Efficiently Counting Linear Extensions of Posets

Linear extensions of partially ordered sets are fundamental in order theory and in algebraic combinatorics, holding high importance in the study of permutations and Young tableaux. Computing the number of linear extensions, in general, takes O(n!) time, given a poset of n elements. However, many efficient algorithms are known for certain large classes of posets. SageMath offers a single function to calculate the linear extensions of a poset. For even small posets, this quickly becomes impossible to use. The goal of this project would be to implement efficient methods of computing linear extensions of several important classes of posets, and introduce uniform sampling of linear extensions. This includes tree posets, d-complete posets, skew diagrams, mobile posets, and more.