Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: The Redheffer matrix of a partially ordered set. | The Redheffer matrix of a partially ordered set Herbert S. Wilf University of Pennsylvania Philadelphia PA 19104-6395 Submitted Sep 22 2004 Accepted Nov 2 2004 Published Nov 22 2004 Mathematics Subject Classifications 05A15 05E99 Dedicated to Richard Stanley on his sixtieth birthday Abstract R. Redheffer described an n X n matrix of 0 s and 1 s the size of whose determinant is connected to the Riemann Hypothesis. We describe the permutations that contribute to its determinant and its permanent in terms of integer factorizations. We generalize the Redheffer matrix to finite posets that have a 0 element and find the analogous results in the more general situation. 1 Introduction In 1977 R. Redheffer described a matrix that is closely connected to the Riemann Hypothesis RH . Let Rn be the n X n matrix whose i j entry is 1 if i j or if j 1 and otherwise is 0 for 1 i j n. For example 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 p R8 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 He showed that the proposition for every e 0 we have I det Rn O n2 e is equivalent to RH. More precisely he showed that n det Rn X h k k 1 1 THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2 2004 R10 1 where ụ is the classical Mobius function and the equivalence of the O n1 e growth bound of the right side of 1 to the RH is well known see . 10 Thm. C . Here we will hrst describe the permutations of n letters that contribute to the determinant of Rn . the permutations that do not hit any 0 entries in the matrix. Then we will count those permutations which is to say that we will evaluate the permanent of the Redheffer matrix. It turns out that this permanent is also nicely expressible in terms of well known number theoretic functions. After that we will generalize the Redheffer matrix to posets other than the positive integers under divisibility and hnd an application in the case of the Boolean lattice. 2 The permanent Which permutations of n letters .
