Asymptotic bounds for permutations containing many different patterns

Abstract

We say that a permutation σ ∈ S n contains a permutation π ∈ S k 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 2 n − O (n 2 2 n− √ 2n ) pat(n) 2 n − Θ(n2 n− √ 2n ) by estimating the amount of redundancy due to patterns that are contained multiple times in a given permutation. We also consider the question of ksuperpatterns, which are permutations that contain all patterns of a given length k. We give a simple construction that shows that L k , the length of the shortest k-superpattern, is at most. This may lend evidence to a conjecture of Eriksson et al. that L k ∼ k 2 2 .

Superpatterns and Universal Point Sets

“…Any two point sets with the same two sorted orders may be used as the basis for a dominance drawing combinatorially equivalent to D. In particular, if π D appears as a pattern in another permutation σ, then the subset of the points (i, σ i ) corresponding to elements of π D may be used to draw the same graph. This gives us the following result: Combining this result with Miller's bound on superpatterns [16] shows that dominance drawings have universal point sets of size n 2 /2 + Θ(n), half the size of the point sets given by previous methods based on n × n grids.…”

“…For instance, {3, 4, 5} forms a block in 14352. (Our definition of rows and columns differs from that of Miller [16]: for our definition, the intersection of a row and column is a block that could contain more than one element, whereas in Miller's definition a row and column intersect in at most one element. )…”

“…2 A permutation σ is said to contain the pattern π (also a permutation) if σ has a (not necessarily contiguous) subsequence whose elements are in the same relative order with respect to each other as the elements of π. The permutations that do not contain any pattern in a given set F of forbidden patterns are said to be F-avoiding; we define S n (F) to be the length-n permutations avoiding F. Researchers in permutation patterns have defined a superpattern to be a permutation that contains all length-n permutations among its patterns, and have studied bounds on the lengths of these patterns [14,15], culminating in a proof by Miller that there exist superpatterns of length n 2 /2 + Θ(n) [16]. We generalize this concept to an S n (F)-superpattern, a permutation that contains all possible patterns in S n (F); we prove that for certain sets F, the S n (F)-superpatterns are much shorter than Miller's bound.…”

Small Superpatterns for Dominance Drawing

“…The chessboard representation of a permutation σ is a matrix M where M i,j is the number of elements of σ belonging to the intersection of the i th row and j th column. (This differs from a related definition of the chessboard representation by Miller [26], using descending rows rather than ascending rows, for which the intersection of a row and a column can contain at most one element.) As defined here, the chessboard representation respects the dominance relations in plot(σ).…”

Universal Layered Permutations

“…To date, the best bounds on the length of the shortest n-universal permutation are that it lies between n 2 {e 2 (a consequence of Stirling's Formula, because if such a permutation has length m, the inequality`m n˘ě n! must hold) and`n`1 2˘, which was established by Miller [5]. Even before Miller's result was established, Eriksson, Eriksson, Linusson, and Wästlund [2] conjectured that this length is asymptotic to n 2 {2.…”