Permanents, Pfaffian orientations, and even directed circuits.
Positive Diagonals of ...1-Matrices.
Power indices of trace zero symmetric Boolean matrices
The power index of a square Boolean matrix A is the least integer d such that Ad is a linear combination of previous nonnegative powers of A. We determine the maximum power indices for the class of n×n primitive symmetric Boolean matrices of trace zero, the class of n×n irreducible nonprimitive symmetric Boolean matrices, and the class of n×n reducible symmetric Boolean matrices of trace zero, and characterize the extreme matrices respectively.
Proof of the alternating sign matrix conjecture.
Proof of the refined alternating sign matrix conjecture.
Quantum permutations, Hadamard matrices, and the search for matrix models
This is a presentation of recent work on quantum permutation groups, complex Hadamard matrices, and the connections between them. A long list of problems is included. We include as well some conjectural statements about matrix models.
Ranks of Incidence Matrices of Steiner Triple Systems.
Ranks of permutative matrices
A new type of matrix, termed permutative, is defined and motivated herein. The focus is upon identifying circumstances under which square permutative matrices are rank deficient. Two distinct ways, along with variants upon them are given. These are a special kind of grouping of rows and a type of partition in which the blocks are again permutative. Other, results are given, along with some questions and conjectures.
Regular elements of the semigroup of all binary relations.
Relationships between conbinatorics and 0-simple semigroups.
Remark to a problem on 0-1 matrices
Seriation with applications in philology
Small forbidden configurations. II.
Some applications of a theorem of Schur to graphs and to a class of endomorphisms
Spectrum Of The Total Graph Of A Graph
Standard monomials for q-uniform families and a conjecture of Babai and Frankl
Let n, k, α be integers, n, α>0, p be a prime and q=p α. Consider the complete q-uniform family We study certain inclusion matrices attached to F(k,q) over the field . We show that if l≤q−1 and 2l≤n then This extends a theorem of Frankl [7] obtained for the case α=1. In the proof we use arguments involving Gröbner bases, standard monomials and reduction. As an application, we solve a problem of Babai and Frankl related to the size of some L-intersecting families modulo q.
Subdeterminants and subgraphs
Symmetric Hadamard matrices of order 116 and 172 exist
We construct new symmetric Hadamard matrices of orders 92, 116, and 172. While the existence of those of order 92 was known since 1978, the orders 116 and 172 are new. Our construction is based on a recent new combinatorial array (GP array) discovered by N. A. Balonin and J. Seberry. For order 116 we used an adaptation of an algorithm for parallel collision search. The adaptation pertains to the modification of some aspects of the algorithm to make it suitable to solve a 3-way matching problem....
Tactical decompositions of designs.
The (16,6,2) designs.