A bijective proof of the ASM theorem Part I: the operator formula

Alternating sign matrices are known to be equinumerous with descending plane partitions, totally symmetric self-complementary plane partitions and alternating sign triangles, but no bijective proof for any of these equivalences has been found so far. In this paper we provide the first bijective proof of the operator formula for monotone triangles, which has been the main tool for several non-combinatorial proofs of such equivalences. In this proof, signed sets and sijections (signed bijections) play a fundamental role. Mathematics Subject Classifications: 05A15


Introduction
An alternating sign matrix (ASM) is a square matrix with entries in {0, 1, −1} such that in each row and each column the non-zero entries alternate and sum to 1. Robbins and Rumsey introduced alternating sign matrices in the 1980s [RR86] when studying their λ-determinant (a generalization of the classical determinant) and showing that the λdeterminant can be expressed as a sum over all alternating sign matrices of fixed size. The classical determinant is obtained from this by setting λ = −1, in which case the sum reduces so that it extends only over all ASMs without −1's, i.e., permutation matrices, and the well-known formula of Leibniz is recovered. Numerical experiments led Robbins and Rumsey to conjecture that the number of n × n alternating sign matrices is given by the surprisingly simple product formula n−1 i=0 (3i + 1)! (n + i)! .
(1) Back then the surprise was even bigger when they learned from Stanley (see [BP99,Bre99]) that this product formula had recently also appeared in Andrews' paper [And79] on his proof of the weak Macdonald conjecture, which in turn provides a formula for the number of cyclically symmetric plane partitions. As a byproduct, Andrews had introduced descending plane partitions and had proven that the number of descending plane partitions (DPPs) with parts at most n is also equal to (1). Since then, the problem of finding an explicit bijection between alternating sign matrices and descending plane partitions has attracted considerable attention from combinatorialists. To many, it is remarkable that a bijection has not yet been found-all the more so because Mills, Robbins and Rumsey had also introduced several "statistics" on alternating sign matrices and on descending plane partitions for which they had strong numerical evidence that the joint distributions coincide as well, see [MRR83].
There were a few further surprises yet to come. Robbins introduced a new operation on plane partitions, complementation, and had strong numerical evidence that totally symmetric self-complementary plane partitions (TSSCPPs) in a 2n × 2n × 2n-box are also counted by (1). Again this was further supported by statistics that have the same joint distribution as well as certain refinements, see [MRR86,Kra96,Kra16,BC16]. We still lack an explicit bijection between TSSCPPs and ASMs, as well as between TSSCPPs and DPPs.
In his collection of bijective proof problems [Sta09,Problem 226], Stanley says the following about the problem of finding all these bijections: "This is one of the most intriguing open problems in the area of bijective proofs." In Krattenthaler's survey on plane partitions [Kra16] he expresses his opinion by saying: "The greatest, still unsolved, mystery concerns the question of what plane partitions have to do with alternating sign matrices." Many of the above mentioned conjectures have since been proved by non-bijective means: Zeilberger [Zei96a] was the first who proved that n × n ASMs are counted by (1). Kuperberg gave another, shorter proof [Kup96] based on the remarkable observation that the six-vertex model (which had been introduced by physicists several decades earlier) with domain wall boundary conditions is equivalent to ASMs, see [EKLP92a,EKLP92b], and he used the techniques that had been developed by physicists to study this model. Andrews enumerated TSSCPPs in [And94]. The equidistribution of certain statistics for ASMs and of certain statistics for DPPs has been proved in [BDFZJ12,BDFZJ13], while for ASMs and TSSCPPs see [Zei96b,FZJ08], and note in particular that already in Zeilberger's first ASM paper [Zei96a] he could deal with an important refinement. Further work including the study of symmetry classes has been accomplished; for a more detailed description of this we defer to [BFK17]. Then, in very recent work, alternating sign triangles (ASTs) were introduced in [ABF20], which establishes a fourth class of objects that are equinumerous with ASMs, and also in this case nobody has so far been able to construct a bijection.
Another aspect that should be mentioned here is Okada's work [Oka06](see also [Str]), which hints at a connection between ASMs and representation theory that has not yet been well understood. He observed that a certain multivariate generating function (a specialization at a root of unity of the partition function that had been introduced by physicists in their study of the six-vertex model) can be expressed-up to a power of 3-by a single Schur polynomial. Since Schur polynomials are generating functions of semistandard tableaux, this establishes yet another challenging open problem for combinatorialists inclined to find bijections.
The proofs of the results briefly reviewed above contain rather long and complicated computations, and include hardly any arguments of a combinatorial flavor; in this paper we refer to such proofs as "computational" proofs. In fact, it seems that all ASM-related identities for which there exists a bijective proofs are trivial, with the exception of the rotational invariance of fully packed loop configurations. This was proved bijectively by Wieland [Wie00] and is also used in the celebrated proof of the Razumov-Stroganov (ex-)conjecture [CS11].
We come now to the purpose of the current paper. This is the first paper in a planned series that seeks to give the first bijective proofs of several results described so far. The seed of the idea to do so came from a brief discussion of the first author with Zeilberger on the problem of finding such bijections at the AMS-MAA Joint Mathematics Meetings in January 2019. Zeilberger mentioned that such bijections can be constructed from existing "computational" proofs but that, most likely, these bijections would be complicated. The authors of the current paper agree-in fact, the first author gave her "own" proof of the ASM theorem in [Fis06,Fis07,Fis16] and expressed some speculations in this direction in the final section of the last paper. There is obviously no guarantee that there exists a simple, satisfactory bijective proof of the ASM theorem that does not involve the Garsia-Milne involution principle. This is how the authors of the current paper decided to work on converting the proof in [Fis16] into a bijective proof. After having figured out how to actually convert computations and also having shaped certain useful fundamental concepts related to signed sets (see Section 2), the translation of several steps became quite straightforward; other steps were quite challenging. Then a certain type of (exciting) dynamics evolved, where the combinatorial point of view led to simplifications and other modifications, and after this process the original "computational" proof is in fact rather difficult to recognize. For several obvious reasons, we find it essential to check all our constructions with computer code (for details see the final section and [FKb]); to name one it can possibly be used to identify new equivalent statistics in future work.
After the above mentioned simplifications, it seems that signs are unavoidable. After all, if there would be a simple bijective proof that avoided signs, would it not also be plausible that such a proof could be converted into a simple "computational" proof that avoids signs? Such a proof has also not been found so far.

