Baxter Permutations, Snow Leopard Permutations, and Domino Tilings of Aztec Diamonds Ben Caffrey, Gregory Michel, Kailee Rubin, & Jonathan Ver Steegh∗ March 3, 2014 Abstract In [2], Elkies, Kuperberg, Larsen, and Propp introduce a bijection between domino tilings of Aztec diamonds and pairs of alternating sign matrices whose sizes differ by one. In this paper we first study those smaller permutations (viewed as matrices) which are paired with doubly alternating Baxter permutations. We call these permutations snow leopard permutations, and we use a recursive decomposition to show they are counted by the Catalan numbers. We also describe a specific set of transpositions which generates these permutations. Lastly, we investigate the general case, where we conjecture that the number of larger permutations paired with a given smaller permutation is a product of Fibonacci numbers. 1 Introduction and Background An alternating sign matrix (ASM) is an n × n matrix with entries in f0; 1; −1g whose nonzero entries in each row and column alternate in sign and sum to 1. An Aztec diamond of order n is a two dimensional array of unit squares with 2i squares in rows i ≤ n and 2(2n − i + 1) squares in rows n < i ≤ 2n; in which the squares are centered on each row. Figure1 shows the Aztec diamond of order 3. Figure 1: The Aztec diamond of order 3. Aztec diamonds can be tiled using 2×1 domino rectangles, which is to say they can be completely covered by disjoint dominoes whose union is the entire diamond. We abbreviate a tiling of an Aztec ∗Current address: c/o Eric Egge, Department of Mathematics, Carleton College, Northfield, MN 55057; Email: [email protected] 1 diamond as a TOAD. In [2], Elkies, Kuperberg, Larsen, and Propp describe how to construct, for each TOAD of order n, a pair of ASMs of size n × n and (n + 1) × (n + 1). We can describe this construction most easily with an example. In Figure2, the vertices that compose the tiled Aztec diamond fall naturally into two matrices: the red vertices form an (n + 1) × (n + 1) matrix while the blue vertices form an n × n matrix. The tops of these matrices align with the northeast edge of the diamond. To construct an ASM on the red vertices, label each vertex of degree 4 with a 1, each vertex with degree 3 with a 0, and each vertex with degree 2 with a −1. The n × n matrix on the blue vertices can be constructed in the same way except the degree 4 and degree 2 rules are reversed. Given a TOAD T , we write LASM(T ) to denote the larger of the ASMs and SASM(T ) to denote the smaller of the two ASMs. Following [2], we say an (n + 1) × (n + 1) ASM A and an n × n ASM B are compatible whenever there exists a proper tiling T such that A = LASM(T ) and B = SASM(T ). Figure 2: A domino tiling of the Aztec diamond of order 3 According to the rules defined above, the TOAD in Figure2 has 20 0 0 13 20 0 13 1 0 0 0 LASM(T ) = 6 7 ; SASM(T ) = 1 0 0 : 60 0 1 07 4 5 4 5 0 1 0 0 1 0 0 In one-line notation, these matrices refer to the permutations 4132 and 312, respectively. In [2], Elkies, Kuperberg, Larsen, and Propp show that an order n+1 ASM with k entries equal to −1 is compatible with 2k ASMs of size n. As a corollary, each permutation matrix, which is an ASM with no −1 entries, is compatible with exactly one smaller ASM. Canary [1] gives an algorithm to construct the unique smaller ASM compatible with any given permutation. To implement this algorithm, suppose π is a permutation of length n + 1. Label the red vertices in a diagram for an Aztec diamond of the appropriate size with the entries of the matrix for π. For each blue vertex, if the two red vertices immediately to the northwest, and all red vertices northwest of those, are labeled with 0, then label the blue vertex 0. Now repeat this process in each of the other three directions (northeast, southeast, and southwest). Canary shows that each row and column of blue vertices will now contain an odd number of unlabeled vertices; there is a unique way to label these vertices with 1s and −1s to create an ASM. 2 Canary proves that this smaller ASM will also be a permutation matrix if and only if the original permutation matrix is Baxter. A Baxter permutation is a permutation that avoids 2 − 41 − 3 and avoids 3 − 14 − 2, which means that, in one-line notation, it has no subsequence ai; aj; aj+1; ak such that aj+1 < ai < ak < aj (for 3 − 