Cyclic Pattern Containment and Avoidance

Abstract

The study of pattern containment and avoidance for linear permutations is a well-established area of enumerative combinatorics. A cyclic permutation is the set of all rotations of a linear permutation. Callan initiated the study of permutation avoidance in cyclic permutations and characterized the avoidance classes for all single permutations of length 4. We continue this work. In particular, we establish a cyclic variant of the Erdős-Szekeres Theorem that any linear permutation of length mn + 1 must contain either the increasing pattern of length m + 1 or the decreasing pattern of length n + 1. We then derive results about avoidance of multiple patterns of length 4. We also determine generating functions for the cyclic descent statistic on these classes. Finally, we end with various open questions and avenues for future research.

Vincular Pattern Avoidance on Cyclic Permutations

“…In particular, in Section 4.1 we find one of the doubleton sets is equinumerous to the set of updown permutations, and in Section 4.2 we find the unique nonzero Wilf equivalence class of tripleton sets of patterns of length 3 is equinumerous to the set of total extensions of a certain partial cyclic order, a circular analog of a poset, for which a recurrence is known. In Section 5, we enumerate six of the eight trivial Wilf equivalence classes of vincular cyclic patterns of length 4 with a single vinculum, demonstrating that there are five Wilf equivalence classes; combining this with the result by Domagalski, Liang, Minnich, Sagan, Schmidt, and Sietsema [10], this leaves only one of the eight trivial Wilf equivalence classes unresolved. Notably, we find two more trivial Wilf equivalence classes that are enumerated by the Catalan numbers.…”

“…resolving the second half of [10,Conjecture 6.4]. Elizalde and Sagan [12,Corollary 4] also concurrently and independently resolved this second half of the conjecture by proving a more general result that implies Eq.…”

“…In particular, these vincular notions apply to cyclic patterns and permutations as well, without change. For example, Domagalski, Liang, Minnich, Sagan, Schmidt, and Sietsema [10] proved that Av n [1324] , the number of cyclic permutations of [n] avoiding [1324], equals C n−1 , where C n denotes the nth Catalan number.…”

“…This resolves the first half of [10,Conjecture 6.4], although we had to change the indices of the exponential generating function to use Av n+1 [123] in order for the conjectured differential equation to hold. Elizalde and Sagan [12, Corollary 2] concurrently and independently proved a more general result that implies Eq.…”

“…In 2002, Callan [7] initiated the study of pattern avoidance in cyclic permutations, enumerating the avoidance classes for all patterns of length 4. In 2011, Domagalski, Liang, Minnich, Sagan, Schmidt, and Sietsema [10] extended Callan's work, enumerating the avoidance classes for all sets of patterns of length 4, where a permutation avoids a set of patterns if it avoids all patterns in this set. Domagalski et al additionally initiated the study of vincular pattern avoidance in cyclic permutations, showing that the Catalan numbers appear as the sizes of a particular avoidance class of cyclic permutations.…”