Continued fractions, statistics, and generalized patterns

Abstract

Recently, Babson and Steingrimsson (see \cite{BS}) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Following \cite{BCS}, let $e_k\pi$ (respectively; $f_k\pi$) be the number of the occurrences of the generalized pattern $12\mn3\mn...\mn k$ (respectively; $21\mn3\mn...\mn k$) in $\pi$. In the present note, we study the distribution of the statistics $e_k\pi$ and $f_k\pi$ in a permutation avoiding the classical pattern $1\mn3\mn2$. Also we present an applications, which relates the Narayana numbers, Catalan numbers, and increasing subsequences, to permutations avoiding the classical pattern $1\mn3\mn2$ according to a given statistics on $e_k\pi$, or on $f_k\pi$.

8 pages

References in corpus (3)

• Generalised Pattern Avoidance
• On multi-avoidance of generalized patterns
• Permutations with restricted patterns and Dyck paths

Cited by in corpus (7)

• Simultaneous avoidance of generalized patterns
• On multi-avoidance of generalized patterns
• Generalized pattern avoidance with additional restrictions
• Some statistics on restricted 132 involutions
• Restricted 132-alternating permutations and Chebyshev polynomials
• Restricted 132-Dumont permutations
• Restricted Motzkin permutations, Motzkin paths, continued fractions, and Chebyshev polynomials