The 4 × 8-rectangle centered at the point (4, 5) in the graph of σ = 471569283. 

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... this paper, we study 1-box patterns, a particular case of (a, b)-rectangular patterns in- troduced in [7]. That is, let σ = σ 1 · · · σ n be a permutation written in one-line notation. Then we will consider the graph of σ, G(σ), to be the set of points (i, σ i ) for i = 1, . . . , n. For example, the graph of the permutation σ = 471569283 is pictured in Figure 1. Then if we draw a coordinate system centered at a point (i, σ i ), we will be interested in the points that lie in the 2a×2b rectangle centered at the origin. That is, the (a, b)-rectangle pattern centered at (i, σ i ) equals the set of points (i ± r, σ i ± s) such that r ∈ {0, . . . , a} and s ∈ {0, . . . , b}. Thus σ i matches the (a, b)-rectangle pattern in σ, if there is at least one point in the 2a × 2b-rectangle centered at the point (i, σ i ) in G(σ) other than (i, σ i ). For example, when we look for matches of the (2,3)-rectangle patterns, we would look at 4 × 6 rectangles centered at the point (i, σ i ) as pictured in Figure 2. We shall refer to the (k, k)-rectangle pattern as the k-box pattern. For example, if σ = 471569283, then the 2-box centered at the point (4, 5) in G(σ) is the set of circled points pictured in Figure 3. Hence, σ i matches the k-box pattern in σ, if there is at least one point in the k-box centered at the point (i, σ i ) in G(σ) other than (i, σ i ). For example, σ 4 matches the pattern k-box for all k ≥ 1 in σ = 471569283 since the point (5,6) is present in the k-box centered at the point (4, 5) in G(σ) for all k ≥ 1. However, σ 3 only matches the k-box pattern in σ = 471569283 for k ≥ 3 since there are no points in 1-box or 2-box centered at (3, 1) in G(σ), but the point (1, 4) is in the 3-box centered at (3, 1) in G(σ). For k ≥ 1, we let k-box(σ) denote the set of all i such that σ i matches the k-box pattern in σ = σ 1 · · · σ n . Note that σ i matches the 1-box pattern in σ if either |σ i − σ i+1 | = 1 or |σ i−1 − σ i | = 1. For example, the distribution of 1-box(σ) for S 2 , S 3 , and S 4 is given below, where S n is the set of all permutations of length n. (σ) 12 2 21 2 σ 1-box(σ) 123 3 132 2 213 2 231 2 312 2 321 3 The notion of k-box patterns is related to the mesh patterns introduced by Brändén and Claesson [2] to provide explicit expansions for certain permutation statistics as, possibly infinite, linear combinations of (classical) permutation patterns. This notion was further studied in [1,4,5,8,9,10,12]. In particular, Kitaev and Remmel [5] initiated the system- atic study of distribution of marked mesh patterns on permutations, and this study was extended to 132-avoiding permutations by Kitaev, Remmel and Tiefenbruck in ...