The inversion number of the alternating sign matrix.
If we denote the entries of the alternating sign matrix as $a_{i,j}$, the inversion number is defined as
$$\sum_{i > k}\sum_{j < \ell} a_{i,j}a_{k,\ell}.$$
When restricted to permutation matrices, this gives the usual inversion number of the permutation.

The gyration of an alternating sign matrix.
This is the alternating sign matrix obtained by applying the gyration action to the height function.
Gyration is a bijective map on ASMs defined in [1].

One may turn an ASM into a square ice configuration (Figure 1), then turning that into a colored graph with boundary conditions (Figure 2). Then fully convert the ASM into a colored graph (Figure 4). The gyration is then defined using this colored graph.

Return the Dyck path as an alternating sign matrix.
This is an inclusion map from Dyck words of length $2n$ to certain
$n \times n$ alternating sign matrices.