Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: Dodgson’s Determinant-Evaluation Rule Proved by. | Dodgson s Determinant-Evaluation Rule Proved by TWO-TIMING MEN and WOMEN Doron ZEILBERGER1 Submitted April 15 1996 Accepted May 15 1996 Bijections are where it s at Herb Wilf Dedicated to Master Bijectionist Herb Wilf on finishing 13 24 of his life I will give a bijective proof of the Reverend Charles Lutwidge Dodgson s Rule D det ai j 1 i n det ai j 2 i n 1 1 j n J 2 j n 1_ 1 i n 1 det ãi j 2 i n 1 j n 1 2 j n 1 i n 1 det i j 2 i n n 1 j n 1 Alice Consider n men 1 2 . n and n women 1 2 . n each of whom is married to exactly one member of the opposite sex. For each of the n possible perfect matchings n let n weight n sign n n di n i i 1 where sign n is the sign of the corresponding permutation and for i 1 . n Mr. i is married to Ms. n i . Except for Mr. 1 Mr. n Ms. 1 and Ms. n all the persons have affairs. Assume that each of the men in 2 . n 1 has exactly one mistress amongst 2 . n 1 and each of the women in 2 . n 1 has exactly one lover amongst 2 . n 1 2. For each of the n 2 possible perfect matchings Ơ let n 1 weight ơ sign ơ n Oi a i i 2 where sign ơ is the sign of the corresponding permutation and for i 2 . n 1 Mr. i is the lover of Ms. ơ i . 1 Department of Mathematics Temple University Philadelphia PA 19122 USA. zeilberg@ http zeilberg ftp pub zeilberg . Supported in part by the NSF. Version of Dec 6 1996. First Version April 15 1996. Thanks are due to Bill Gosper for several corrections. 2 Somewhat unrealistically a man s wife may also be his mistress and equivalently a woman s husband may also be her lover. 1 THE ELECTRONIC .JOURNAL OF COMBINATORICS 4 2 1997 R22 2 Let A n be the set of all pairs n ơ as above and let weight n ơ weight n weight ơ . The left side of Alice is the sum of all the weights of the elements of A n . Let B n be the set of pairs n ơ where now n and n are unmarried but have affairs . n is a matching of 1 . n 1 to 1 . n 1 and Ơ is a matching of 2 . n to 2 . n and define the .
