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Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture
The American Mathematical Monthly
... In Figure 8, we can easily obtain the Ehrhart series for O(P 1 ), O(P 2 ), O(P 3 ), and O(P 4 ) as follows: 6 . We obtained the following results using Stembridge's software package [44]: ...
... □ Now let us consider the second non-intersecting lattice paths model. This model is inspired by the Jacobi-Trudi identity [6,Proposition 4.2]. Before that, we need to introduce the relevant knowledge of symmetric functions [43,Chapter 7]. ...
... In [6,Proposition 4.2], the proof of the Jacobi-Trudi identity was completed using the Lindström-Gessel-Viennot lemma. We observed that RSSYT requires weakly decreasing in every row and strictly decreasing in every column, with entries chosen from {1, 2, . . . ...
In this paper, we provide an overview of Ehrhart polynomials associated with order polytopes of finite posets, a concept first introduced by Stanley. We focus on their combinatorial interpretations for many sequences listed on the OEIS. We begin by exploring the Ehrhart series of order polytopes resulting from various poset operations, specifically the ordinal sum and direct sum. We then concentrate on the poset associated with the Ferrers diagram of a partition . When , the Ehrhart polynomial is a shifted Hankel determinant of the well-known Catalan numbers; when , the Ehrhart polynomial is solved by Stanley's hook content formula and is used to prove conjectures for the sequence [A140934] on the OEIS. When solving these problems, we rediscover Kreweras' determinant formula for the Ehrhart polynomial through the application of the Lindstr\"om-Gessel-Viennot lemma on non-intersecting lattice paths.
... The representation of the Schur functions makes it possible to apply the well-developed theory of the symmetric functions to calculate of the correlation functions [1][2][3] for quantum integrable models. In turn, the correlation functions allow to obtain and to interpret connections between symmetric functions [4], plane partitions [5], and enumerative combinatorics [6]. The correlation function of strongly correlated bosons [7], as well as the partition function of inhomogeneous four-vertex model [8], lead to a new approach to derivation of the normtrace generating functions of plane partitions with the fixed traces [10,11]. ...
... The star C k with deviation k is the nest of paths introduced by Definition 1. The star C k with deviation k is a nest of N self-avoiding lattice paths that connect (see Fig. 1), the starting points C i = (i, N − i) with the nonequidistant ending points (N, μ i ), where μ i are the parts of the strict partition μ defined in (5) while upward steps are absent along the lines x 1 , x 2 , . . . , x k (see Fig.1). ...
... A watermelon with deviation k (see Fig. 2) is a nest of paths obtained by 'gluing' the stars C k and B along the dissection line determined by μ (5). with parts respecting (1) so that the points (N + 1, μ i ) and the points (1, μ i ) are identified. ...
The amplitudes of the leading asymptotics of the XX0 Heisenberg spin chain depend on the generating function of plane partitions with the additional conditions. In our paper we apply the Cauchy–Binet determinantal identity for derivation of the generating function of plane partitions with the fixed conjugate trace in a high box.
... Enumeration of plane partitions subject to various constraints is a special part of the enumerative combinatorics [24][25][26][27][28][29] and fixed volume of diagonal parts is the case of particular interest [30,31]. For instance, the norm-trace generating function of plane partitions [30] arises in the study of the temporal evolution of the first moment of particles distribution of the phase model [32]. ...
... In the present paper our approach is based on the theory of symmetric functions [34]. The Bethe vectors, which are the state-vectors of the model, are considered both as onand off-shell, allowing to establish connection between the partitions, the lattice walks, and the plane partitions [25,35]. We consider the generating exponential operator, the operator defined by the sum of projectors to the spin "down" states taken with inhomogeneous weights. ...
... is an array (π ij ) i,j≥1 of non-negative integers that satisfy π ij ≤ M, π ij ≥ π i+1,j and π ij ≥ π i,j+1 for all i, j ≥ 1. Furthermore, π ij = 0 whenever i or j exceed N (Figure 3), [24,25]. There exists bijection between the watermelon configuration of non-intersecting lattice paths ( Figure 2 ...
Relations between the mean values of distributions of flipped spins on periodic Heisenberg XX chain and some aspects of enumerative combinatorics are discussed. The Bethe vectors, which are the state-vectors of the model, are considered both as on- and off-shell. It is this approach that makes it possible to represent and to study the correlation functions in the form of non-intersecting nests of lattice walks and related plane partitions. We distinguish between two types of walkers, namely lock step models and random turns. Of particular interest is the connection of random turns walks and a circulant matrix. The determinantal representation for the norm-trace generating function of plane partitions with fixed height of diagonal parts is obtained as the expectation of the generating exponential over off-shell N-particle Bethe states. The asymptotics of the dynamical mean value of the generating exponential is calculated in the double scaling limit provided that the evolution parameter is large. It is shown that the amplitudes of the leading asymptotics depend on the number of diagonally constrained plane partitions.
... The genealogy of these families extends back to 1979 in a classic paper by Andrews [2], where we encounter the first result of the kind that we will see in this paper, namely, that the determinant of a matrix from one of the families has a closed form and counts certain combinatorial objects (more precisely: descending plane partitions). For more background on plane partitions and their connections to determinants up to the year 1999, see [4]. ...
... This identity is very useful in some determinant evaluations, particularly whenever there is a need to establish a link between determinants with parameters s and t that are closely related (see Section 7). The proof of this identity can be found in [4]. We refer the reader to [1] for an entertaining discussion and excellent explanation of its use. ...
... This is because an even sign implies that all of the possible paths/tilings that should be counted are included in the summation. (2,2,4,3). All regions are hexagonal after the forced tiling removal, except for the left-most one, which becomes triangular. ...
We present and prove closed form expressions for some families of binomial determinants with signed Kronecker deltas that are located along an arbitrary diagonal in the corresponding matrix. They count cyclically symmetric rhombus tilings of hexagonal regions with triangular holes. We extend a previous systematic study of these families, where the locations of the Kronecker deltas depended on an additional parameter, to families with negative Kronecker deltas. By adapting Zeilberger's holonomic ansatz to make it work for our problems, we can take full advantage of computer algebra tools for symbolic summation. This, together with the combinatorial interpretation, allows us to realize some new determinantal relationships. From there, we are able to resolve all remaining open conjectures related to these determinants, including one from 2005 due to Lascoux and Krattenthaler.
