A solution to the ASM-TSSCPP-DPP bijection problem in the permutation case

In this talk, I will give partial progress on a bijective problem that has been open for a few decades, that of finding an explicit bijection between alternating sign matrices (ASM) and either of two types of plane partitions, descending (DPP) or totally symmetric self-complementary (TSSCPP). I will show natural bijections from the set of permutations (a subset of ASMs), to subsets of DPPs and TSSCPPs. We will discuss poset structures and generating functions on these objects as well as the outlook of the general bijection problem.