arXiv:math/9801067v3 [math.CO] 21 Jun 1999 nexactly in ue mgn ieo lp unn hog h center the through running through passing 1 with “spine”), squares, slope (the diamond of Aztec line the of a Imagine ture. and barriers, th with with barriers. compatible 8 of is Figure placement that order -tiling of a graph.) shows diamond 1(b) dual tan- Figure Aztec is the an tiling from shows the edge in 1(a) an barrier removing a to down match- tamount a putting having perfect then grid-squares a of edge, pairs as shared grid-square to to region correspond correspond prefers edges a vertices whose one whose of and graph (If tiling dual a domino cross. of a ing to of permitted think are is These to domino diamond. no Aztec that an edges with associated grid square the 1 htteAtcdaodo order of diamond Aztec the that [1] A hsmtra a o ecpe rrpse ihu ex- without reposted or permission.) copied plicit be not may material This or Electron access at Digital Library) also International Access can (the You IDEAL via Press. article this Academic therein by rights retained ac- all and and are Copyright certified review. been peer has after that cepted content definitive the only of the A, repository Series Theory, Combinatorial of nal et(r“otws” ouprrgt(r“otes” ssqu as “northeast”) through (or 1 right upper to “southwest”) (or left 2 o sasakof stack a as ion n eew study we Here h are-ofiuaino iue1a a pca struc- special has 1(a) Figure of barrier-configuration The An result. of Statement I. NT:Ti ril a enpbihdi h Jour- the in published been has article This (NOTE: domino ( n http://www.europe.idealibrary.com +1) ze imn forder of diamond Aztec 2 , ntsurs ragdi iaeal ymti fash- symmetric bilaterally in arranged squares, unit 2 2 4 http://www.idealibrary.com sa1b- o -y1 etnl.I a hw in shown was It rectangle. 2-by-1) (or 1-by-2 a is n k , ( 6 o ahsc qae emypaebarriers place may we square, such each For . n h ares ed o nwo ipecmiaoilexplan combinatorial simple a of constridentity. know in not of do number “barriers” We the placing barriers. Remarkably, the of barrier. a ways cross certain may domino at look we particular, k nti ae,w otnetesuyo oiotlnso Azt of domino-tilings of study the continue we paper, this In ..., , +1) = 2 n / barriers 2 2 ⌈ oso qae,terw aiglengths having rows the squares, of rows n n/ ways. − 2 ⌉ 2 ubrteesursfo lower from squares these Number . , 2 niae ydree de of edges darkened by indicated , n, 2 n, n 2 n 2 n ascuet nttt fTcnlg,Cmrde A0213 MA Cambridge, Technology, of Institute Massachusetts sargo opsdof composed region a is a etldb dominoes by tiled be can − 2 ..., , 1 nvriyo icni,Mdsn I53706 WI Madison, Wisconsin, of University nmmr fGa-al Rota Gian-Carlo of memory In oiotlnswt barriers with tilings Domino ae Propp James 6 , 4 , 2 . 2 k . grid- ares 1 ic is n ihr Stanley Richard and s 1 teeyohrlcto ln h pn a nFgr () or 1(a)) Figure zags in and (as zigs spine puts the along only the location one along other whether every zags true at and holds zigs This placing the of incompatible spine. domino-tiling, way is each compatible square for unique that that follows a at It zig tiling. a the righ placing with its then to square it, the oth with below the domino or a On shares tiling. square placing the the then with if it, incompatible hand, above is or square left, that sha its at square to zag that square If the with spine. domino the a along square particular a sider proof. the for Preliminaries II. even. is hc the barrier-configurations which consider we th if so true remains spine, Theorem the along squares even-indexed and odd-indexed tilings. lie oeflyi eto I.Hne umn h for- get the one summing configurations, barriers Hence, all ex-2 over II). be Theorem the section will in in mula (this fully configuration more barrier plained one exactly with ible compatible of numbers equal have to tilings. claimed are which of all ail ihti lcmn is com- placement domino-tilings this of with number patible the above, described as mond barriers 1: Theorem the of placement form. the follow special that the this has of assume statement we the Theorem) in ing particular in (and Henceforth i,zg i,zg”Ntc hti hseape o all for example, this in that Notice za zip, zig.” zig, Figur zip, “zip, Thus decisions zag, (“zip”). of zip, all sequence at the a to barriers left corresponds no its 1 or on barriers “zag”), “zig”), (a (a edges edges top right and bottom its on ln h pn,teeare there spine, the along i ina l ftepriua atr fzg n asmani- zags and zigs of are pattern there Since particular barriers. the the by of fested all at tion hsur a i razgif zag a or zig a has square th k osdrapriua iigo nAtcdaod n con- and diamond, Aztec an of tiling particular a Consider 3 8-erertto fteAtcdaodsice the switches diamond Aztec the of rotation 180-degree (3) 2 ahdmn-iigo h ze imn scompat- is diamond Aztec the of domino-tiling Each (2) 1 h oml o h ubro iig ae omen- no makes tilings of number the for formula The (1) Theorem: the on remarks Some · to fti at u ro ssteJacobi-Trudi the uses proof our fact; this no of ation that constraint the with diamond, Aztec the ie iig sidpneto h lcmn of placement the of independent is tilings ained 2 n cdaod itoue n[]ad[].In [2]). and [1] in (introduced diamonds ec ( n +1) i hsur a i razgif zag a or zig a has square th / ie lcmn fbrir nteAtcdia- Aztec the in barriers of placement a Given 2 2 / 2 k hc is which , 9 2 k ifrn barrier-configurations, different 2 2 n n ( ( i n n see n i if zip and even is +1) +1) / / k 2 2 h oa ubrof number total the , / vnidxdsquares even-indexed 2 i k sodadzpif zip and odd is . i sodd. is i the , eis re res nd in er g, t, a e e s i - (a) (b)

