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<p>We completely classify the asymptotic behavior of the number of alternating sign matrices classically avoiding a single permutation pattern, in the sense of Johansson and Linusson (Ann Combin 11:471–480, 2007). In particular, we give a uniform proof of an exponential upper bound for the number of alternating sign matrices classically avoiding one of eleven particular patterns, and a super-exponential lower bound for all other single-pattern avoidance classes. We also show that for any fixed integer <i>k</i>, there is an exponential upper bound for the number of alternating sign matrices that classically avoid any single permutation pattern and contain precisely <i>k</i> negative ones. Finally, we prove that there must be at most 3 negative ones in an alternating sign matrix which classically avoids both 2143 and 3412, and we exactly enumerate the number of them with precisely 3 negative ones.</p>

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Enumeration of Pattern-Avoiding Alternating Sign Matrices: An Asymptotic Dichotomy

  • 期刊论文

  • 发表日期：

    2025年9月20日

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  • Mathilde Bouvel,
  • Eric S. Egge,
  • Rebecca N. Smith,
  • Jessica Striker,
  • Justin M. Troyka

摘要

We completely classify the asymptotic behavior of the number of alternating sign matrices classically avoiding a single permutation pattern, in the sense of Johansson and Linusson (Ann Combin 11:471–480, 2007). In particular, we give a uniform proof of an exponential upper bound for the number of alternating sign matrices classically avoiding one of eleven particular patterns, and a super-exponential lower bound for all other single-pattern avoidance classes. We also show that for any fixed integer k, there is an exponential upper bound for the number of alternating sign matrices that classically avoid any single permutation pattern and contain precisely k negative ones. Finally, we prove that there must be at most 3 negative ones in an alternating sign matrix which classically avoids both 2143 and 3412, and we exactly enumerate the number of them with precisely 3 negative ones.