The operator formula
We use the well-known correspondence between order n×n ASMs and monotone triangles with bottom row 1, 2, . . . , n. A monotone triangle is a triangular array (a i,j ) 1≤j≤i≤n of integers, where the elements are usually arranged as follows a 1,1 a 2,1 a 2,2 . . . . . . . . . a n−2,1 . . . . . . a n−2,n−2 a n−1,1 a n−1,2 . . . . . . a n−1,n−1 a n,1 a n,2 a n,3 . . . . . . a n,n (2) such that the integers increase weakly along ↗-diagonals and ↘-diagonals, and increase strictly along rows, i.e., a i,j ≤ a i−1,j ≤ a i,j+1 and a i,j < a i,j+1 for all i, j with 1 ≤ j < i ≤ n. In order to convert an ASM into the corresponding monotone triangle, add to each entry all the entries that are in the same column above it, and record then row by row the positions of the 1's, see Figure 1 for an example.
The following operator formula for the number of monotone triangles with prescribed bottom row was first proved in [Fis06] (see [Fis10,Fis16] for simplifications and generalizations). Note that we allow arbitrary strictly increasing bottom rows.
the electronic journal of combinatorics 27(3) (2020), #3.35 The purpose of this paper is to provide a bijective proof of Theorem 1, following the approach suggested in [Fis16]. While the operator formula is an interesting result in its own right, it has also been the main tool for proofs of several results mentioned above. This will be reviewed in the final section of this paper along with indications for future projects on converting also these proofs into bijective proofs.
In order to be able to construct a bijective proof of Theorem 1, we need to interpret (3) combinatorially. Recall that Gelfand-Tsetlin patterns are defined as monotone triangles with the condition on the strict increase along rows being dropped, see [Sta99,p. 313] or [GC50,(3)] for the original reference 1 . It is well known that the number of Gelfand-Tsetlin patterns with bottom row which is the operand in the formula (3).
kn (keeping a copy for each multiplicity), (3)is a signed enumeration of certain Gelfand-Tsetlin patterns, where each monomial causes a deformation of the bottom row k 1 , . . . , k n . It is useful to encode these deformations by arrow patterns as defined in Section 5, where we choose arrows in a triangular manner so that the arrows coming from E kp + E −1 kq − E kp E −1 kq are situated in the p-th ↙-diagonal and the q-th ↘-diagonal, and placing k 1 , . . . , k n in the bottom row will allow us to describe the deformation coming from a particular monomial in a convenient way. The combinatorial objects associated with (3) then consist of a pair of such an arrow pattern and a Gelfand-Tsetlin pattern where the bottom row is a deformation of k 1 , . . . , k n as prescribed by the arrow pattern. This will lead directly to the definition of shifted Gelfand-Tsetlin patterns. Let us clarify that "shifted" refers to the shift operator, and not to shifted tableaux or some kind of B or C type Gelfand-Tsetlin patterns.
A sign comes from picking −E kp E −1 kq , but there is also a more subtle appearance. The deformation induced by the arrow pattern can cause a deformation of the increasing bottom row k 1 , k 2 , . . . , k n into a sequence that is not increasing. Therefore we are in need of an extension of the combinatorial interpretation of (4) to any sequence k 1 , . . . , k n of integers. Such an interpretation was given in [Fis05] and is repeated below in Section 4.

