Fibonacci numbers and resolutions of domino ideals
Keywords:
Fibonacci numbers, Monomial ideals, Domino tilings
Abstract
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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P. K. Benedetto, A. N. Loehr, Domino tiling graphs, Ars Combin. 109 (2013), 3–29.
R. R. Bouchat, H. T. Hà, 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, T. M. Brown, Multi-graded Betti numbers of path ideals of trees, J. Algebra Appl. 16(1) (2017) 1750018.
R. R. Bouchat, T. M. Brown, Minimal free resolutions of $2times n$ domino tilings, J. Algebra Appl. online ready.
S. Butler, P. Horn, E. Tressler, Intersection domino tilings, Fibonacci Quart. 48(2) (2010) 114–120.
A. Conca, E. De Negri, M-sequences, graph ideals, and ladder ideals of linear type, J. Algebra 211(2) (1999) 599–624.
N. Erey, 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, M. Stillman, Macaulay2, a software system for research in algebraic geometry. Available at https://faculty.math.illinois.edu/Macaulay2/.
H. T. Hà, A. Van Tuyl, Monomial ideals, edge ideals of hyper graphs, and their graded Betti numbers, J. Algebr. Comb. 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.
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H. N. V. Temperley, M. E. Fisher, Dimer problem in statistical mechanics-an exact result, Philosophical Magazine, 6(68) (1961) 1061–1063.
