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Bounded Affine Permutations II. Avoidance of Decreasing Patterns

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Abstract

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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  1. Proportional to one-dimensional Lebesgue measure.

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Acknowledgements

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

Funding

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

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Authors and Affiliations

  1. Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON, M3J 1P3, Canada

    Neal Madras

  2. Department of Mathematics and Computer Science, Davidson College, Davidson, NC, 28035, USA

    Justin M. Troyka

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  1. Neal Madras
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  2. Justin M. Troyka
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Correspondence to Neal Madras.

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Communicated by David Bevan.

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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.

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Madras, N., Troyka, J.M. Bounded Affine Permutations II. Avoidance of Decreasing Patterns. Ann. Comb. 25, 1007–1048 (2021). https://doi.org/10.1007/s00026-021-00553-4

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