Outline of the bijective proof
Given a sequence k 1 < ⋅ ⋅ ⋅ < k n , it suffices to find an injective map from the set of monotone triangles with bottom row k 1 , . . . , k n to our shifted Gelfand-Tsetlin patterns associated with k 1 , . . . , k n so that the images under this map have positive signs, along with a signreversing involution on the set of shifted Gelfand-Tsetlin patterns that are not the image of a monotone triangle.
We will accomplish something more general, as we will also consider an extension of monotone triangles to all integer sequences k 1 , . . . , k n , see Section 5, along with a sign function on these objects, and prove that the operator formula also holds in this instance. To do that, we will construct a sign-reversing involution on a subset of monotone triangles, another sign-reversing involution on a subset of shifted Gelfand-Tsetlin patterns, and a sign-preserving bijection between the remaining monotone triangles and the remaining shifted Gelfand-Tsetlin patterns. Note that this is actually equivalent to the construction of a bijection between the (disjoint) union of the "positive" monotone triangles and the "negative" shifted Gelfand-Tsetlin patterns, and the (disjoint) union of the "negative" monotone triangles and the "positive" shifted Gelfand-Tsetlin patterns. We call such maps sijections for general signed sets. For an illustration, see Figure 1. In the figure, S + (resp. S − ) refers to positive (resp. negative) monotone triangles, and T + (resp. T − ) refers to positive (resp. negative) shifted Gelfand-Tsetlin patterns. Furthermore, there is a sign-reversing involution on the blue (resp. green) part of S (resp. T ), and a bijection between light (resp. dark) gray parts of S + and T + (resp. S − and T − ). It is clear that this The actual construction here will make use of the recursion underlying monotone triangles. For a monotone triangle with bottom row k 1 , . . . , k n , the eligible penultimate rows l 1 , . . . , l n−1 are those with and l 1 < l 2 < ⋅ ⋅ ⋅ < l n−1 . This establishes a recursion that can be used to construct all monotone triangles. Phrased differently, "at" each k i we need to sum over all l i−1 , l i such that l i−1 ≤ k i ≤ l i and l i−1 < l i . 2 However, we can split this into the following three cases: 3. Combining (1) and (2), we have done some double counting, thus we need to subtract the intersection, i.e., all This can be written as a recursion. The arrow rows in Section 5 are used to describe this recursion: we choose ↖ "at" k i if we are in Case (1), ↗ in Case (2), and ↖ ↗ in Case (3).
Our main effort will be to show "sijectively" that shifted Gelfand-Tsetlin patterns also fulfill this recursion.

