  We investigate analogues of alternating sign matrices, called partial
alternating sign matrices. We prove bijections between these matrices and
several other combinatorial objects. We use an analogue of Wieland's gyration
on fully-packed loops, which we relate to the study of toggles and order
ideals. Finally, we show that rowmotion on order ideals of a certain poset and
gyration on partial fully-packed loop configurations are in equivariant
bijection.
