Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: On the number of possible row and column sums of 0,1-matrices. | On the number of possible row and column sums of 0 1-matrices Daniel Goldstein and Richard Stong Center for Communications Research 4320 Westerra Court San Diego CA 92121 dgoldste@ Department of Mathematics Rice University Houston TX 77005 stong@ Submitted Aug 9 2005 Accepted Apr 4 2006 Published Apr 18 2006 Mathematics Subject Classification 05A15 Abstract For n a positive integer we show that the number of of 2n-tuples of integers that are the row and column sums of some n X n matrix with entries in 0 1 is evenly divisible by n 1. This confirms a conjecture of Benton Snow and Wallach. We also consider a ợ-analogue for m X n matrices. We give an efficient recursion formula for this analogue. We prove a divisibility result in this context that implies the n 1 divisibility result. 1 Introduction We study the number p m n of m n -tuples of integers that are the row and column sums of some m X n matrix with entries in 0 1 . For each n 1 the sequence p m n 1 1 is a linear recursion of degree n. Moreover this recursion is annihilated by the polynomial T n 1 n. It follows that if 1 n m then p m n is evenly divisible by n 1 m-n 1. This confirms a conjecture of Benton Snow and Wallach. For positive integers m and n let M Mmn be the set of m X n matrices with entries in 0 1 . For M in M we write M Mij . We have two vector-valued functions on M the vector x M x1 . xm of row sums where xi 52 i_ j n Mij for 1 i m and the vector y M y1 . yn of column sums where yj mi m Mij for 1 j n. Define RC RCm n to be the set of pairs of row and column sums x M y M as M ranges over M. Our main result concerns the cardinality p m n of RCmn. THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 N8 1 Theorem 1 We have 1. p 1 1 2. 2. p m n p n m for m n 1. 3. If 1 n m then p m n VKi n 1 i 1 n n 1 ip m i n . Of these statements part 1 is clear and part 2 follows by taking transpose for x Mt y M and y Mt x M . Part 3 says that for each n 1 the sequence p m n 1 1 is a linear .
