New bijective proofs pertaining to alternating sign matrices

2023/06/01 by Takuya Inoue, Inoue, Takuya · 1 citation
Mathematics · Engineering · Computer Science · #Advanced Combinatorial Mathematics #graph theory and CDMA systems #Coding theory and cryptography

paper · pdf · doi:10.48550/arxiv.2306.00413

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 ASM_n × DPP_n-1 and DPP_n × 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.].

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