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function H satisfying (i) and (ii), we can place a domino covering every edge wv of G with |H(u) — H(v)| = 3, obtaining thereby a tiling of the Aztec diamond, which will coincide with the original tiling T in the event H = Hr. Thus there is a bijection between tilings of the Aztec diamond and height functions H(.) on the graph G that satisfy (i) and (ii). For a geometric interpretation of H(-), see f21).

Figure 2 function H satisfying (i) and (ii), we can place a domino covering every edge wv of G with |H(u) — H(v)| = 3, obtaining thereby a tiling of the Aztec diamond, which will coincide with the original tiling T in the event H = Hr. Thus there is a bijection between tilings of the Aztec diamond and height functions H(.) on the graph G that satisfy (i) and (ii). For a geometric interpretation of H(-), see f21).

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Fig. 3. Extending a tiling to the whole plane. is an edge of G whose standard orientation is from u to v. Call v = (a,b) a boundary vertex of G if |a| + |b] = n or n +1, and let the boundary cycle be the closed zigzag path (—n — 1, 0), (—n, 0), (—n, 1), (-n +1, 1), (-n +1, 2),..., (-1, n), (0, n), (0, n+1), (0, n), (1, n),..., (n+1, 0),..., (0, -n—-1),..., (-n- 1,0). Call the vertex v = (a, b) even if it is the upper-left corner of a white square (i.e., if a+b+n+1 is even) and odd otherwise, so that in particular the four corner vertices (—n — 1, 0), (n + 1, 0), (0, -n — 1), (0, n + 1) are even.
Fig. 3. Extending a tiling to the whole plane. is an edge of G whose standard orientation is from u to v. Call v = (a,b) a boundary vertex of G if |a| + |b] = n or n +1, and let the boundary cycle be the closed zigzag path (—n — 1, 0), (—n, 0), (—n, 1), (-n +1, 1), (-n +1, 2),..., (-1, n), (0, n), (0, n+1), (0, n), (1, n),..., (n+1, 0),..., (0, -n—-1),..., (-n- 1,0). Call the vertex v = (a, b) even if it is the upper-left corner of a white square (i.e., if a+b+n+1 is even) and odd otherwise, so that in particular the four corner vertices (—n — 1, 0), (n + 1, 0), (0, -n — 1), (0, n + 1) are even.
Fig. 5. Height functions of the all-horizontals and all-verticals tilings.
Fig. 5. Height functions of the all-horizontals and all-verticals tilings.
Note that the matrix elements on the boundary of A’ and B’ are independent of the particular tiling T. Also note that in both matrices consecutive elements in any row or column differ by exactly 2. Therefore, under suitable normalization A’ and B’ can be seen as skewed summations of alternating-sign matrices A and B. Specifically, by setting a}, = (a;; — 1)/2 and bj; = b;,/2, we arrive at matrices A*, B*, which, under the inverse of the skewed summation operation, yield the for 0 <i, 7 <n +1; thus, the tiling of Figure 6 gives the matrices
Note that the matrix elements on the boundary of A’ and B’ are independent of the particular tiling T. Also note that in both matrices consecutive elements in any row or column differ by exactly 2. Therefore, under suitable normalization A’ and B’ can be seen as skewed summations of alternating-sign matrices A and B. Specifically, by setting a}, = (a;; — 1)/2 and bj; = b;,/2, we arrive at matrices A*, B*, which, under the inverse of the skewed summation operation, yield the for 0 <i, 7 <n +1; thus, the tiling of Figure 6 gives the matrices
Conversely, A and B determine A’ and B’, which determine Hy, which determines 7 matrices A, B that we desire:
Conversely, A and B determine A’ and B’, which determine Hy, which determines 7 matrices A, B that we desire:
Fig. 6. An Aztec tiling converts into two alternating-sign matrices.
Fig. 6. An Aztec tiling converts into two alternating-sign matrices.
