an example which gives the flavour of our approach without requiring much additional machinery.

Example \#1: Avoiding 4213 and 3142 {#sec-4213-3142}
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Before describing our first example we need to introduce a family of simple permutations and quote a result. A [*parallel alternation*]{} is a permutation whose plot can be divided into two parts, by a single horizontal or vertical line, so that the points on either side of this line are both either increasing or decreasing and for every pair of points from the same part there is a point from the other part which [*separates*]{} them, i.e., there is a point from the other part which lies either horizontally or vertically between them. It is easy to see that a parallel alternation of length at least four is simple if and only if its length is even, it does not begin with its smallest entry, and it does not end with its greatest entry. Thus there are precisely four simple parallel alternations of each even length at least six, shown in Figure \[fig-par-alts\], and no simple parallel alternations of odd length.

[ccccccc]{}

(0,0)(80,80) (0,0)(80,80) (10,20)[0.04in]{} (20,40)[0.04in]{} (30,60)[0.04in]{} (40,80)[0.04in]{} (50,10)[0.04in]{} (60,30)[0.04in]{} (70,50)[0.04in]{} (80,70)[0.04in]{}

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(0,0)(80,80) (0,0)(80,80) (10,50)[0.04in]{} 