... Our approach to the investigation of correlation functions is based on the theory of symmetric functions [31], which allows us to establish natural connection with the different types of the directed lattice walks, partitions and plane partitions [32,33]. In [34,35] it was shown that the multi-spin correlation functions over the ferromagnetic vacuum are in one-to-one correspondence with the path configuration of the random turns walkers [36,37]. ...
... where (15) and (20) are taken into account, N 2 (e iθ N /2 ) is given by (33), and summation is over all independent solutions to (26). ...
... x1 x2 x3 x4 x5 x6 A boxed plane partition π is an array (π ij ) i,j≥1 of non-negative integers that satisfy π ij ≥ π i+1,j and π ij ≥ π i,j+1 for all i, j ≥ 1, [33,41]. A boxed plane partition is contained in L × N × M box, if π ij ≤ M for all i and j, and π ij = 0, whenever i > L or j > N . ...
The mean values of non-homogeneously parameterized generating exponential are obtained and investigated for the periodic Heisenberg XX model. The norm-trace generating function of boxed plane partitions with fixed volume of their diagonal parts is obtained as N-particles average of the generating exponential. The generating function of self-avoiding walks of random turns vicious walkers is obtained in terms of the circulant matrices that leads to generalizations of the Ramus's identity. Under various specifications of the generating exponential, the N-particles averages arise for a set of inconsecutive flipped spins and for powers of the first moment of flipped spins distribution at large length of the chain. These averages are expressed through the numbers of closed trajectories with constrained initial/final positions. The estimates at large temporal parameter are expressed through the numbers of diagonally restricted plane partitions characterized by fixed values of the main diagonal trace or by fixed heights of the diagonal columns in one-to-one correspondence with the flipped spins positions.
(Preliminary of https://iopscience.iop.org/article/10.1088/1751-8121/ac5363)
... Zielberger famously proved that alternating sign matrices (ASMs) are equinumerous with totally symmetric self-complimentary plane partitions (TSSCPPs) [33]. The first seven numbers in this sequence are 1,2,7,42,429,7436,218348 and the general formula is ...
... This connects our poset extension problem to the illustrious family of alternating sign matrices. See [6,7], respectively, for a brief or an extended recounting of the history of the famous alternating sign matrix conjecture. Magog triangles of M n are in bijection with totally the symmetric self-complementary plane partitions (TSSCPP) in a 2n × 2n × 2n box. ...
... There are many combinatorial manifestations of the ASM sequence (2), see [7,23]. A natural bijective proof between TSSCPPs and ASMs (or equivalently, between magog and gog triangles) remains elusive, though progress on subfamilies has been achieved [4,31]. ...
Let denote the order ideal of the boolean lattice consisting of all subsets of size at most 2. Let denote the poset extension of induced by the rule: implies and . We give an elementary bijection from the set of linear extensions of to the set of shifted standard Young tableau of shape , which are counted by the strict-sense ballot numbers. We find a more surprising result when considering the set of poset extensions so that each singleton is comparable with all of the doubletons. We show that is in bijection with magog triangles, and therefore is equinumerous with alternating sign matrices. We adopt our proof techniques to show that row reversal of an alternating sign matrix corresponds to a natural involution on gog triangles.
... We keep using variable y as it is suitable for producing integral solutions to Pascal's hexagon relations. In particular, one finds following relations between polynomials F m,n and the numbers given by formulas (A.5) and (A.6) 3) is given a family of gauge equivalent solutions {αa, α ′ a ′ , βb, β ′ b ′ , γc, γ ′ c ′ } is parametrized by six scaling numbers α, α ′ , β, β ′ , γ and γ ′ satisfying constraints αα ′ = ββ ′ = γγ ′ = 0. On a trigonal lattice this gauge transformation is uniquely defined provided one fixes 6 mutually independent scaling parameters for the vertices situated as shown on a figure Finally, it is remarkable that the solutions of Pascal's hexagon relation reproduce in several ways numbers which were discovered in counting various symmetry classes of the alternating sign matrices (see [23,24,25] F 4,2 = 2xy + y 2 F 6,3 = 11x 2 y + 12xy 2 + 3y 3 F 5,2 = 3x 2 + 6xy + 2y 2 F 7,3 = 26x 3 + 78x 2 y + 55xy 2 + 11y 3 F 6,2 = 11x 2 + 12xy + 3y 2 F 8,3 = 170x 3 + 294x 2 y + 156xy 2 + 26y 3 F 7,2 = 50x 2 + 30xy + 5y 2 F 9,3 = 646x 3 + 816x 2 y + 350xy 2 + 50y 3 F 8,2 = 85x 2 + 42xy + 6y 2 F 9,2 = 133x 2 + 56xy + 7y 2 F 8,4 = 170x 3 y + 294x 2 y 2 + 156xy 3 + 26y 4 F 9,4 = 646x 4 + 2584x 3 y + 2839x 2 y 2 + 1190xy 3 + 170y 4 ...
... On enumeration of various symmetry classes of alternating sign matrices see[23,24] 6 Note that S (a) 2p−1 is just the number of cyclically symmetric transpose complement plane partitions in a (2p) 3 box (see, e.g.,[25], p.199). It is usually denoted as N S (2p). ...
The raise and peel model of a one-dimensional fluctuating interface (model A) is extended by considering one source (model B) or two sources (model C) at the boundaries. The Hamiltonians describing the three processes have, in the thermodynamic limit, spectra given by conformal field theory. The probability of the different configurations in the stationary states of the three models are not only related but have interesting combinatorial properties. We show that by extending Pascal's triangle (which gives solutions to linear relations in terms of integer numbers), to an hexagon, one obtains integer solutions of bilinear relations. These solutions give not only the weights of the various configurations in the three models but also give an insight to the connections between the probability distributions in the stationary states of the three models. Interestingly enough, Pascal's hexagon also gives solutions to a Hirota's difference equation.
