Edward T. H. Wang

The weights of Hadamard matrices

Abstract Let Ω n denote the set of all n × n Hadamard matrices. For H ∈ Ω n , define the weight of H to be w ( H ) = number of 1s in H , and w(n) = max {w(H); H ∈ Ω n } . In this paper, we derive upper and lower bounds for w ( n ).

On a conjecture of M. Marcus and H. Minc

In this note, we prove that the following conjecture of Marcus and Minc is true for n=3. For all n×n doubly stochastic matrices .

On permanents of (1, − 1)-matrices

A preliminary study on permanents of (1, − 1)-matrices is given. Some inequalities are derived and a few unsolved problems, believed to be new, are mentioned.

Permanental pairs of doubly stochastic matrices. II

Abstract The n ×n doubly stochastic matrices A, B form a permanental pair if the permanent of every convex linear combination λA+(1−λ)B(0⩽λ⩽1) is independent of λ A, B are called mates. In this article we show that the direct sum of any number, k, of matrices Ji (of varying individual dimension) cannot have a mate. Here Ji is the ni×ni matrix with every entry equal to 1 n i ;∑ni=n.

A simple proof of a curious congruence by Sun

In this note, we give a simple and elementary proof of the following curious congruence which was established by Zhi-Wei Sun: (p−1)/2 X k=1 1 k · 2k ≡ [3p/4] X k=1 (−1)k−1 k (mod p). In [4], the following curious congruence for odd prime p was established by ZhiWei Sun: (p−1)/2 ∑ k=1 1 k · 2k ≡ [3p/4] ∑ k=1 (−1)k−1 k (mod p). (1) The author’s proof, using Pell sequences, is fairly complicated. In fact, a recent article [3] on congruence modulo p ends in the remark that “It seems unlikely that (1) can be proved with the simple approach that we have used here.” In the present note, we give a simple and elementary proof of (1). Throughout, p denotes an odd prime. First of all, it is well known (e.g. [1], [2]) that for k = 0, 1, 2, . . . , p− 1, ( p− 1 k ) ≡ (−1)k (mod p). (2) From (2) we get 2p−1 − 1 2 = (1 + 1) − 2 2p = 1 2p p−1 ∑ k=1 ( p k ) = 1 2 p−1 ∑ k=1 1 k ( p− 1 k − 1 ) ≡ 1 2 p−1 ∑ k=1 (−1)k−1 k (mod p). (3) Received by the editors August 13, 1997. 1991 Mathematics Subject Classification. Primary 11A07, 11A41. c ©1999 American Mathematical Society 1289 1290 ZUN SHAN AND EDWARD T. H. WANG Let e = e. Then (1 + e) + (1− e) = 2 + 2 ∑ 1≤k≤p k even ( p k ) e = 2 + 2p ∑ 1≤k≤p k even 1 k ( p− 1 k − 1 ) e ≡ 2− 2p ∑ 1≤k≤p k even e k (mod p) = 2− 2p [ p−1 4 ] ∑ k=1 (−1)k 4k + i [ p+1 4 ] ∑ k=1 (−1)k−1 4k − 2  = 2− p 2 [ p−1 4 ] ∑ k=1 (−1)k k + ip [ p+1 4 ] ∑ k=1 (−1)k 2k − 1 = 2− p 2 A + ipB (4)

Diagonal sums of doubly stochastic matrices

In this paper, we study some combinatorial problems involving diagonal sums of d.s. (doubly stochastic) matrices. We prove, among others, that if a d.s. matrix A has more than (n−1)(n−1)! diagonals...

A combinatorial application of the Frobenius inequality on rank function to maximum set of commuting nilpotent matrices

In this note we show how the Frobenius inequality on rank function can be applied to determine the maximum cardinality of a setS of pairwise-commutative n×n nilpotent matrices with the property that no finite product of distinct elements of S equals zero.