Picture of neon light reading 'Open'

Discover open research at Strathprints as part of International Open Access Week!

23-29 October 2017 is International Open Access Week. The Strathprints institutional repository is a digital archive of Open Access research outputs, all produced by University of Strathclyde researchers.

Explore recent world leading Open Access research content this Open Access Week from across Strathclyde's many research active faculties: Engineering, Science, Humanities, Arts & Social Sciences and Strathclyde Business School.

Explore all Strathclyde Open Access research outputs...

(a,b)-rectangle patterns in permutations and words

Kitaev, Sergey and Remmel, Jeffrey (2015) (a,b)-rectangle patterns in permutations and words. Discrete Applied Mathematics, 186. pp. 128-146. ISSN 0166-218X

[img] PDF (Kitaev-Remmel-DAM-2015-a-b-rectangle-patterns-in-permutations-and-words)
Kitaev_Remmel_DAM_2015_a_b_rectangle_patterns_in_permutations_and_words.pdf - Accepted Author Manuscript
License: Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 logo

Download (168kB)

Abstract

In this paper, we introduce the notion of an (a,b)(a,b)-rectangle pattern on permutations which is closely related to the notion of successive elements (bonds) in permutations and to mesh patterns introduced recently by Brändén and Claesson. We call the (k,k)(k,k)-rectangle pattern the kk-box pattern. We show that we can derive an explicit formula for the number of permutations of SnSn which have the maximum possible occurrences of the 1-box pattern by using a new enumerative result on pattern-avoidance in signed permutations. We also study the notion of (a,b)(a,b)-rectangle patterns in words. In particular, we give a general method for computing the generating function for the distribution of (1,b)(1,b)-rectangle patterns on words over an alphabet of size kk for b∈{1,2}b∈{1,2}. Our method requires to invert a certain matrix depending on bb and kk, and can be used to give explicit formulas for such generating functions for k=2,…,7k=2,…,7. We also provide similar results for the distribution of bonds over words. As a corollary to our studies, we prove a conjecture of Mathar on the number of “stable LEGO walls” of width 7, as well as prove three conjectures due to Hardin and a conjecture due to Barker. We provide generating functions for two sequences published by Hardin in the On-Line Encyclopedia of Integer Sequences.