FIG. 1. Barriers and tiling. at every location along the spine. In the case of the tiling de- is independent of A∗ 2, 4, ..., 2k , where the (A, B) in the picted in Figure 1(b), the full sequence of zigs and zags goes sum ranges over all balanced⊆{ partitions} of 1, 2, ..., 2k such “zag, zig, zig, zag, zig, zag, zag, zig.” that A 2, 4, ..., 2k = A∗. Note that in{ this formulation,} Each such sequencemust contain equal numbersof zigs and n has disappeared∩{ from} the statement of the result, as has the zags. For, suppose we color the unit squares underlying the Aztec diamond itself. Aztec diamond in checkerboard fashion, so that the squares along the spine are white and so that each white square has III. Restatement in terms of . four only neighbors (and vice versa). The barriers divide the Aztec diamond into two parts, each of which must have equal We can interpret the left-hand side of (2) using Schur func- numbers of black and white squares (since each part can be tions and apply the Jacobi-Trudi identity. The expression tiled by dominoes). It follows that the white squares along the spine must be shared equally by the part northwest of the a a j − i (3) diagonal and the part southeast of the diagonal. j i 1≤i

aj ai bj bi  −   −  (2) j i j i (A,BX) 1≤i

2 Trudi identity, this is equal to the is simple to check that regardless of whether n is even or odd, the exponent k(k 1)+ k′(k′ + 1) is equal to n(n + 1)/2 k, − − hak−k hak−2 hak−1 as was to be shown. h ··· h h ak−1−k ak−1−2 ak−1−1 . ··· . . . . . IV. Completion of proof. . . .

ha1−k ha1−2 ha1−1 ··· We can deduce the desired identity as a special case of a where hm is 0 if m < 0 and otherwise is equal to the sum general formula on products of determinants. This formula of all monomials in x1, ..., xk with total degree m (so that appears as formula II on page 45 (chapter 3, section 9) of [9], 2 2 h0 =1, h1 = x1 + ... + xk, h2 = x1 + x1x2... + xk, etc.). where it is attributed to Sylvester. However, we give our own Thus, if we let v(m) denote the length-k row-vector proof below. Suppose (A∗,B∗) is a fixed partition of 2, 4, ..., 2k into (hm−k, ,hm−2,hm−1), { } ··· two sets, and let v(1), ..., v(2k) be any 2k row-vectors of we see that the summand in (2) is the determinantal product length k. Given A 1, ..., 2k with A = k, let A denote the determinant of⊆{ the k-by-k } | | || || v(ak) v(bk) v(a ) . . 1 . . v(a ) · 2 v(a1) v(b1) . , .

specialized to x1 = x2 = ... = xk = 1. To prove Theo- v(ak) rem 1, it will suffice to show that this product, summed over all balanced partitions with ∗, where A = a1,a2, ..., ak with a1 < a2 < ... < ak. Abus- (A, B) A 2, 4, ..., 2k = A { } yields ∩{ } ing terminology somewhat, we will sometimes think of A as a set of vectors v(ai), rather than as a set of integers ai. v(2k) v(2k 1) − Theorem 2: v(2k 2) v(2k 3) .− .− . . · . . . A B = 1, 3, ..., 2k 1 2, 4, ..., 2k X || || · || || ||{ − }|| · ||{ }|| v(2) v(1) (A,B) (4) For, since this expression is independent of A∗, and since the sum of this expression over all 2k possible values of where (A, B) ranges over all balanced partitions of A∗ 2, ..., 2k is 2n(n+1)/2 (by the result proved in [1]), 1, 2, ..., 2k with A 2, 4, ..., 2k = A∗, B 2, 4, ..., 2k ⊆ { } − the value of the expression must be 2n(n+1)/2 k, as claimed {= B∗. } ∩{ } ∩{ } in Theorem 1. It is interestingto note that one can also evaluate the preced- Remark: This yields as a corollary the desired formula of ing determinantal product directly. Appealing to the Jacobi- the last section, with an extra sign-factor everywhere to take Trudi identity, we see that the product is account of the fact that we are stacking row-vectors the other way. sσ(x1, ..., xk)sτ (x1, ..., xk) Proof of Theorem 2: Every term on the left is linear in v(1), ..., v(2k), as is the term on the right; hence it suffices where σ = (k, k 1, ..., 1) and τ = (k 1, k 2, ..., 0). It is to check the identity when all the v(i)’s are basis vectors for − − − known that the k-dimensional row-space. First, suppose that the list v(1), ..., v(2k) does not contain s (x , ..., x )= x x (x + x ) σ 1 k 1 ··· k i j each basis vector exactly twice. Then it is easy to see that Yi

