New bijective proofs pertaining to alternating sign matrices

Abstract

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 $\text{ASM}_n \times \text{DPP}_{n-1}$ and $\text{DPP}_n \times \text{ASM}_{n-1}$. 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.].