... Eqs. (1) and (2) were proved by Zeilberger [13,14], who in the refined case of (2) (which contains (1) as a special case, since A n,1 = A n−1 , and took longer to prove) used the square-ice techniques introduced by Kuperberg [8] in his simplified proof of (1). An alternative proof of (2) using entirely different methods was found by Fischer [5,6]. ...
... It is well-known (see [1,10]) that ASMs correspond under a simple bijection to complete monotone triangles, which are monotone triangles with the numbers 1, 2, ..., n in the bottom row. Formally, the bijection maps the ASM M = (m i,j ) to the monotone triangle T = (t i,j ) such that t i,1 < t i,2 < t i,3 < . . . ...
We study a further refinement of the standard refined enumeration of alternating sign matrices (ASMs) according to their first two rows instead of just the first row, and more general "d-refined" enumerations of ASMs according to the first d rows. For the doubly-refined case of d=2, we derive a system of linear equations satisfied by the doubly-refined enumeration numbers A_{n,i,j} that enumerate such matrices. We give a conjectural explicit formula for A_{n,i,j} and formulate several other conjectures about the sufficiency of the linear equations to determine the A_{n,i,j}'s and about an extension of the linear equations to the general d-refined enumerations.
... If the sum of all entries, i,j π ij = n, then π is referred to as a plane partition of n. Regarding the history of plane partitions, please refer to [5,8] and [22, P. 440-442]. ...
... See [14] for a proof. By (8), the enumerative generating function of the set π(P 9 ) in Figure 15 is PF(π(P 9 )) = F (q n , q n+1 ; P ) − qF (q n+1 , q n ; P ) (q; q) n (q; q) n−1 = 1 (q; q) n (q; q) n−1 1 (q n+1 ; q) r+1 (q n ; q) k+2 − q (q n ; q) r+1 (q n+1 ; q) k+2 ...
Inspired by Gansner's elegant k-trace generating function for rectangular plane partitions, we introduce two novel operators, and , along with their combinatorial interpretations. Through these operators, we derive a new formula for P-partitions of posets extended by two-rowed plane partitions. This formula allows us to compute explicit enumerative generating functions for various classes of P-partitions. Our findings encompass skew plane partitions, diamond-related two-rowed plane partitions, an extended V-poset, and ladder poset extensions, enriching the theory of P-partitions.
... The left seven matrices in Fig. 1 are the matrices of ASM 3 . Alternating sign matrices were initially studied by Robbins and Rumsey in relation to the lambda determinant [8]. The enumeration formula for n × n ASMs, ...
... was initially conjectured by Mills, Robbins, and Rumsey [20] and was subsequently proven independently by Zeilberger [34], Kuperberg [17], and Fischer [16]. For more details on the history of ASMs, see [8]. Striker studied ASMs geometrically by taking the convex hull of n × n ASMs to define the alternating sign matrix polytope [29]. ...
Plane partitions in the totally symmetric self-complementary symmetry class (TSSCPPs) are known to be equinumerous with alternating sign matrices, but no explicit bijection is known. In this paper, we give a bijection from these plane partitions to -matrices we call magog matrices, some of which are alternating sign matrices. We explore enumerative properties of these matrices related to natural statistics such as inversion number and number of negative ones. We then investigate the polytope defined as their convex hull. We show that all the magog matrices are extreme and give a partial inequality description. Finally, we define another TSSCPP polytope as the convex hull of TSSCPP boolean triangles and determine its dimension, inequalities, vertices, and facets.
... This method was famously used by one of us (DZ) to prove the Mills-Robbins-Rumsey alternating sign matrix conjecture [26] (see also [3]). In that article the author first found a (complicated) constant term expression for the desired quantity (the number of alternating sign matrices) and another (almost as complicated) constant term expression for another quantity (totally symmetric self-complementary plane partitions), already proved by guru George Andrews to be equal to the desired expression. ...
... , c − 1. This is explained in Section 3.3 of Bressoud's book [3]. An example for such a family of paths for a = 6 and b = c = 3 is displayed in the figure below. ...
In this {\it case study}, we hope to show why Sheldon Axler was not just wrong, but {\em wrong}, when he urged, in 1995: ``Down with Determinants!''. We first recall how determinants are useful in enumerative combinatorics, and then illustrate three versatile tools (Dodgson's condensation, the holonomic ansatz and constant term evaluations) to operate in tandem to prove a certain intriguing determinantal formula conjectured by the first author. We conclude with a postscript describing yet another, much more efficient, method for evaluating determinants: `ask determinant-guru, Christian Krattenthaler', but advise people only to use it as a last resort, since if we would have used this last method right away, we would not have had the fun of doing it all by ourselves.
... Alternating sign matrices (ASMs) and their equinumerous friends, descending plane partitions (DPPs) and totally symmetric self-complementary plane partitions (TSSCPPs), have been bothering combinatorialists for decades by the lack of an explicit bijection between any two of the three sets of objects. (See [7] [8] [1] [12] [6] for these enumerations and bijective conjectures and [4] for the story behind these papers.) In [9], we gave a bijection between permutation matrices (which are a subclass of ASMs) and descending plane partitions with no special parts in such a way that the inversion number of the permutation matrix equals the number of parts of the DPP. ...
... It is well-known that monotone triangles of order n are in bijection with n × n alternating sign matrices via the following map [4]. For each row of the ASM note which columns have a partial sum (from the top) of 1 in that row. ...
International audience
We define a subclass of totally symmetric self-complementary plane partitions (TSSCPPs) which we show is in direct bijection with permutation matrices. This bijection maps the inversion number of the permutation, the position of the 1 in the last column, and the position of the 1 in the last row to natural statistics on these TSSCPPs. We also discuss the possible extension of this approach to finding a bijection between alternating sign matrices and all TSSCPPs. Finally, we remark on a new poset structure on TSSCPPs arising from this perspective which is a distributive lattice when restricted to permutation TSSCPPs.
... The simplest way of presenting the workings appears to be to arrange the series of matrices one under another, as it is displayed below; it will then be found very easy to pick out the divisors (in the interior matrices): Dodgson's condensation method, being interesting and excellently suited to hand-computations, is in the first place remarkable for its exceedingly great briefness, lucidity and accuracy. It is also noteworthy as it involves the evaluation of only 2nd order determinants, the elements of which are adjacent to one another [19], [21], [20]. ...
