Home

# The Infinite limit of random permutations avoiding patterns of length three

## Abstract

For $$\tau \in {S_3}$$ , let $$\mu _n^\tau$$ denote the uniformly random probability measure on the set of $$\tau$$ -avoiding permutations in $${S_n}$$ . Let $${\mathbb {N}^*} = {\mathbb {N}} \cup \{ \infty \}$$ with an appropriate metric and denote by $$S({\mathbb{N}},{\mathbb{N}^*})$$ the compact metric space consisting of functions $$\sigma {\rm{ = }}\{ {\sigma _i}\} _{i = 1}^\infty {\rm{ }}$$ from $$\mathbb {N}$$ to $${\mathbb {N}^ * }$$ which are injections when restricted to $${\sigma ^{ - 1}}(\mathbb {N})$$ ; that is, if $${\sigma _i}{\rm{ = }}{\sigma _j}$$ , $$i \ne j$$ , then $${\sigma _i} = \infty$$ . Extending permutations $$\sigma \in {S_n}$$ by defining $${\sigma _j} = j$$ , for $$j \gt n$$ , we have $${S_n} \subset S({\mathbb{N}},{{\mathbb{N}}^*})$$ . For each $$\tau \in {S_3}$$ , we study the limiting behaviour of the measures $$\{ \mu _n^\tau \} _{n = 1}^\infty$$ on $$S({\mathbb{N}},{\mathbb{N}^*})$$ . We obtain partial results for the permutation $$\tau = 321$$ and complete results for the other five permutations $$\tau \in {S_3}$$ .

## References

Hide All
[1] Bassino, F., Bouvel, M., Féray, V., Gerin, L. and Pierrot, A. (2018) The Brownian limit of separable permutations. Ann. Probab. 46 21342189.
[2] Bona, M. (2004) Combinatorics of Permutations, Chapman & Hall/CRC.
[3] Gnedenko, B. V. and Kolmogorov, A. N. (1968) Limit Distributions for Sums of Independent Random Variables, revised edition, Addison-Wesley.
[4] Gnedin, A. and Olshanski, G. (2010) q-exchangeability via quasi-invariance. Ann. Probab. 38 21032135.
[5] Gnedin, A. and Olshanski, G. (2012) The two-sided infinite extension of the Mallows model for random permutations. Adv. Appl. Math. 48 615639.
[6] Hoffman, C., Rizzolo, D. and Slivken, E. (2017) Pattern-avoiding permutations and Brownian excursion, I: Shapes and fluctuations. Random Structures Algorithms 50 394419.
[7] Janson, S. (2017) Patterns in random permutations avoiding the pattern 132. Combin. Probab. Comput. 26 2451.
[8] Miner, S. and Pak, I. (2014) The shape of random pattern-avoiding permutations. Adv. Appl. Math. 55 86130.
[9] Pinsky, R. G. (2014) Problems from the Discrete to the Continuous: Probability, Number Theory, Graph Theory, and Combinatorics, Universitext, Springer.
[10] Pitman, J. and Tang, W. (2019) Regenerative random permutations of integers. Ann. Probab. 47 13781416.

# The Infinite limit of random permutations avoiding patterns of length three

## Metrics

### Full text viewsFull text views reflects the number of PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 0 *