Structure of random 312‐avoiding permutations

Abstract

We evaluate the probabilities of various events under the uniform distribution on the set of 312-avoiding permutations of 1, . . . , N . We derive exact formulas for the probability that the i th element of a random permutation is a specific value less than i, and for joint probabilities of two such events. In addition, we obtain asymptotic approximations to these probabilities for large N when the elements are not close to the boundaries or to each other. We also evaluate the probability that the graph of a random 312-avoiding permutation has k specified decreasing points, and we show that for large N the points below the diagonal look like trajectories of a random walk.

Large Deviations and Ratio Limit Theorems for Pattern-Avoiding Permutations

“…Complementing this result, Proposition 4.2 shows that the number of points well below the diagonal is o(N) with high probability. Madras and Pehlivan [16] consider the properties of random 312-avoiding permutations below the diagonal in more detail. Theorem 1.2 also follows directly from recent independent work of Miner and Pak [19], who investigate fine asymptotics of P τ N (π i = j) for τ ∈ S 3 .…”

“…In fact, they showed that L(4231) 9.47 while (k − 1) 2 = 9 for k = 4. For several years, the best published upper bound for L(4231) was 288 [7], until Claesson, Jelínek and Steingrímsson [11] recently proved that L(4231) 16; a more recent preprint of Bóna [9] lowers the upper bound to 7 + 4 √ 3 ≈ 13.93. So it remains an active open problem to find the exact value or an accurate estimate of L(4231).…”

Permutations with fixed pattern densities

“…For densities in the interior of we study the shape of a typical permutation with those densities, again in the large n limit. We note that the typical shape of pattern‐avoiding permutations (which necessarily lie on the boundary of the feasible region ) has also recently been investigated .…”

“…Specifically, we consider the densities of one or more patterns and consider the feasible region, or phase space F, of possible values of densities of the chosen patterns for permutations in S n in the limit of large n. For densities in the interior of F we study the shape of a typical permutation with those densities, again in the large n limit. We note that the typical shape of pattern-avoiding permutations (which necessarily lie on the boundary of the feasible region F) has also recently been investigated [1,10,15,[26][27][28].…”

Patterns in random permutations avoiding the pattern 321

“…There is also a number of papers studying other properties of random -avoiding permutations. Some examples, in addition to those mentioned above, are consecutive patterns [6]; descents and the major index [5]; number of fixed points [21,22,25,26,39,44]; position of fixed points [25,26,39]; exceedances [21,22]; longest increasing subsequence [19]; shape and distribution of individual values i [24,37,38].…”