... However, it is evident that, when zeros (which Dodgson called ciphers in his paper [20] ) appear in the interior of the original matrix or any one of the derived matrices, the process cannot be continued because of the emergence of division by zero [20]. A solution to this problem, as Dodgson suggests, is to recommence the operation by first rearranging the original matrix by transferring the top row to the bottom or the bottom row to the top so that the zero, when it occurs, is now found in an exterior row [19], [21], [22]. The merit of this solution is that "there is only one new row to be computed; the other rows are simply copied from the work already done" [20]. ...
This article furnishes a new and simple matrix inversion method which makes full use of the condensation technique of the author of Alice's Adventures in Wonderland, Charles Dodgson. A special feature of this article is the adoption of Bhaskara's Law of Impending Operation on Zero in overcoming the problem of division by zero whenever zero appears as a divisor in the condensation technique of Dodgson.
... See Figure 1 below for an example of a Young diagram of a partition. Let us now recall Sylvester's bijection [5,8]. Define O n (resp. ...
In this paper, we count the total number of hooks of length two in all odd partitions of n and all distinct partitions of n with a bound on the largest part of the partitions. We generalize inequalities of Ballantine et al by showing there is a bias in the number of hooks of length two in all odd partitions over all distinct partitions of n in presence of a bound on the largest part. To establish such a bias, we use a variant of Sylvester's map. Then, we conjecture a similar finite bias for a weighted count of hooks of length two and prove it when we remove the bound on the largest part.
... See Figure 1 below for an example of a Young diagram of a partition. Let us now recall Sylvester's bijection [5,8]. Define O n (resp. ...
In this paper, we count the total number of hooks of length two in all odd partitions of n and all distinct partitions of n with a bound on the largest part of the partitions. We generalize inequalities of Ballantine et al by showing there is a bias in the number of hooks of length two in all odd partitions over all distinct partitions of n in presence of a bound on the largest part. To establish such a bias, we use a variant of Sylvester's map. Then, we conjecture a similar finite bias for a weighted count of hooks of length two and prove it when we remove the bound on the largest part.
... Mills, Robbins, and Rumsey [19] introduced alternating sign matrices and posed the problem of counting the n × n alternating sign matrices. An interesting account of what led to the ASM conjecture of the enumeration of n × n ASM s proposed by Mills, Robbins and Rumsey [19], and proved by Zeilberger [22] can be found in Bressoud's Proofs and Confirmations, The Story of the Alternating Sign Matrix Conjecture [8]. The number of n × n alternating sign matrices is n−1 j=0 (3j + 1)! (n + j)! . ...
The polytope , the convex hull of the alternating sign matrices, was introduced by Striker and by Behrend and Knight. A face of corresponds to an elementary flow grid defined by Striker, and each elementary flow grid determines a doubly directed graph defined by Brualdi and Dahl. We show that a face of is symmetric if and only if its doubly directed graph has all vertices of even degree. We show that every face of is a 2-level polytope. We show that a d-dimensional face of has at most vertices and facets, for . We show that a d-dimensional face of satisfies , where v and f are the numbers of vertices and edges of the face. If the doubly directed graph of a d-dimensional face is 2-connected, then . We describe the facets of a face and a basis for the subspace parallel to a face in terms of the elementary flow grid of the face. We prove that no face of has the combinatorial type of the Birkhoff polytope . We list the combinatorial types of faces of that have dimension 4 or less.
... The XX0 model has diverse relations with recent topics of research in mathematical physics such as combinatorial and probabilistic models such as alternating sign matrices [35], random tilings, theory of random walks in lattice and random matrix theory [22], plane partitions and theory of symmetric functions [36], [37], and also topological string theory [38]. ...
We study finite size and temperature XX0 Heisenberg spin chain in weak and strong coupling regimes. By using an elegant connection of the model to integrable combinatorics and probability, we explore and interpret a possible phase structure of the model in asymptotic limit: the limit of large inverse temperature and size. First, partition function and free energy of the model are derived by using techniques and results from random matrix models and nonintersecting Brownian motion. We show that, in the asymptotic limit, partition function of the model, written in terms of matrix integral, is governed by the Tracy-Widom distribution. Second, the exact analytic results for the free energy, which is obtained by the asymptotic analysis of the Tracy-Widom distribution, indicate a completely new and sophisticated phase structure of the model. This phase structure consists of second- and third-order phase transitions. Finally, to shed light on our new results, we provide a possible interpretation of the phase structure in terms of dynamical behavior of magnons in the spin chain. We demonstrate distinct features of the phases with schematic spin configurations which have definite features in each region of the phase diagram.
... An important result was obtained in [9] where an exact determinant formula for the partition function was obtained, see also [10]. Subsequently, this result was found of fundamental importance in the proof of long-standing conjectures in enumerative combinatorics, due to the close connection of the model with alternating sign matrices (ASMs) [11][12][13][14][15][16], see also [17] for a review. It should be mentioned that ASM enumerations appear to be in turn deeply related with quantum spin chains and some loop models, via Razumov-Stroganov conjecture [18]; for recent results, see for instance [19][20][21] and references therein. ...
The six-vertex model with domain wall boundary conditions (DWBC) on an N x N square lattice is considered. The two-point correlation function describing the probability of having two vertices in a given state at opposite (top and bottom) boundaries of the lattice is calculated. It is shown that this two-point boundary correlator is expressible in a very simple way in terms of the one-point boundary correlators of the model on N x N and (N-1) x (N-1) lattices. In alternating sign matrix (ASM) language this result implies that the doubly refined x-enumerations of ASMs are just appropriate combinations of the singly refined ones.
... Alternating sign matrices (ASMs) are square matrices of 0s, 1s, and −1s with row and column-sums equal to 1 and with the restriction that the non-zero entries alternate signs across each row and down each column. An example is These are rich combinatorial objects with connections to many problems in algebraic combinatorics (see [2], [3], [12]). They also have many different representations. ...