14 − 2) or aj < ak < ai < aj+1 (for 2 − 41 − 3). For example, 174962835 is not Baxter because the subsequence 4−62−5 is an instance of 2−41−3. In contrast, 879164325 is Baxter because it contains no instances of 2 − 41 − 3 or 3 − 14 − 2. An interesting class of Baxter permutations are the doubly alternating Baxter permutations. We call a permutation π = a1a2 ··· an alternating whenever ai < ai+1 if i is odd and ai > ai+1 if i is even. This is to say that π starts with an ascent and the ascents and descents alternate. A doubly alternating permutation is an alternating permutation whose inverse is also alternating. We call permutations that are both doubly alternating and Baxter doubly alternating Baxter permutations th (DABPs). Guibert and Linusson show in [3] that Cn, the n Catalan number, counts both the DABPs of length 2n and the DABPs of length 2n + 1. The Catalan numbers are ubiquitous in 1 2n combinatorics and have a number of definitions. The simplest of these is the formula Cn = n+1 n , but they can also be thought of as the number of Catalan paths, which are paths from (0; 0) to (n; n) along an n × n square grid using north and east steps and not crossing below the main diagonal. Pn The Catalan numbers also satisfy the recursive definition C0 = 1 and Cn+1 = j=1 Cj−1Cn−j for n ≥ 0. In this paper, we introduce snow leopard permutations (SLPs), which are the permutations whose matrices are compatible with doubly alternating Baxter permutation matrices. In Section 2, we characterize these permutations recursively, and by doing so, we show that the compatibility relation in this case is one-to-one, which implies that the snow leopard permutations of length 2n are also counted by Cn, as are the snow leopard permutations of length 2n + 1. Then, we give a bijection between snow leopard permutations and Catalan paths in Section 3, and in Section 4 we describe a construction of snow leopard permutations based on transpositions. Lastly, in Section 5 we show that the identity permutation of length n is compatible with Fn permutations of length th n + 1, where Fn denotes the n Fibonacci number. We conclude by conjecturing that the number of length n + 1 permutations with which a length n permutation is compatible is a product of Fibonacci numbers. 2 Snow Leopard Permutations: a Recursive Characterization In order to construct snow leopard permutations recursively, it is important to first understand how to construct DABPs and TOADs recursively. To understand this process, we first describe how to decompose DABPs and TOADs into smaller DABPs and TOADs, respectively. 2.1 Definitions and Notation Before we begin, we introduce some key definitions and specific notation for working with permu- tations in one-line notation. C Definition 2.1. The complement of a permutation π = a1 ··· an is π = n − a1 + 1 n − a2 + 1 ··· n − an + 1. Lemma 2.2. The complement of a Baxter permutation is Baxter. 3 Proof. Assume π is a Baxter permutation. Since the patterns 2 − 41 − 3 and 3 − 14 − 2 are complements of each other, πC will avoid 2 − 41 − 3 since π avoids 3 − 14 − 2 and vice versa. Definition 2.3. For any two sequences of positive integers, π1 = x1 ··· xi and π2 = y1 ··· yj, we write π1 ⊕ π2 to denote the concatenated sequence x1 ··· xiy1 ··· yj. Definition 2.4. Given a sequence of positive integers π = x1 ··· xi, and two indices m and n with 1 ≤ m ≤ n ≤ i, we define πm!n to be the subsequence of consecutive entries given by xmxm+1 ··· xn. Definition 2.5. Given a sequence of positive integers π and a constant b, we define AUGC(π; b) to be the sequence of positive integers created by taking the complement of π and then increasing the value of each entry by b. Example 2.6. Let π1 = 14325 and π2 = 768. We have π1 ⊕ π2 = 14325768, π12!4 = 432, and AUGC(π1; 5) = 10 7 8 9 6. 2.2 Doubly Alternating Baxter Permutation Decomposition In this subsection, we describe a decomposition algorithm that subdivides a single DABP into two smaller DABPs. In order to build to this algorithm, we must first prove several facts about the structure of DABPs. We begin a Lemma of Ouchterlony from [4]. Lemma 2.7. Suppose π is a DABP of length n. Then the following hold. 1. If n is odd then π(1) = 1. 2. If n is even and π(1) = 1 then π(n) = n. 3. If n is even, π(1) > 1, and k is the smallest integer for which π(k) < π(1), then i < k ≤ j implies π(i) < π(j).