Counting extreme U 1 matrices and characterizing quadratic doubly stochastic operators

The U 1 matrix and extreme U 1 matrix were successfully used to study quadratic doubly stochastic operators by R. Ganikhodzhaev and F. Shahidi [Linear Algebra Appl., 2010, 432: 24–35], where a necessary condition for a U 1 matrix to be extreme was given. S. Yang and C. Xu [Linear Algebra Appl., 2013, 438: 3905–3912] gave a necessary and sufficient condition for a symmetric nonnegative matrix to be an extreme U 1 matrix and investigated the structure of extreme U 1 matrices. In this paper, we count the number of the permutation equivalence classes of the n × n extreme U 1 matrices and characterize the structure of the quadratic stochastic operators and the quadratic doubly stochastic operators.