Thus, in all but one of the six possible cases for the submatrix The legality constraint (ii) from the Section 2 tells us that for 1 <i, j <n, the internal entries 6, of the matrix B’ must be equal to Also note that each such vertex is incident with 2, 3, or 4 dominoes belonging to the tiling T; if we mark each such site with a 1, 0, or —1 (respectively), we get the entries of A, where the upper-left corner of each matrix corresponds to positions near the left corner of the diamond. Similarly, the odd vertices of the Aztec diamond (including those on the boundary) form a tilted (n + 1)-by-(n + 1) square. If we mark each such site with a —1, 0, or 1 according to whether it is incident with 2, 3, or 4 dominoes of the extended tiling T+, we get the entries of B. (We omit the proof that this construction agrees with the one we gave earlier, since it is only the first one that we actually need.) Thea lagality ennctraint (11\ fram the Cactinn 9 telle ne that far 1o“3 fg om
Thus, in all but one of the six possible cases for the submatrix The legality constraint (ii) from the Section 2 tells us that for 1 <i, j <n, the internal entries 6, of the matrix B’ must be equal to Also note that each such vertex is incident with 2, 3, or 4 dominoes belonging to the tiling T; if we mark each such site with a 1, 0, or —1 (respectively), we get the entries of A, where the upper-left corner of each matrix corresponds to positions near the left corner of the diamond. Similarly, the odd vertices of the Aztec diamond (including those on the boundary) form a tilted (n + 1)-by-(n + 1) square. If we mark each such site with a —1, 0, or 1 according to whether it is incident with 2, 3, or 4 dominoes of the extended tiling T+, we get the entries of B. (We omit the proof that this construction agrees with the one we gave earlier, since it is only the first one that we actually need.) Thea lagality ennctraint (11\ fram the Cactinn 9 telle ne that far 1o“3 fg om
Table 1. From domino tilings to alternating-sign matrices. where N_(-) gives the number of —1’s in an alternating-sign matrix. Replacing n by n—1 and B by A in (2), we get
Table 1. From domino tilings to alternating-sign matrices. where N_(-) gives the number of —1’s in an alternating-sign matrix. Replacing n by n—1 and B by A in (2), we get
Fig. 9. The tray in which the cubes sit. Let A* be the skewed summation of an n-by-n alternating-sign matrix A. Notice that the ith row (0 <i < n) begins with an i and ends with an n —i, so that reading from left to right we must see i descents and n -i ascents; that is, there are exactly 7 values of j in {1, 2,..., n} satisfying aj, = — 1, and a, j-1 the remaining n — 1 values of 9 satisfy aj; = a, ,, +1. Forma triangular array
Fig. 9. The tray in which the cubes sit. Let A* be the skewed summation of an n-by-n alternating-sign matrix A. Notice that the ith row (0 <i < n) begins with an i and ends with an n —i, so that reading from left to right we must see i descents and n -i ascents; that is, there are exactly 7 values of j in {1, 2,..., n} satisfying aj, = — 1, and a, j-1 the remaining n — 1 values of 9 satisfy aj; = a, ,, +1. Forma triangular array
Fig. 10. Cubes stacked in the domino-tiling fashion.
Fig. 10. Cubes stacked in the domino-tiling fashion.
whose ith row (1 <i < n) consists of those values of j for which aj ; =a; . .— i,j-1 e.g., for the first ¢ entries in column j of the alternating-sign matrix is 1. A monotone triangle of size n is a triangular array of natural numbers with strict increase from left to right along its n rows and with nonstrict increase from left to right along its diagonals, as in the array above. If the bottom row of a monotone triangle is 1 2 --- n, we call the array a complete monotone triangle. It is not difficult to show that the preceding construction gives a bijection between the n-by-n alternating-sign matrices and the complete monotone triangles of size n. Moreover, the +1’s in the alternating-sign matrix correspond to entries in some row of the triangle that do not occur in the preceding row. (This correspondence, as well as the notion of a monotone triangle, was introduced in [13].)
