TY - JOUR

T1 - Dimer Coverings of 1-Slab Cubic Lattices

AU - Oh, Seungsang

N1 - Funding Information:
This work was supported by Institute for Information and communications Technology Planning and Evaluation (IITP) grant funded by the Korea government (MSIT) (No. 2019-0-00033, Study on Quantum Security Evaluation of Cryptography based on Computational Quantum Complexity).
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Japan KK, part of Springer Nature.

PY - 2022/8

Y1 - 2022/8

N2 - Lattice dimer statistics in two-dimension gained momentum in 1961 when the exact solution of the enumeration of pure dimer coverings of a rectangular square lattice was found by Kasteleyn (Physica 27:1209–1225) and Temperley and Fisher (Philos Magn 6:1061–1063). Recently the author introduced the state matrix recursion method to generate the partition function for monomer–dimer coverings in the square lattice with monomer and dimer activities. In this paper, stepping up a dimension, we apply this method to the enumeration of pure dimer coverings in the 1-slab cubic m× n× 2 lattice. Its asymptotic behavior is also derived. As a pure dimer covering in two-dimension is known as a perfect matching or a domino tiling, a pure dimer covering in the 1-slab cubic lattice is considered as a perfect matching or a packing of a three-dimensional 2-layer box with dicubes which are formed by two unit cubes joined face to face. We further discuss a new bijective relation of this model to the set of so-called matrix entry neighbor-permutations of 1 , 2 , ⋯ , mn in Smn rearranging each element at most to a neighboring position in the shape of the matrix whose entries are 1 , 2 , ⋯ , mn in usual order.

AB - Lattice dimer statistics in two-dimension gained momentum in 1961 when the exact solution of the enumeration of pure dimer coverings of a rectangular square lattice was found by Kasteleyn (Physica 27:1209–1225) and Temperley and Fisher (Philos Magn 6:1061–1063). Recently the author introduced the state matrix recursion method to generate the partition function for monomer–dimer coverings in the square lattice with monomer and dimer activities. In this paper, stepping up a dimension, we apply this method to the enumeration of pure dimer coverings in the 1-slab cubic m× n× 2 lattice. Its asymptotic behavior is also derived. As a pure dimer covering in two-dimension is known as a perfect matching or a domino tiling, a pure dimer covering in the 1-slab cubic lattice is considered as a perfect matching or a packing of a three-dimensional 2-layer box with dicubes which are formed by two unit cubes joined face to face. We further discuss a new bijective relation of this model to the set of so-called matrix entry neighbor-permutations of 1 , 2 , ⋯ , mn in Smn rearranging each element at most to a neighboring position in the shape of the matrix whose entries are 1 , 2 , ⋯ , mn in usual order.

KW - 1-slab cubic lattice

KW - Dicube packing

KW - Dimer covering

KW - Perfect matching

UR - http://www.scopus.com/inward/record.url?scp=85133944782&partnerID=8YFLogxK

U2 - 10.1007/s00373-022-02522-x

DO - 10.1007/s00373-022-02522-x

M3 - Article

AN - SCOPUS:85133944782

SN - 0911-0119

VL - 38

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

IS - 4

M1 - 117

ER -