This paper highlights three known identities, each of which involves sums over alternating sign matrices. While proofs of all three are known, the only known derivations are as corollaries of difficult results. The simplicity and natural combinatorial interpretation of these identities, however, suggest that there should be direct, bijective proofs.
... We have three explicit formulas for the expression lim b→−1 (1 − b)F (b, 0; b) coming from Fine's work: Note that the equality of the right sides of the identities (4.6)-(4.8) can be proved in a purely combinatorial manner with the aid of Sylvester's bijection [13] and Franklin's involution [8]. The equality of (4.6) and (4.8) will be used later in the proof of the Theorem 5.1. ...
We utilize false theta function results of Nathan Fine to discover three new partition identities involving weights. These relations connect G\"ollnitz--Gordon type partitions and partitions with distinct odd parts, partitions into distinct parts and ordinary partitions, and partitions with distinct odd parts where the smallest positive integer that is not a part of the partition is odd and ordinary partitions subject to some initial conditions, respectively. Some of our weights involve new partition statistics, one is defined as the number of different odd parts of a partition larger than or equal to a given value and another one is defined as the number of different even parts larger than the first integer that is not a part of the partition. Dedicated to our friend, Krishna Alladi, on his 60th birthday.
... Mills, Robbins, and Rumsey [22] (see also [5,Eq. (6.15), p. 199]) showed that the number of objects of type (a) is ...
Michael Somos conjectured a relation between Hankel determinants whose entries count ternary trees and the number of certain plane partitions and alternating sign matrices. Tamm evaluated these determinants by showing that the generating function for these entries has a continued fraction that is a special case of Gauss's continued fraction for a quotient of hypergeometric series. We give a systematic application of the continued fraction method to a number of similar Hankel determinants. We also describe a simple method for transforming determinants using the generating function for their entries. In this way we transform Somos's Hankel determinants to known determinants, and we obtain, up to a power of 3, a Hankel determinant for the number of alternating sign matrices. We obtain a combinatorial proof, in terms of nonintersecting paths, of determinant identities involving the number of ternary trees and more general determinant identities involving the number of r-ary trees.
... A 6 × 6 FPL with gyration orbit of length 84, and its link pattern alternating sign matrices, which was proved by D. Zeilberger [Ze96] and G. Kuperberg [Ku96] (cf. [Br99] for a detailed exposition of this history). There is a poset A n whose order ideals are in bijection with n × n alternating sign matrices (denote this set as ASM n ), such that gyration of Definition 5.3 is equivalent to the action of the toggle group element Gyr of Definition 3.21. ...
We introduce a new concept of resonance on discrete dynamical systems. This concept formalizes the observation that, in various combinatorially-natural cyclic group actions, orbit cardinalities are all multiples of divisors of a fundamental frequency. Our main result is an equivariant bijection between plane partitions in a box (or order ideals in the product of three chains) under rowmotion and increasing tableaux under K-promotion. Both of these actions were observed to have orbit sizes that were small multiples of divisors of an expected orbit size, and we show this is an instance of resonance, as K-promotion cyclically rotates the set of labels appearing in the increasing tableaux. We extract a number of corollaries from this equivariant bijection, including a strengthening of a theorem of [P. Cameron--D. Fon-der-Flaass '95] and several new results on the order of K-promotion. Along the way, we adapt the proof of the conjugacy of promotion and rowmotion from [J. Striker--N. Williams '12] to give a generalization in the setting of n-dimensional lattice projections. Finally we discuss known and conjectured examples of resonance relating to alternating sign matrices and fully-packed loop configurations.
... Plane partitions were a hot topic back in the 1970's and 1980's (as beautifully described in [4]), and they still keep combinatorialists busy. For example, the q-enumeration formula of totally symmetric plane partitions, conjectured independently by David Robbins and George Andrews in 1983, remained open for almost 30 years and was finally proved in 2011 [8] using massive computer algebra calculations. ...
We evaluate a curious determinant, first mentioned by George Andrews in 1980 in the context of descending plane partitions. Our strategy is to combine the famous Desnanot-Jacobi-Dodgson identity with automated proof techniques. More precisely, we follow the holonomic ansatz that was proposed by Doron Zeilberger in 2007. We derive a compact and nice formula for Andrews's determinant, and use it to solve a challenge problem that we posed in a previous paper. By noting that Andrews's determinant is a special case of a two-parameter family of determinants, we find closed forms for several one-parameter subfamilies. The interest in these determinants arises because they count cyclically symmetric rhombus tilings of a hexagon with several triangular holes inside.
... In this paper, we are concerned exclusively with the domain wall This model was introduced in Ref. [5] in connection with the calculation of the correlation functions for exactly solvable 1+1 dimensional models [6]. It appears that some problems from the theory of alternating sign matrices [7,8] and domino tilings [9] may be reformulated in terms of this model. Aperiodic boundary conditions are of interest since they demonstrate the influence of the boundaries and internal defects of real physical systems on their bulk properties. ...
Vertical-arrow fluctuations near the boundaries in the six-vertex model on the two-dimensional square lattice with the domain wall boundary conditions are considered. The one-point correlation function (`boundary polarization') is expressed via the partition function of the model on a sublattice. The partition function is represented in terms of standard objects in the theory of orthogonal polynomials. This representation is used to study the large N limit: the presence of the boundary affects the macroscopic quantities of the model even in this limit. The logarithmic terms obtained are compared with predictions from conformal field theory.
... 05E05. Work supported by the Australian Research Council. of (1.1a) to prove the famous MacMahon conjecture in the theory of plane partitions [3,17]. ...
We use Rogers-Szego polynomials to unify some well-known identities for Hall-Littlewood symmetric functions due to Macdonald and Kawanaka.
... Plane partitions, along with their many generalizations and their cousins the standard and semistandard Young tableaux, are frequently-studied objects in combinatorics (see, for instance, [Bre99]). As is often the case with combinatorial objects of interest, they arise in other areas of mathematics; relevant to our work is a particular use in computing the equivariant Calabi-Yau topological vertex in Pandharipande-Thomas theory and Donaldson-Thomas theory (PT and DT). ...
