Asymptotic bounds for permutations containing many different patterns

We say that a permutation σ∈Sn contains a permutation π∈Sk as a pattern if some subsequence of σ has the same order relations among its entries as π. We improve on results of Wilf, Coleman, and Eriksson et al. that bound the asymptotic behavior of pat(n), the maximum number of distinct patterns of any length contained in a single permutation of length n. We prove that by estimating the amount of redundancy due to patterns that are contained multiple times in a given permutation. We also consider the question of k-superpatterns, which are permutations that contain all patterns of a given length k. We give a simple construction that shows that Lk, the length of the shortest k-superpattern, is at most . This may lend evidence to a conjecture of Eriksson et al. that .     

q-Distributions on boxed plane partitions

We introduce elliptic weights of boxed plane partitions and prove that they give rise to a generalization of MacMahon’s product formula for the number of plane partitions in a box. We then focus on the most general positive degenerations of these weights that are related to orthogonal polynomials; they form three 2-D families. For distributions from these families, we prove two types of results. First, we construct explicit Markov chains that preserve these distributions. In particular, this leads to a relatively simple exact sampling algorithm. Second, we consider a limit when all dimensions of the box grow and plane partitions become large and prove that the local correlations converge to those of ergodic translation invariant Gibbs measures. For fixed proportions of the box, the slopes of the limiting Gibbs measures (that can also be viewed as slopes of tangent planes to the hypothetical limit shape) are encoded by a single quadratic polynomial.     

Clusters,generating functions and asymptotics for consecutive patterns in permutations

We use the cluster method to enumerate permutations avoiding consecutive patterns. We reprove and generalize in a unified way several known results and obtain new ones, including some patterns of lengths 4 and 5, as well as some infinite families of patterns of a given shape. By enumerating linear extensions of certain posets, we find a differential equation satisfied by the inverse of the exponential generating function counting occurrences of the pattern. We prove that for a large class of patterns, this inverse is always an entire function.We also complete the classification of consecutive patterns of length up to 6 into equivalence classes, proving a conjecture of Nakamura. Finally, we show that the monotone pattern asymptotically dominates (in the sense that it is easiest to avoid) all non-overlapping patterns of the same length, thus proving a conjecture of Elizalde and Noy for a positive fraction of all patterns.     

Asymptotic behaviour of the containment of certain mesh patterns

We present some results on the proportion of permutations of length n containing certain mesh patterns as n grows large, and give exact enumeration results in some cases. In particular, we focus on mesh patterns where entire rows and columns are shaded. We prove some general results which apply to mesh patterns of any length, and then consider mesh patterns of length four. An important consequence of these results is to show that the proportion of permutations containing a mesh pattern can take a wide range of values between 0 and 1.     

The q-exponential generating function for permutations by consecutive patterns and inversions

The inverse of Fedou's insertion-shift bijection is used to deduce a general form for the q-exponential generating function for permutations by consecutive patterns (overlaps allowed) and inversion number from a result due to Jackson and Goulden for enumerating words by distinguished factors. Explicit q-exponential generating functions are then derived for permutations by the consecutive patterns 12…m, 12…(m−2)m(m−1), 1m(m−1)…2, and by the pair of consecutive patterns (123,132).     

Multinomial permutations on a circle

Multinomial permutations on a circle are considered in the framework of combinatorics. Different cases are presented and shown to agree with previously derived formula for the number of cyclic necklaces. Two applied examples are discussed with a view to illustrate the implications of derived formulas. Copyright © 2006 John Wiley & Sons, Ltd.     

Non-nudgable subgroups of permutations

Motivated by a problem from behavioral economics, we study subgroups of permutation groups that have a certain strong symmetry. Given a fixed permutation, consider the set of all permutations with disjoint inversion sets. The group is called non-nudgable if the cardinality of this set always remains the same when replacing the initial permutation with its inverse. It is called nudgable otherwise. We show that all full permutation groups, standard dihedral groups, half of the alternating groups, and any abelian subgroup are non-nudgable. In the right probabilistic sense, it is thus quite likely that a randomly generated subgroup is non-nudgable. However, the other half of the alternating groups are nudgable. We also construct a smallest possible nudgable group, a 6-element subgroup of the permutation group on 4 elements.     

Cross-intersecting families of permutations

For positive integers r and n with r?n, let Pr,n be the family of all sets {(1,y₁),(2,y₂),…,(r,yr)} such that y₁,y₂,…,yr are distinct elements of [n]={1,2,…,n}. Pn,n describes permutations of [n]. For r<n, Pr,n describes permutations of r-element subsets of [n]. Families A₁,A₂,…,Ak of sets are said to be cross-intersecting if, for any distinct i and j in [k], any set in Ai intersects any set in Aj. For any r, n and k?2, we determine the cases in which the sum of sizes of cross-intersecting sub-families A₁,A₂,…,Ak of Pr,n is a maximum, hence solving a recent conjecture (suggested by the author).     

The pure descent statistic on permutations

We introduce a new statistic based on permutation descents which has a distribution given by the Stirling numbers of the first kind, i.e., with the same distribution as for the number of cycles in permutations. We study this statistic on the sets of permutations avoiding one pattern of length three by giving bivariate generating functions. As a consequence, new classes of permutations enumerated by the Motzkin numbers are obtained. Finally, we deduce results about the popularity of the pure descents in all these restricted sets.