Sign-balance identities of Adin–Roichman type on 321-avoiding alternating permutations

Adin and Roichman proved a set of refined sign-balance identities on 321-avoiding permutations respecting the last descent of the permutations, which we call the identities of Adin–Roichman type. In this work, we construct a new involution on plane trees that proves refined sign-balance properties on 321-avoiding alternating permutations respecting the first and last entries of the permutations respectively and obtain two sets of identities of Adin–Roichman type.