Outline of the paper
The remainder of this paper is devoted to the bijective proof of Theorem 1 (or rather, the more general version with the increasing condition on k 1 , . . . , k n dropped). In Section 2 we lay the groundwork by defining concepts like signed sets and sijections, and we extend known concepts such as disjoint union, Cartesian product and composition for ordinary sets and bijections to signed sets and sijections. The composition of sijections will use a variation of the well-known Garsia-Milne involution principle [GM81,And86]. Many of the signed sets we will be considering are signed boxes (Cartesian products of signed intervals) or at least involve them, and we define some sijections on them in Section 3. These sijections will be the building blocks of our bijective proof later on. In Section 4 we introduce the extended Gelfand-Tsetlin patterns and construct some related sijections.
In Section 5, we finally define the extended monotone triangles as well as the shifted Gelfand-Tsetlin patterns (i.e., the combinatorial interpretation of (3)), and use all the preparation to construct the sijection between monotone triangles and shifted Gelfand-Tsetlin patterns. In the final section, we discuss further projects.
To emphasize that we are not merely interested in the fact that two signed sets have the same size, but want to use the constructed signed bijection later on, we will be using a convention that is slightly unorthodox in our field. Instead of listing our results as lemmas and theorems with their corresponding proofs, we will be using the Problem-Construction terminology. See for instance [Voe] and [Bau].

Outline of future work
This is the first in a series of papers that will deal with bijective proofs of results in the theory of alternating sign matrices. The second paper [FKa] will give the first bijective proofs of the enumeration formula and of the relation between ASMs and DPPs. We expect Part III to cover the relation between DPPs and ASTs, and Part IV the relation between ASTs and TSSCPPs. The constructions presented in this paper will be heavily used in all these follow-up papers, but we expect them to be more or less independent of one another. See Section 6 for more details.

Signed sets and sijections Signed sets
A signed set is a pair of disjoint finite sets: S = (S + , S − ) with S + ∩ S − = ∅. Equivalently, a signed set is a finite set S together with a sign function sign∶ S → {1, −1}. While we will mostly avoid the use of the sign function altogether (with the exception of monotone triangles defined in Section 5), it is useful to keep this description at the back of one's mind. Note that throughout the paper, signed sets are underlined. We will write i ∈ S to mean i ∈ S + ∪ S − .
The size of a signed set S is S = S + − S − . The opposite signed set of S is −S = (S − , S + ). We have −S = − S . The Cartesian product of signed sets S and T is and we can similarly (or recursively) define the Cartesian product of a finite number of signed sets. We have The intersection of signed sets S and T is defined as we can extend these definitions to a finite family of signed sets.
Example. One of the crucial signed sets is the signed interval We will also see many signed boxes, Cartesian products of signed intervals. Note that S + = ∅ or S − = ∅ for every signed box S.
Signed subsets T ⊆ S are defined in an obvious manner, in particular, for s ∈ S, we have The disjoint union of signed sets S and T is the signed set with elements (s, 0) for s ∈ S and (t, 1) for t ∈ T . If S and T are signed sets with More generally, we can define the disjoint union of a family of signed sets S t , where the family is indexed with a signed set T : The usual properties such as associativity (S ⊔ T ) ⊔ U = S ⊔ (T ⊔ U ) and distributivity (S ⊔ T ) × U = S × U ⊔ T × U also hold. Strictly speaking, the = sign here and sometimes later on indicates that there is an obvious and natural sign-preserving bijection between the two signed sets. We summarize a few more basic properties that will be needed in the following and that are easy to prove.

