(a,b)rectangle patterns in permutations and words
Abstract
In this paper, we introduce the notion of a $(a,b)$rectangle pattern on permutations that not only generalizes the notion of successive elements (bonds) in permutations, but is also related to mesh patterns introduced recently by Brändén and Claesson. We call the $(k,k)$rectangle pattern the $k$box pattern. To provide an enumeration result on the maximum number of occurrences of the 1box pattern, we establish an enumerative result on patternavoiding signed permutations. Further, we extend the notion of $(k,\ell)$rectangle patterns to words and binary matrices, and provide distribution of $(1,\ell)$rectangle patterns on words; explicit formulas are given for up to 7 letter alphabets where $\ell \in \{1,2\}$, while obtaining distributions for larger alphabets depends on inverting a matrix we provide. We also provide similar results for the distribution of bonds over words. As a corollary to our studies we confirm 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 also enumerate two sequences published by Hardin in the OnLine Encyclopedia of Integer Sequences.
 Publication:

arXiv eprints
 Pub Date:
 April 2013
 arXiv:
 arXiv:1304.4286
 Bibcode:
 2013arXiv1304.4286K
 Keywords:

 Mathematics  Combinatorics