Pandharipande-Thomas theory and Donaldson-Thomas theory (PT and DT) are two branches of enumerative geometry in which particular generating functions arise that count plane-partition-like objects. That these generating functions differ only by a factor of MacMahon's function was proven recursively by Jenne, Webb, and Young using the double dimer model. We bijectivize two special cases of the result by formulating these generating functions using vertex operators and applying a particular type of local involution known as a toggle, first introduced in the form we use by Pak.
... The vertex models with fixed boundary conditions of two-dimensional statistical mechanics play an important role in contemporary studies of integrable systems [1][2][3][4][5][6][7][8]. There are intriguing connections of these models with the problems of enumerative combinatorics [9][10][11][12], the theory of symmetric functions [13], and the limit shapes phenomena [14][15][16][17]. ...
The scalar product of the state-vectors of the exactly solvable five-vertex model with the fixed boundary conditions is considered. Various relations including those in terms of complete symmetric polynomials are derived. The limiting forms of the obtained answers may be interpreted in terms of random walks on a square grid.
... Alternating sign matrices have a very interesting history, enumeration, and connection to other areas of mathematics and science [3]. There exist beautiful bijections between alternating sign matrices and many other combinatorial objects, including monotone triangles, height-function matrices, fully-packed loop configurations, square ice configurations and order ideals of a particular poset. ...
We investigate analogues of alternating sign matrices, called partial alternating sign matrices. We prove bijections between these matrices and several other combinatorial objects. We use an analogue of Wieland's gyration on fully-packed loops, which we relate to the study of toggles and order ideals. Finally, we show that rowmotion on order ideals of a specific poset and gyration on partial fully-packed loop configurations have the same orbit structure.
... where X , is constructed from X by removing its th and th rows and th and th columns. It is noted that the Desnanot-Jacobi Identity [43,Theorem 3.12] gives det X , det X 1,1 − det X det X 1, 1 = det X 1, det X ,1 . ...
Transmit power consumption is practically important in future massive connectivity scenarios, especially when in-band inter-cell interference and user-transmit-power budget are considered. In this paper, the transmit-power estimation for the uplink of multiple-input multiple-output (MIMO) non-orthogonal multiple access (NOMA) is investigated in which the user precoding and the base station detection are included. For the MIMO-NOMA scheme with group-wise successive interference cancellation (SIC), closed-form expressions for the estimated individual user-transmit-powers are derived for given signal-to-interference-plus-noise ratio (SINR) requirements. Two metrics of average transmit-power are obtained to predict how the user-transmit-powers are distributed within the cell coverage. Numerical experiments show that the analyzed and simulated results are well matched and the gap is smaller than 2 dB for most cases, thus validating the conducted performance analysis. The analysis is further extended to imperfect SIC case. For this case, sufficient and necessary condition for the feasibility of the SINR-constrained user-precoder-design problem is obtained. Using the condition, bounds on system parameters such as scheduled user number and SINR target are derived for feasible MIMO-NOMA implementation.
... The relation between boxed plane partitions and non-intersecting lattice paths is well known, see, e.g., [56] and reference therein. ...
We consider the four-vertex model with a particular choice of fixed boundary conditions, closely related to scalar products of off-shell Bethe states. In the scaling limit, the model exhibits the limit shape phenomenon, with the emergence of an arctic curve separating a central disordered region from six frozen `corners' of ferroelectric or anti-ferroelectric type. We determine the analytic expression of the interface by means of the Tangent Method. We extend the method to situations in which the curve separates disorder from anti-ferroelectric order. We supplement this heuristic method with an alternative, rigorous derivation of the arctic curve. This is based on the exact evaluation of suitable correlation functions, discriminating spatial transition from order to disorder, in terms of the partition function of some discrete log-gas associated to Hahn polynomials. As a by-product, we also deduce that the arctic curve's fluctuations are governed by the Tracy-Widom distribution.
... The fact that not only the number of plane partitions that fit in a box (equivalently, lozenge tilings 1 of a hexagon), but also all the symmetry classes (a total of ten) are given by simple product formulas, is of singular beauty in enumerative combinatorics. 2 This has been a rich source of inspiration for many researchers over the last four decades. Just to skim the surface, we mention [1,4,36,42,47,48] and the survey [39] for more recent developments. Works of the first author inspired by this include [7-11, 15, 20, 23]. ...
We consider regions obtained from 120 degree rotationally invariant hexagons by removing a core and three equal satellites (all equilateral triangles) so that the resulting region is both vertically symmetric and 120 degree rotationally invariant, and give simple product formulas for the number of their lozenge tilings. We describe a new method of approach for proving these formulas, and give the full details for an illustrative special case. As a byproduct, we are also able to generalize this special case in a different direction, by finding a natural counterpart of a twenty year old formula due to Ciucu, Eisenkölbl, Krattenthaler, and Zare, which went unnoticed until now. The general case of the original problem will be treated in a subsequent paper. We then work out consequences for the correlation of holes, which were the original motivation for this study.
... For a proof and historical remarks on the Desnanot-Jacobi identity, we refer the reader to [5,Theorem 3.12]. If p a ℓ ℓ divides b ′ 3,3 , we write det(B ′ ) as its expansion along row 1 plus its expansion along row 2 minus its expansion along row 3 to get ...
An arithmetical structure on a finite, connected graph without loops is given by an assignment of positive integers to the vertices such that, at each vertex, the integer there is a divisor of the sum of the integers at adjacent vertices, counted with multiplicity if the graph is not simple. Associated to each arithmetical structure is a finite abelian group known as its critical group. Keyes and Reiter gave an operation that takes in an arithmetical structure on a finite, connected graph without loops and produces an arithmetical structure on a graph with one fewer vertex. We study how this operation transforms critical groups. We bound the order and the invariant factors of the resulting critical group in terms of the original arithmetical structure and critical group. When the original graph is simple, we determine the resulting critical group exactly.
... Proof. For a proof of the previous lemma, we refer to [12] Page 111. ...
In this thesis, we consider the discrete moment problem that is the
problem associated with discrete measures. First, we extend the idempotent
approach from the truncated case to the full one. The main
results obtained establish a bridge between the study of discrete measures
and the existence of some bases in the Hilbert space associated
with the moment problem.
