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.
Highlights
Monomial ideals have been studied using mechanisms from several different areas of mathematics, including combinatorics, graph theory, algebra, and topology
This paper considers a class of monomial ideals, called domino ideals, whose generating sets correspond to the sets of domino tilings of a 2 × n tableau
It is well-known that the number of domino tilings of a 2 × n tableau is given by a Fibonacci number
Summary
Monomial ideals have been studied using mechanisms from several different areas of mathematics, including combinatorics, graph theory, algebra, and topology. Conca and De Negri [8] introduced the study of edge ideals These edge ideals are squarefree monomial ideals generated from the edges of a graph. We consider a class of monomial ideals arising from the set of domino tilings of a 2 × n tableau. The generating set of the domino ideals associated to the set of all 2 × n domino tilings can be viewed as paths of a graph. Consider a 2 × 3 tableau, the domino ideal I is generated by the domino monomials corresponding to the tilings in the set: x1 x3. We give the necessary background from topology and commutative algebra
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
