The classical Tutte polynomial is a two-variate polynomial  ( , ) associated to graphs or more generally, matroids. In this paper, we introduce a polynomial  ˜ ( , ) associated to a bipartite graph   that we call the permutation Tutte polynomial of the graph  . It turns out that  ( , ) and  ˜ ( , ) share many properties, and the permutation Tutte polynomial serves as a tool to study the classical Tutte polynomial. We discuss the analogues of Brylawsi’s identities and Conde–Merino–Welsh type inequalities. In particular, we will show that if   does not contain isolated vertices, then ˜ (3,0) ˜ (0,3)≥ ˜ (1,1)2,which gives a short proof of the analogous result of Jackson: (3,0) (0,3)≥ (1,1)2for graphs without loops and bridges. We also give improvement on the constant 3 in this statement by showing that one can replace it with 2.9243.