoya index
UR - http://www.scopus.com/inward/record.url?scp=85061524548&partnerID=8YFLogxK
U2 - 10.1016/j.disc.2019.01.022
DO - 10.1016/j.disc.2019.01.022
M3 - Article
AN - SCOPUS:85061524548
SN - 0012-365X
VL - 342
SP - 1434
EP - 1445
JO - Discrete Mathematics
JF - Discrete Mathematics
IS - 5
ER -