Consecutive patterns in permutations

Abstract

In their study of cyclic pattern containment, Domagalski et al. [4] conjecture differential equations for the generating functions of circular permutations avoiding consecutive patterns of length 3. In this note, we prove and significantly generalize these conjectures. We show that, for every consecutive pattern σ beginning with 1, the bivariate generating function counting occurrences of σ in circular permutations can be obtained from the generating function counting occurrences of σ in (linear) permutations. This includes all the patterns for which the latter generating function is known.

Limit distribution of the number of consecutive records

“…For simplicity, we call such occurrences (possibly overlapping) r consecutive records. For some recent literature on other consecutive patterns in permutations, see Johnson [13], Elizalde and Noy [9], and the references therein.…”

Classifying Descents According to Parity

“…There have been many papers in the literature that have studied the number of τ-avoiding permutations of S n or the distribution of τ-matches for S n (e.g., see [2,6,7] and references therein). Now we can generalize the notion of τ-avoiding permutations or τ-matches by adding parity type conditions.…”

Generating Trees for Permutations Avoiding Generalized Patterns

“…Note that for the classes that we consider in this section, the enumeration of the permutations by their length has already been done by different authors [8,9,11,12,15,18]. Our contribution is a refined enumeration of these permutations by several parameters, and also the fact that our results are obtained using the unifying framework of rightward generating trees.…”

“…We omit this result here because a more direct way to enumerate these permutations was already given in [12].…”