Counting occurrences of a pattern of type (1,2) or (2,1) in permutations

Abstract

Babson and Steingrímsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Claesson presented a complete solution for the number of permutations avoiding any single pattern of type (1, 2) or (2, 1). For eight of these twelve patterns the answer is given by the Bell numbers. For the remaining four the answer is given by the Catalan numbers.With respect to being equidistributed there are three different classes of patterns of type (1, 2) or (2, 1). We present a recursion for the number of permutations containing exactly one occurrence of a pattern of the first or the second of the aforementioned classes, and we also find an ordinary generating function for these numbers. We prove these results both combinatorially and analytically. Finally, we give the distribution of any pattern of the third class in the form of a continued fraction, and we also give explicit formulas for the number of permutations containing exactly r occurrences of a pattern of the third class when r ∈ {1, 2, 3}. and s 1 132 (n) = 2n−3 n−3 . The latter conjecture was proved by Bóna in [7]. A conjecture of Noonan and Zeilberger states that s r σ (n) is P -recursive in n for any r and τ . It was proved by Bóna [5] for σ = 132. Mansour and Vainshtein [14] suggested a new approach to this problem in the case σ = 132, which allows one to get an explicit expression for s r 132 (n) for any given r. More precisely, they presented an algorithm that computes the generating function Date: October 31, 2018.

Permutation Tableaux and the Dashed Permutation Pattern 32–1

“…This givesπ = (6,5,7,4,9,2,3,8,1). It can be checked that the number of occurrences of the dashed pattern 32-1 inπ equals 8.…”

“…It follows that inv(T ) = f 3-21 (π), which is equivalent to the statement of the theorem. (9,2,7,8,1,6,5,3,4).…”

“…For example, we say that a permutation π on [n] avoids a dashed pattern 32-1 if there are no subscripts i < k such that π i−1 > π i > π k . Claesson and Mansour [4] found explicit formulas for the number of permutations containing exactly i occurrences of a dashed pattern σ of length 3 for i = 1, 2, 3.…”

New permutation statistics: Variation and a variant

“…For example, the pattern 41523 ∈ S 5 has three occurrences of the pattern 31 − 2, namely 412, 413 and 523. Several papers deal with the enumeration of the number of permutations with fixed number of occurrences of a generalized pattern ab − c (for instance, see [2][3][4]). By definition, the variation statistic can be characterized in terms of counting occurrences of generalized patterns 31 − 2 and 13 − 2 as follows:…”

Equidistributions of Mahonian Statistics over Pattern Avoiding Permutations

“…11. A permutation σ ∈ S n (231) is uniquely determined by any of the following data:(i) The values and positions of right-to-left minima.…”