Fibonacci numbers and resolutions of domino ideals
Keywords:
Fibonacci numbers, Monomial ideals, Domino tilingsAbstract
This paper considers a class of monomial ideals, called domino ideals, whose generating sets correspond to the sets of domino tilings of a $2\times n$ tableau. The multi-graded Betti numbers are shown to be in one-to-one correspondence with equivalence classes of sets of tilings. It is well-known that the number of domino tilings of a $2\times n$ tableau is given by a Fibonacci number. Using the bijection, this relationship is further expanded to show the relationship between the Fibonacci numbers and the graded Betti numbers of the corresponding domino ideal.
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References
A. Alilooee and S. Faridi, On the resolution of path ideals of cycles, Comm. Algebra 43(12) (2015) 5413–5433.
F. Ardila and R. P. Stanley, Tilings, Math. Intelligencer 32(4) (2010) 32–43.
P. K. Benedetto and A. N. Loehr, Domino tiling graphs, Ars Combin. 109 (2013) 3–29.
R. R. Bouchat, H. T. Hà, and A. O’Keefe, Path ideals of rooted trees and their graded Betti numbers, J. Combin. Theory Ser. A 118(8) (2011) 2411–2425.
R. R. Bouchat and T. M. Brown, Multi-graded Betti numbers of path ideals of trees, J. Algebra Appl. 16(1) (2017) 1750018.
R. R. Bouchat and T. M. Brown, Minimal free resolutions of $2 \times n$ domino tilings, J. Algebra Appl. 18(7) (2019) 1950118.
S. Butler, P. Horn, and E. Tressler, Intersecting domino tilings, Fibonacci Quart. 48(2) (2010) 114–120.
A. Conca and E. De Negri, M-sequences, graph ideals, and ladder ideals of linear type, J. Algebra 211(2) (1999) 599–624.
N. Erey and S. Faridi, Multigraded Betti numbers of simplicial forests, J. Pure Appl. Algebra 218(10) (2014) 1800–1805.
S. Faridi, The facet ideal of a simplicial complex, Manuscripta Math. 109(2) (2002) 159–174.
D. Grayson and M. Stillman, Macaulay2, a software system for research in algebraic geometry, available at https://faculty.math.illinois.edu/Macaulay2/.
H. T. Hà and A. Van Tuyl, Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers, J. Algebraic Combin. 27(2) (2008) 215–245.
P. W. Kasteleyn, The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice, Physica 27(12) (1961) 1209–1225.
E. Miller and B. Sturmfels, Combinatorial Commutative Algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005.
N. J. A. Sloane (Ed.), The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org (2018).
H. N. V. Temperley and M. E. Fisher, Dimer problem in statistical mechanics-an exact result, Philos. Mag. 6(68) (1961) 1061–1063.
