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”. 1 Alternating sign matrices Recall that a square matrix A = (aij)1≤i,j≤n is an order n alternating sign matrix if aij ∈ {1, 0,−1} and if, in each row and each column, the non-zero entries alternate in sign, beginning and ending with a 1. Thus, the entries of each row and of each column add up to 1. The entries in the first row of an alternating sign matrix are all 0 except for one, which must be a 1. It will be called the first 1. In their paper [MRR], Mills, Robbins and Rumsey defined the following parameters on order n alternating sign matrices A = (aij): • r(A) is the number of entries to the left of the first 1. We have 0 ≤ r(A) ≤ n− 1. • s(A) is the number of entries that are equal to −1.