Quantum entanglement is the most surprising feature of quantum mechanics, and plays a crucial role in quantum computation. Ground states of quantum many-body systems typically exhibit the area law behavior in the entanglement entropy, which measures the amount of entanglement between a subsystem and the rest of the system. Recently, a class of solvable one-dimensional spin models with local interactions has been constructed by Mavassagh and Shor and by Salberger and Korepin, in which the ground state is expressed as a superposition of random walks, and has much larger entanglement. Its entanglement entropy is shown to be proportional to the square root of the volume. In this talk, after a brief review of the models, we construct extensions of these models based on the symmetric inverse semigroup, and discuss properties of ground states with the entanglement entropy. As a new feature arising by the extension, there are excited states with Anderson localization properties.