Equipopularity Classes in the Separable Permutations

Abstract

When two patterns occur equally often in a set of permutations, we say that these patterns are equipopular. Using both structural and analytic tools, we classify the equipopular patterns in the set of separable permutations. In particular, we show that the number of equipopularity classes for length n patterns in the separable permutations is equal to the number of partitions of n − 1.These operations are more naturally understood graphically: the graph of σ ⊕ τ (resp., σ ⊖ τ) is obtained by stacking the graph of τ above (resp., below) and to the right of that of σ. Both operations are individually, but not jointly, associative and are noncommutative. This paper is concerned with separable permutations which are defined as the class of all permutations that can be formed from the length one permutation, 1, by iterated applications of direct and skew sums. For instance, π = 543612 is separable, as:A permutation class is a set of permutations that forms a downset in the pattern ordering, i.e., a set C for which π ∈ C and σ ≺ π implies σ ∈ C. The class of all permutations is denoted S. Enumerating

The Brownian limit of separable permutations

“…• the algorithmic problem of PERMUTATION PATTERN MATCHING is NP-hard in general, but polynomial on separable permutations [13]; • from an enumerative combinatorics point of view, in addition to being simple to count, separable permutations display remarkable equipopularity properties, see [2]. Besides, separable permutations appear naturally in several problems, at first sight independent from permutation pattern theory:…”

Popularity of patterns over d-equivalence classes of words and permutations

“…When τ ωa < π ω i − 1, since the entry τ ωa occurs twice in τ we can find as previously a pattern τ (2) d-equivalent with τ (and so with π) where the entry τ ωa + 1 occurs twice in τ (2) . Iterating this procedure, we obtain a sequence of d-equivalent patterns τ = τ (1) , τ (2) , . .…”

“…u k is a word with the underlying alphabet [m], then there is a ([a, b], [c, d]), then the ([a, b], [c, d])-substitution by u in w is w itself, and we have the following easy to understand fact. See Example 5 where the ( [3,7], [1,4])-substitution by 3321 in w = 21143 61 5441 is v = 21443 61 5441 (the replaced elements are in italic and represented by × in the corresponding diagrams).…”

“…The equidistribution of two patterns implies their equipopularity and recently a growing interest is shown in patterns that have same popularity but not same distribution on particular classes of words or permutations, see for instance [1,2,5,8].…”

On the Number of Group-Separable Preference Profiles

“…Separable permutations are counted by the large Schröder numbers (see the different proofs in West 1995;Stankova 1994;Ehrenfeucht et al 1998). Separable permutations have different tree representations (West 1996;Bose et al 1998;Albert et al 2015;Kitaev 2011), which are applicable for related combinatorial problems.…”