Counting square-free monomial Cremona maps

We give a complete list of square-free monomial Cremona maps of P n - 1 , with n ≤ 6 , up to equivalence classes. Also, we present a sufficient condition for all the ideals associated to a square-free Cremona transformation with same degree d in a fixed projective space P n - 1 to have eight two. Finally, we introduce an algorithm to count them. Using this algorithm, we obtain the complete list of square-free Cremona maps of P n - 1 when the degree is two for n = 7 , n = 8 , n = 9 , n = 10 , n = 11 and when the degree is three for n = 7 .