3 and t B or vice versa. But then (A s,t ,B s,t ) of this assertion, we may say that the events s A and t A (where∈ denotes symmetric difference) is△{ another} good△{ bal-} are uncorrelated with one another when s and ∈t are both even∈ anced partition.△ We claim that it cancels the contribution of (or both odd, by symmetry). (A, B). For, if one simply switches the row-vectors v(s) and Theorem 3: For 1 m k, let N be the random vari- v(t), one introduces t s 1 inversions, relative to the pre- ≤ ≤ m scribed ordering of the− rows− in the determinant; specifically, able A 1, ..., m , where (A, B) is a random partition of 1, ...,| 2k∩{in the sense}| defined above. Then N has mean each i with ss) or s t 1 (if s>t) inversions, relative− to the− pre- function”associated with a random domino-tilingof the Aztec scribed ordering− − of the rows. Since both quantities are even, diamond (see [1]). Theorem 3 tells us that if one looks along we may re-order the rows in the determinants so that indices the spine, the sequence of differences between heights of con- increase from top to bottom, without changing the sign of secutive vertices satisfies a weak law of large numbers. the product of the two determinants. We now recognize the modified term as the sole non-vanishing term associated with (A∗ s ,B∗ s ). Since this term has the same sign as the VI. Open problems. term\{ associated} ∪{ with} (A∗,B∗), and since the sign is correct in the base case ( 1, ..., 2k 1 , φ), the correctness of the sign One open problem is to find a combinatorial (preferably for all partitions{ of 1, ...,−2k} 1 follows by induction. bijective) proof of Theorem 1. For instance, one might be This concludes the{ proof of− Theorem} 2, which in turn im- able to find a bijection between the tilings compatible with plies Theorem 1. ✷ (A∗,B∗) and the tilings compatible with some other partition REMARK. An identity equivalent to summing both sides of of 2, 4, ..., 2k . k ∗ { } equation (4) for all 2 sets A (with row v(m) specialized to Also, recall the variables x1, x2, ..., xm that made a brief (hm−k,...,hm−2,hm−1) as needed for Theorem 1) is a spe- appearance in section III before getting swallowed up by the cial case of an identity proved combinatorially by M. Fulmek notation. Leaving aside our appeal to the explicit formulas [3] using a nonintersecting lattice path argument. It is eas- for sσ(x1, ..., xk, 0, ...) and sτ (x1, ..., xk, 0, ...), we may use ily seen that Fulmek’s proof applies equally well to prove our the linear algebra formalism of section IV to derive a Schur Theorem 2. Thus Fulmek’s paper contains an implicit bijec- function identity in infinitely many variables, expressing the tive proof of Theorem 2. product sσsτ as a sum of products of other pairs of Schur functions. It would be desirable to have a combinatorial ex- V. Probabilistic application. planation of these identities at the level of Young tableaux. In section V, we made use of the fact that if (A, B) is cho- sen randomly from among the balanced ordered partitions of One can define a probability distribution on ordered par- 1, 2, ..., 2k , and if s,t 1, 2, ..., 2k have the same par- titions of 1, 2, ..., 2k into two sets of size k, where the {ity, then the} events s A∈and { t A are} independent of one probability{ of the partition} (A, B) is another. We conjecture,∈ based on∈ numerical evidence, that if s,t 1, 2, ..., 2k have opposite parity, then the events aj −ai bj −bi ∈ { } 1≤i

4 the variance of Nm, which empirically is on the order of log k Bibliography or perhaps even smaller. Finally, fix 1 k n. Define 0-1 random variables [1] N. Elkies, G. Kuperberg, M. Larsen, and J. Propp, Al- ≤ ≤ n ternating sign matrices and domino tilings, part 1, Journal of X1,X2, ..., Xn such that for all (x1, x2, ..., xn) 0, 1 , n ∈ { } Algebraic Combinatorics 1, 111-132 (1992). Prob[Xi = xi for all i]=0 unless i=1 xi = k, in which case P [2] N. Elkies, G. Kuperberg, M. Larsen, and J. Propp, Al- ternating sign matrices and domino tilings, part 2, Journal of Prob[Xi = xi for all i]= Algebraic Combinatorics 1, 219-234 (1992).

aj −ai bj −bi 1≤i

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