Alternating sign matrices with one −1 under vertical reflection

We define a bijection that transforms an alternating sign matrix A with one −1 into a pair ( N , E ) where N is a (so called) neutral alternating sign matrix (with one −1) and E is an integer. The bijection preserves the classical parameters of Mills, Robbins and Rumsey as well as three new parameters (including E). It translates vertical reflection of A into vertical reflection of N. A hidden symmetry allows the interchange of E with one of the remaining two new parameters. A second bijection transforms ( N , E ) into a configuration of lattice paths called “mixed configuration.”