Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: 2-ADIC BEHAVIOR OF NUMBERS OF DOMINO TILINGS. | 2-ADIC BEHAVIOR OF NUMBERS OF DOMINO TILINGS HENRY COHN DEPARTMENT OF MATHEMATICS HARVARD UNIVERSITY CAMBRIDGE MA 02138 USA Dedicated to my grandparents Garnette Cohn 1907-1998 and Lee Cohn 1908-1998 Abstract. We study the 2-adic behavior of the number of domino tilings of a 2n 2n square as n varies. It was previously known that this number was of the form 2nf n 2 where f n is an odd positive integer. We show that the function f is uniformly continuous under the 2-adic metric and thus extends to a function on all of z. The extension satisfies the functional equation f 1 n f n where the sign is positive iff n 0 3 mod 4 . 2 Wl 2 J 4 cos2 _ 4 cos2 - 2n 1 2n 1 Kasteleyn K and Temperley and Fisher TF proved that the number of tilings of a 2n X 2n square with 1 X 2 dominos is Y Y nil i 1j 1 Although it is by no means obvious at first glance this number is always a perfect square or twice a perfect square see L . Furthermore it is divisible by 2n but no higher power of 2. This fact about 2-divisibility was independently proved by several people see JSZ or see P for a combinatorial proof but there seems to have been little further investigation of the 2-adic properties of these numbers except for JS . Write the number of tilings as 2nf n 2 where f n is an odd positive integer. In this paper we study the 2-adic properties of the function f. In particular we will prove the following theorem which was conjectured by James Propp Theorem 1. The function f is uniformly continuous under the 2-adic metric and its unique extension to a function from z2 to z2 satisfies the functional equation f -1 - n if n 0 3 mod 4 and if n 1 2 mod 4 . John and Sachs JS have independently investigated the 2-adic behavior of f and explicitly determined it modulo 26. Their methods as well as ours can be used to write formulas for f modulo any power of 2 but no closed form is known. The proof of Theorem 1 will not make any use of sophisticated 2-adic machinery. The only .
