- In this paper we consider several variants of the problem of sorting integer permutations with a minimum number of moves, a task with many potential applications ranging from computational biology to logistics. Each problem is formulated as a heuristic search problem, where different variants induce different sets of allowed moves within the search tree. Due to the intrinsic nature of this category of problems, which in many cases present a very large branching factor, classic unidirectional heuristic search algorithms such as A∗ and IDA∗ quickly become inefficient or even infeasible as the problem dimension grows. Therefore, more sophisticated algorithms are needed. To this aim, we propose to combine two recent paradigms which have been employed in difficult heuristic search problems showing good performance: enhanced partial expansion (EPE) and efficient single-frontier bidirectional search (eSBS). We propose a new class of algorithms combining the benefits of EPE and eSBS, named efficient Single-frontier Bidirectional Search with Enhanced Partial Expansion (eSBS-EPE). We then present an experimental evaluation that shows that eSBS-EPE is a very effective approach for this family of problems, often outperforming previous methods on large-size instances. With the new eSBS-EPE class of methods we were able to push the limit and solve the largest size instances of some of the problem domains (the pancake and the burnt pancake puzzles). This novel search paradigm hence provides a very promising framework also for other domains.