The alternating sign matrices-descending plane partitions (ASM-DPP) bijection
problem is one of the most intriguing open problems in bijective combinatorics,
which is also relevant to integrable combinatorics. The notion of a signed set
and a signed bijection is used in [Fischer, I. \& Konvalinka, M., Electron. J.
Comb., 27 (2020) 3-35.] to construct a bijection between ASMnβΓDPPnβ1β and DPPnβΓASMnβ1β. Here, we shall
construct a more natural alternative to a signed bijection between alternating
sign matrices and shifted Gelfand-Tsetlin patterns which is presented in that
paper, based on the notion of compatibility which we introduce to measure the
naturalness of a signed bijection. In addition, we give a bijective proof for
the refined enumeration of an extension of alternating sign matrices with n+3
statistics, first proved in [Fischer, I. \& Schreier-Aigner, F., Advances in
Mathematics 413 (2023) 108831.].Comment: 50 page