On the other hand, we introduce the notion of k-positive matrices,
we show that there is rank propagation phenomena associated with
this family of matrices. A phenomenon already noticed by Stampfli
in his work on moment matrices and by Curto in the study of k-
hyponormal shifts.
Finally, we will almost solve the quintic moment problem in the
complex case, by a new method, based on the notion of generator
polynomials and a systematic study of the complex moment matrix.
... This stems from the fact that they are in one-to-one correspondence with Alternating Sign Matrices (ASMs) of size n, i.e. n × nmatrices with entries in {0, 1, −1} where in each row and column the non-zero entries alternate in sign and sum up to 1. More details can be found in [Bre99]. In Section 3 we show a correspondence between DMTs with bottom row (n, n, n − 1, n − 1, . . . ...
International audience
The number of Monotone Triangles with bottom row is given by a polynomial in n variables. The evaluation of this polynomial at weakly decreasing sequences turns out to be interpretable as signed enumeration of new combinatorial objects called Decreasing Monotone Triangles. There exist surprising connections between the two classes of objects – in particular it is shown that . In perfect analogy to the correspondence between Monotone Triangles and Alternating Sign Matrices, the set of Decreasing Monotone Triangles with bottom row is in one-to-one correspondence with a certain set of ASM-like matrices, which also play an important role in proving the claimed identity algebraically. Finding a bijective proof remains an open problem.
Le nombre de Triangles Monotones ayant pour dernière ligne est donné par un polynôme en n variables. Il se trouve que les valeurs de ce polynôme en les suites décroissantes peuvent s'interpréter comme l'énumération signée de nouveaux objets appelés Triangles Monotones Décroissants. Il existe des liens surprenants entre ces deux classes d'objets – en particulier on prouvera l'identité . En parfaite analogie avec la correspondance entre Triangles Monotones et Matrices à Signe Alternant, l'ensemble des Triangles Monotones Décroissants ayant pour dernière ligne est en correspondance biunivoque avec un certain ensemble de matrices similaires aux MSAs, ce qui joue un rôle important dans la preuve algébrique de l'identité précédente. C'est un problème ouvert que d'en donner une preuve bijective.
... Back to the proposed problem (b). Using the Jacobi triple product identity ( [8], p. 497, [25], p. 50), we have ...
This book is an outgrowth of a collection of sixty-two problems offered in the The American Mathematical Monthly (AMM) the author has worked over the last two decades. Each selected problem has a central theme, contains gems of sophisticated ideas connected to important current research, and opens new vistas in the understanding of mathematics.
The AMM problem section provides one of the most challenging and interesting problem sections among the various journals and online sources currently available. The published problems and solutions have become a treasure trove rife with mathematical gems.
The author presents either his published solution in the AMM or an alternative solution to the published one to present and develop problem-solving techniques.
A rich glossary of important theorems and formulas is included for easy reference. The reader may regard this book as a starter set for AMM problems, providing a jumping of point to new ideas, and extending their personal lexicon of problems and solutions.
This collection is intended to encourage the reader to move away from routine exercises toward creative solutions, as well as offering the reader a systematic illustration of how to organize the transition from problem solving to exploring, investigating and discovering new results.
... The six-vertex model has broad connections to combinatorics. The resolution of the famous Alternating Sign Matrix conjecture is one example [35,41,48,36,7]. Also, the Tutte polynomial on a planar graph at the point (3,3) is precisely 1/2 of Z Six on its medial graph which is also a planar graph with a specific weight assignment [37]. ...
... As in the proof of the corresponding result for the 6V model without reflecting end (see, e.g., Section 7.1 of [6]), we will count arrows. In our case we need to differentiate between the vertices of the ingoing and outgoing rows. ...
... As in the proof of the corresponding result for the 6V model without reflecting end (see e.g. §7.1 of [6]), we will count arrows. In our case we need to differentiate between the vertices of the ingoing and outgoing rows. ...
We study the connection between the three-color model and the polynomials of Bazhanov and Mangazeev, which appear in the eigenvectors of the Hamiltonian of the XYZ spin chain. By specializing the parameters in the partition function of the 8VSOS model with DWBC and reflecting end, we find an explicit combinatorial expression for in terms of the partition function of the three-color model with the same boundary conditions. Bazhanov and Mangazeev conjectured that has positive integer coefficients. We prove the weaker statement that and have positive integer coefficients. Furthermore, for the three-color model, we find some results on the number of states with a given number of faces of each color, and we compute strict bounds for the possible number of faces of each color.
... Lascoux [Las02] introduced an alternate combinatorial model for (double) Schubert polynomials using states of the square-ice ("6-vertex") model from statistical physics. (For background and history of these ideas, see, e.g., [Bax82,Bre99,EKLP92,Kup96,RR86].) Recently, T. Lam, S.-J. ...
The geometric naturality of Schubert polynomials and their combinatorial pipe dream representations was established by Knutson and Miller (2005) via antidiagonal Gr\"obner degeneration of matrix Schubert varieties. We consider instead diagonal Gr\"obner degenerations. In this dual setting, Knutson, Miller, and Yong (2009) obtained alternative combinatorics for the class of "vexillary'' matrix Schubert varieties. We initiate a study of general diagonal degenerations, relating them to a neglected formula of Lascoux (2002) in terms of the 6-vertex ice model (recently rediscovered by Lam, Lee, and Shimozono (2018) in the guise of "bumpless pipe dreams'').
... For the proof of this lemma, we refer to [12]. ...
In the paper we study 𝑘-positive matrices, that is, the class of Hankel matrices,
for which the (𝑘 + 1) × (𝑘 + 1)-block-matrices are positive semi-definite. This notion is
intimately related to a 𝑘-hyponormal weighted shift and to Stieltjes moment sequences.
