2025/02/09 by Seok Hyun Byun, Byun, Seok Hyun, Wayne Goddard +1
Engineering · Mathematics · #05A15 #05A19 #Advanced Materials and Mechanics #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2502.05918
openalex publication_date 2025/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the number of domino tilings of an odd-by-odd rectangle that leave one hole. This problem is equivalent to the number of near-perfect matchings of the odd-by-odd rectangular grid. For any particular position of the vacancy on the (2k+1)× (2k+1) square grid, we show that the number of near-perfect matchings is a multiple of 2k, and from this follows a conjecture of Kong that the total number of near-perfect matchings is a multiple of 2k. We also determine the parity of the number of near-perfect matchings with a particular vacancy for the rectangle case.