Some statistics on restricted 132 involutions

In [GM] Guibert and Mansour studied involutions on n letters avoiding (or containing exactly once) 132 and avoiding (or containing exactly once) an arbitrary pattern on k letters. They also established a bijection between 132-avoiding involutions and Dyck word prefixes of same length. Extending this bijection to bilateral words allows to determine more parameters; in particular, we consider the number of inversions and rises of the involutions onto the words. This is the starting point for considering two different directions: even/odd involutions and statistics of some generalized patterns. Thus we first study generating functions for the number of even or odd involutions on n letters avoiding (or containing exactly once) 132 and avoiding (or containing exactly once) an arbitrary pattern $\tau$ on k letters. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind. Next, we consider other statistics on 132-avoiding involutions by counting an occurrences of some generalized patterns, related to the enumeration according to the number of rises.Comment: 22 page

Introduction to Partially Ordered Patterns

We review selected known results on partially ordered patterns (POPs) that include co-unimodal, multi- and shuffle patterns, peaks and valleys ((modified) maxima and minima) in permutations, the Horse permutations and others. We provide several (new) results on a class of POPs built on an arbitrary flat poset, obtaining, as corollaries, the bivariate generating function for the distribution of peaks (valleys) in permutations, links to Catalan, Narayna, and Pell numbers, as well as generalizations of few results in the literature including the descent distribution. Moreover, we discuss q-analogue for a result on non-overlapping segmented POPs. Finally, we suggest several open problems for further research.Comment: 23 pages; Discrete Applied Mathematics, to appea

Restricted ascent sequences and Catalan numbers

Ascent sequences are those consisting of non-negative integers in which the size of each letter is restricted by the number of ascents preceding it and have been shown to be equinumerous with the (2+2)-free posets of the same size. Furthermore, connections to a variety of other combinatorial structures, including set partitions, permutations, and certain integer matrices, have been made. In this paper, we identify all members of the (4,4)-Wilf equivalence class for ascent sequences corresponding to the Catalan number C_n=\frac{1}{n+1}\binom{2n}{n}. This extends recent work concerning avoidance of a single pattern and provides apparently new combinatorial interpretations for C_n. In several cases, the subset of the class consisting of those members having exactly m ascents is given by the Narayana number N_{n,m+1}=\frac{1}{n}\binom{n}{m+1}\binom{n}{m}.Comment: 12 page

Pattern Avoidance and the Bruhat Order

The structure of order ideals in the Bruhat order for the symmetric group is elucidated via permutation patterns. A method for determining non-isomorphic principal order ideals is described and applied for small lengths. The permutations with boolean principal order ideals are characterized. These form an order ideal which is a simplicial poset, and its rank generating function is computed. Moreover, the permutations whose principal order ideals have a form related to boolean posets are also completely described. It is determined when the set of permutations avoiding a particular set of patterns is an order ideal, and the rank generating functions of these ideals are computed. Finally, the Bruhat order in types B and D is studied, and the elements with boolean principal order ideals are characterized and enumerated by length.Comment: 18 pages, 7 figure