Using elementary determinant techniques, we prove that for a 𝑘-positive matrix, a 𝑘 × 𝑘-
block-matrix has non zero determinant if and only if all 𝑘 × 𝑘-block matrices have non
zero determinant. We provide several applications of our main result. First, we extend the
Curto-Stampfly propagation phenomena for for 2-hyponormal weighted shift 𝑊𝛼 stating
that if 𝛼𝑘 = 𝛼𝑘+1 for some 𝑛 > 1, then for all 𝑛 > 1, 𝛼𝑛 = 𝛼𝑘, to 𝑘-hyponormal weighted
shifts to higher order. Second, we apply this result to characterize a recursively generated
weighted shift. Finally, we study the invariance of 𝑘-hyponormal weighted shifts under
one rank perturbation. A special attention is paid to calculating the invariance interval of
2-hyponormal weighted shift; here explicit formulae are provided.
... Interest in alternating sign matrices intensified following the discovery of these deep connections between apparently disparate fields. A detailed and engaging account of the resolution of the alternating sign matrix conjecture can be found in the book by Bressoud [7]. ...
We investigate a class of 2-edge coloured bipartite graphs known as alternating signed bipartite graphs (ASBGs) that encode the information in alternating sign matrices. The central question is when a given bipartite graph admits an ASBG-colouring; a 2-edge colouring such that the resulting graph is an ASBG. We introduce the concept of a difference-1 colouring, a relaxation of the concept of an ASBG-colouring, and present a set of necessary and sufficient conditions for when a graph admits a difference-1 colouring. The relationship between distinct difference-1 colourings of a particular graph is characterised, and some classes of graphs for which all difference-1 colourings are ASBG-colourings are identified. One key step is Theorem 3.4.6, which generalises Hall's Matching Theorem by describing a necessary and sufficient condition for the existence of a subgraph H of a bipartite graph in which each vertex v of H has some prescribed degree r(v).
... This has been a rich source of inspiration for many researchers over the last four decades. Just to skim the surface, we mention [1,43,38,44,4,33] and the survey [35] for more recent developments. Works of the first author inspired by this include Key words and phrases. ...
We consider regions obtained from 120 degree rotationally invariant hexagons by removing a core and three equal satellites (all equilateral triangles) so that the resulting region is both vertically symmetric and 120 degree rotationally invariant, and give simple product formulas for the number of their lozenge tilings. We describe a new method of approach for proving these formulas, and give the full details for an illustrative special case. As a byproduct, we are also able to generalize this special case in a different direction, by finding a natural counterpart of a twenty year old formula due to Ciucu, Eisenk\"olbl, Krattenthaler and Zare, which went unnoticed until now. The general case of the original problem will be treated in a subsequent paper. We then work out consequences for the correlation of holes, which were the original motivation for this study.
... The quantum inverse scattering method [4,5] was applied to solve the four-vertex model on a finite lattice with various boundary conditions in [6][7][8]. In these papers, the relation of the model to the theory of random walks and plane partitions [9,10] was also discussed. ...
The exactly solvable four-vertex model on a square grid with the fixed boundary conditions in a presence of a special external field is considered. Namely, we study a system in a linear field acting on the central column of the grid. The partition function of the model is calculated by the quantum inverse scattering method. The answer is written in determinantal form.
... The very remarkable fact that all symmetry classes of lozenge tilings of a hexagon are given by equally beautiful formulas (see [1][18] [15][19] [12] and the survey [2] for more recent developments) supplies ample motivation for considering the problem of enumerating symmetry classes of generalizations thereof. For a shamrock shape removed from the center of hexagons, this is treated in [3][5] [6][7] [8], and for fern-cored hexagons in [4], [16] and the present paper. ...
In this paper we enumerate the centrally symmetric lozenge tilings of a hexagon with a fern removed from its center. The proof is based on a variant of Kuo's graphical condensation method. An unexpected connection with the total number of tilings is established~---~when suitably normalized, the number of centrally symmetric tilings is equal to the square root of the total number of tilings. The results we present can be regarded as a new extension of the enumeration of self-complementary plane partitions that fit in a box.
Starting from a one dimensional vector space, we construct a categorification of a deformed Heiserberg algebra by Cautis and Licata's method. The Grothendieck ring of is . As an application, we discuss some related partition functions related to the MacMahon function of 3D Young diagram. We expect further applications of the results of this paper.
We present several generalizations of Cauchy's determinant and Schur's Pfaffian by considering matrices whose entries involve some generalized Vandermonde determinants. Special cases of our formulae include previuos formulae due to S.Okada and T. Sundquist. As an application, we give a relation for the Littlewood--Richardson coefficients involving a rectangular partition.
We present an elementary method for proving enumeration formulas which are polynomials in certain parameters if others are fixed and factorize into distinct linear factors over Z. Roughly speaking the idea is to prove such formulas by ``explaining'' their zeros using an appropriate combinatorial extension of the objects under consideration to negative integer parameters. We apply this method to prove a new refinement of the Bender-Knuth (ex-)Conjecture, which easily implies the Bender-Knuth (ex-)Conjecture itself. This is probably the most elementary way to prove this result currently known. Furthermore we adapt our method to q-polynomials, which allows us to derive generating function results as well. Finally we use this method to give another proof for the enumeration of semistandard tableaux of a fixed shape, which is opposed to the Bender-Knuth (ex-)Conjecture refinement a multivariate application of our method.
We show that recent determinant evaluations involving Catalan numbers and generalisations thereof have most convenient explanations by combining the Lindstr\"om-Gessel-Viennot theorem on non-intersecting lattice paths with a simple determinant lemma from [Manuscripta Math. 69 (1990), 173-202]. This approach leads also naturally to extensions and generalisations.
We investigate a class of 2-edge coloured bipartite graphs known as alternating signed bipartite graphs (ASBGs) that encode the information in alternating sign matrices. The central question is when a given bipartite graph admits an ASBG-colouring; a 2-edge colouring such that the resulting graph is an ASBG. We introduce the concept of a difference-1 colouring, a relaxation of the concept of an ASBG-colouring, and present a set of necessary and sufficient conditions for when a graph admits a difference-1 colouring. The relationship between distinct difference-1 colourings of a particular graph is characterised, and some classes of graphs for which all difference-1 colourings are ASBG-colourings are identified. One key step is Theorem 3.4.6, which generalises Hall's Matching Theorem by describing a necessary and sufficient condition for the existence of a subgraph H of a bipartite graph in which each vertex v of H has some prescribed degree r(v).
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