whose ith row (1 <i < n) consists of those values of j for which aj ; =a; . .— i,j-1 e.g., for the first ¢ entries in column j of the alternating-sign matrix is 1. A monotone triangle of size n is a triangular array of natural numbers with strict increase from left to right along its n rows and with nonstrict increase from left to right along its diagonals, as in the array above. If the bottom row of a monotone triangle is 1 2 --- n, we call the array a complete monotone triangle. It is not difficult to show that the preceding construction gives a bijection between the n-by-n alternating-sign matrices and the complete monotone triangles of size n. Moreover, the +1’s in the alternating-sign matrix correspond to entries in some row of the triangle that do not occur in the preceding row. (This correspondence, as well as the notion of a monotone triangle, was introduced in [13].)
It suffices to show that interchanging any two consecutive arguments of Vi changes the sign of the result. For convenience, we illustrate with n = 4: Having shown that W is skew symmetric in its arguments, we have completed the proof of (7), which yields the formula for AD(n) as a consequence. We have given our own proof of (7) in order to keep this article self-contained, but we should mention that the formula is equivalent to Theorem 2 in [13] and is a special case of the main result of [22]. It can also be derived from identity (5.11) of [11, p. 120] by setting t = —1 and making the observation immediate from [11, p. 104] that Py(x, ..., tn; —1) = sy(21, ..., 2n) []j<j(@i + %j) whenever A, # are partitions satisfying 1 = w+ (n—1, n—2,..., 0). We thank one of the referees for bringing some of these connections to our attention.
It suffices to show that interchanging any two consecutive arguments of Vi changes the sign of the result. For convenience, we illustrate with n = 4: Having shown that W is skew symmetric in its arguments, we have completed the proof of (7), which yields the formula for AD(n) as a consequence. We have given our own proof of (7) in order to keep this article self-contained, but we should mention that the formula is equivalent to Theorem 2 in [13] and is a special case of the main result of [22]. It can also be derived from identity (5.11) of [11, p. 120] by setting t = —1 and making the observation immediate from [11, p. 104] that Py(x, ..., tn; —1) = sy(21, ..., 2n) []j<j(@i + %j) whenever A, # are partitions satisfying 1 = w+ (n—1, n—2,..., 0). We thank one of the referees for bringing some of these connections to our attention.
do satisfy an analogue of (6), and one can exploit this to give streamlined proofs of some formulas in the theory of plane partitions; details will appear elsewhere. of some formulas in the theory of plane partitions; details will appear elsewhere. Fourth, we should note that the function W(a;, a2,..., am) has significance for tilings of the Aztec diamond of order n, even outside the case with m = n and a; = i for 1 <i <m. Suppose m < n and a, < n, and let J be the path in the graph G that starts at (-m,n— _m) whose 23 — 1st and 2jth steps head south and east, respectively, if 7 € {a1,..., an} and otherwise head east and south, respectively, for 1 < j <n, ending at the vertex (n —m, —m); Figure 11(a) shows IJ when n = 4, m = 2, (a), a2) = (2, 3). It is not difficult to show that the number of domino tilings of the portion of the Aztec diamond that lies above JT is 2™™+D?W (a1, ag, ..., Om):
do satisfy an analogue of (6), and one can exploit this to give streamlined proofs of some formulas in the theory of plane partitions; details will appear elsewhere. of some formulas in the theory of plane partitions; details will appear elsewhere. Fourth, we should note that the function W(a;, a2,..., am) has significance for tilings of the Aztec diamond of order n, even outside the case with m = n and a; = i for 1 <i <m. Suppose m < n and a, < n, and let J be the path in the graph G that starts at (-m,n— _m) whose 23 — 1st and 2jth steps head south and east, respectively, if 7 € {a1,..., an} and otherwise head east and south, respectively, for 1 < j <n, ending at the vertex (n —m, —m); Figure 11(a) shows IJ when n = 4, m = 2, (a), a2) = (2, 3). It is not difficult to show that the number of domino tilings of the portion of the Aztec diamond that lies above JT is 2™™+D?W (a1, ag, ..., Om):
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