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Bounded Affine Permutations II. Avoidance of Decreasing Patterns


We continue our study of a new boundedness condition for affine permutations, motivated by the fruitful concept of periodic boundary conditions in statistical physics. We focus on bounded affine permutations of size N that avoid the monotone decreasing pattern of fixed size m. We prove that the number of such permutations is asymptotically equal to \((m-1)^{2N} N^{(m-2)/2}\) times an explicit constant as \(N\rightarrow \infty \). For instance, the number of bounded affine permutations of size N that avoid 321 is asymptotically equal to \(4^N (N/4\pi )^{1/2}\). We also prove a permuton-like result for the scaling limit of random permutations from this class, showing that the plot of a typical bounded affine permutation avoiding \(m\cdots 1\) looks like \(m-1\) random lines of slope 1 whose y intercepts sum to 0.

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We are grateful to Tom Salisbury for a helpful discussion about random measures, and to the anonymous referees for helpful comments.


The research was pursued as part of a Discovery Grant to N. Madras from NSERC Canada.

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Correspondence to Neal Madras.

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N. Madras was supported in part by a Discovery Grant from NSERC Canada, and by a Minor Research Grant from the Faculty of Science at York University.

Communicated by David Bevan.

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Madras, N., Troyka, J.M. Bounded Affine Permutations II. Avoidance of Decreasing Patterns. Ann. Comb. (2021).

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  • Permutation
  • Affine permutation
  • Permutation pattern
  • Asymptotic enumeration
  • Permuton
  • Random measure

Mathematics Subject Classification

  • Primary 05A05
  • 05A16
  • 60C05
  • 60G57