Picture map of Europe with pins indicating European capital cities

Open Access research with a European policy impact...

The Strathprints institutional repository is a digital archive of University of Strathclyde's Open Access research outputs. Strathprints provides access to thousands of Open Access research papers by Strathclyde researchers, including by researchers from the European Policies Research Centre (EPRC).

EPRC is a leading institute in Europe for comparative research on public policy, with a particular focus on regional development policies. Spanning 30 European countries, EPRC research programmes have a strong emphasis on applied research and knowledge exchange, including the provision of policy advice to EU institutions and national and sub-national government authorities throughout Europe.

Explore research outputs by the European Policies Research Centre...

Quadrant marked mesh patterns in 132-avoiding permutations II

Kitaev, Sergey and Remmel, Jeffrey and Tiefenbruck, Mark (2015) Quadrant marked mesh patterns in 132-avoiding permutations II. Integers: Electronic Journal of Combinatorial Number Theory, 15. ISSN 1553-1732

[img]
Preview
Text (Kitaev-etal-IEJCNT-2015-Quadrant-marked-mesh-patterns-in-132-avoiding)
Kitaev_etal_IEJCNT_2015_Quadrant_marked_mesh_patterns_in_132_avoiding.pdf - Accepted Author Manuscript

Download (337kB) | Preview

Abstract

Given a permutation σ = σ1...... σn in the symmetric group Sn, we say that σi matches the marked mesh pattern MMP (a,b,c,d) in σ if there are at least a points to the right of σi in σ which are greater than σi, at least b points to the left of σi in σ which are greater than σi, at least c points to the left of σi in σ which are smaller than σi, and at least d points to the right of σi in σ which are smaller than σi. This paper is continuation of the systematic study of the distributions of quadrant marked mesh patterns in 132-avoiding permutations started by the present authors where we mainly studied the distribution of the number of matches of MMP(a,b,c,d) in 132-avoiding permutations where exactly one of a,b,c,d is greater than zero and the remaining elements are zero. In this paper, we study the distribution of the number of matches of MMP(a,b,c,d) in 132-avoiding permutations where exactly two of a,b,c,d are greater than zero and the remaining elements are zero. We provide explicit recurrence relations to enumerate our objects which can be used to give closed forms for the generating functions associated with such distributions. In many cases, we provide combinatorial explanations of the coefficients that appear in our generating functions. The case of quadrant marked mesh patterns MMP(a,b,c,d) where three or more of a,b,c,d are constrained to be greater than 0 will be studied in a future article by the present authors.