Refined sign-balance on 321-avoiding permutations

AbstractThe number of even 321-avoiding permutations of length n is equal to the number of odd ones if n is even, and exceeds it by the n−12th Catalan number otherwise. We present an involution that proves a refinement of this sign-balance property respecting the length of the longest increasing subsequence of the permutation. In addition, this yields a combinatorial proof of a recent analogous result of Adin and Roichman dealing with the last descent. In particular, we answer the question of how to obtain the sign of a 321-avoiding permutation from the pair of tableaux resulting from the Robinson–Schensted–Knuth algorithm. The proof of the simple solution is based on a matching method given by Elizalde and Pak

Random walk generated by random permutations of {1,2,3, ..., n+1}

We study properties of a non-Markovian random walk $X^{(n)}_l$, $l =0,1,2, >...,n$, evolving in discrete time $l$ on a one-dimensional lattice of integers, whose moves to the right or to the left are prescribed by the \text{rise-and-descent} sequences characterizing random permutations $\pi$ of $[n+1] = \{1,2,3, ...,n+1\}$. We determine exactly the probability of finding the end-point $X_n = X^{(n)}_n$ of the trajectory of such a permutation-generated random walk (PGRW) at site $X$, and show that in the limit $n \to \infty$ it converges to a normal distribution with a smaller, compared to the conventional P\'olya random walk, diffusion coefficient. We formulate, as well, an auxiliary stochastic process whose distribution is identic to the distribution of the intermediate points $X^{(n)}_l$, $l < n$, which enables us to obtain the probability measure of different excursions and to define the asymptotic distribution of the number of "turns" of the PGRW trajectories.Comment: text shortened, new results added, appearing in J. Phys.

The feasible regions for consecutive patterns of pattern-avoiding permutations

We study the feasible region for consecutive patterns of pattern-avoiding permutations. More precisely, given a family $\mathcal C$ of permutations avoiding a fixed set of patterns, we study the limit of proportions of consecutive patterns on large permutations of $\mathcal C$. These limits form a region, which we call the \emph{pattern-avoiding feasible region for $\mathcal C$}. We show that, when $\mathcal C$ is the family of $\tau$-avoiding permutations, with either $\tau$ of size three or $\tau$ a monotone pattern, the pattern-avoiding feasible region for $\mathcal C$ is a polytope. We also determine its dimension using a new tool for the monotone pattern case, whereby we are able to compute the dimension of the image of a polytope after a projection. We further show some general results for the pattern-avoiding feasible region for any family $\mathcal C$ of permutations avoiding a fixed set of patterns, and we conjecture a general formula for its dimension. Along the way, we discuss connections of this work with the problem of packing patterns in pattern-avoiding permutations and to the study of local limits for pattern-avoiding permutations.Comment: New version before submission to journa