Sijections
The role of bijections for signed sets is played by "signed bijections", which we call sijec- where ⊔ refers to the disjoint union for ordinary ("unsigned") sets. It follows that also ϕ(S − ⊔ T + ) = S + ⊔ T − . There is an obvious sijection id S ∶ S ⇒ S. We can think of a sijection as a collection of a sign-reversing involution on a subset of S, a sign-reversing involution on a subset of T , and a sign-preserving matching between the remaining elements of S with the remaining elements of T . When S − = T − = ∅, the signed sets can be identified with ordinary sets, and a sijection in this case is simply a bijection.
A sijection is a manifestation of the fact that two signed sets have the same size. Indeed, if there exists a sijection ϕ∶ S ⇒ T , we have S For a signed set S, there is a natural sijection ϕ from S ⊔ (−S) to the empty signed set ∅ = (∅, ∅). Indeed, the involution should be defined on , and so we can take ϕ((s, 0), 0) = ((s, 1), 0), ϕ((s, 1), 0) = ((s, 0), 0). Note that in general, a sijection from a signed set S to ∅ is simply a sign-reversing involution on S, in other words, a bijection between S + and S − .
If we have a sijection ϕ∶ S ⇒ T , there is a natural sijection −ϕ∶ −S ⇒ −T (as a map, it is actually precisely the same).
If we have sijections ϕ i ∶ S i ⇒ T i for i = 0, 1, then there is a natural sijection ϕ∶ S 0 ⊔S 1 ⇒ T 0 ⊔T 1 . More interesting ways to create new sijections are described below in Proposition 1, but we will need this in our first construction for the special case S 0 = T 0 and ϕ 0 = id S 0 .
To motivate our first result, note that if . This map will be the crucial building block for more complicated sijections.

(Cartesian product) Suppose we have sijections
the electronic journal of combinatorics 27(3) (2020), #3.35 3. (Disjoint union) Suppose we have signed sets T ,T and a sijection ψ∶ T ⇒T . Furthermore, suppose that for every t ∈ T ⊔T , we have a signed set S t and a sijection ϕ t ∶ S t ⇒ S ψ(t) satisfying ϕ ψ(t) = ϕ −1 t . Then ϕ = ⊔ t∈T ⊔T ϕ t , defined by One important special case of Proposition 1 (3) is T =T and ψ = id. We have two sets of signed sets indexed by T , S (t,0) =∶ S 0 t and S (t,1) =∶ S 1 t , and sijections ϕ t ∶ S 0 t ⇒ S 1 t . By the proposition, these sijections have a disjoint union that is a sijection ⊔ t∈T S 0 t ⇒ ⊔ t∈T S 1 t . By the proposition, the relation S ≈ T ⇐⇒ there exists a sijection from S to T is an equivalence relation on signed sets.

Example. A signed set
Other examples of elementary signed sets appear in the statements of Problems 2 and 3 (in both cases, they are of dimension n − 1).
Normality is preserved under Cartesian product, disjoint union etc. For example, the sijection obtained by using α a 1 ,b 1 ,c 1 × α a 2 ,b 2 ,c 2 and distributivity on disjoint unions, is normal.
The main reason normal sijections are important is that they give a very natural special case of Proposition 1 (3). Suppose that T andT are elementary signed sets of dimension n, and that ψ∶ T ⇒T is a normal sijection. Furthermore, suppose that we have a signed set S k for every k ∈ Z n . Then we have a sijection Indeed, Proposition 1 gives us a sijection provided that we have a sijection ϕ t ∶ S ξ(t) ⇒ S ξ(ψ(t)) satisfying ϕ ψ(t) = ϕ −1 t for every t ∈ T ⊔T . But since ξ(ψ(t)) = ξ(t), we can take ϕ t to be the identity.
Note that the 0 as the second coordinate in the example comes from the fact that a sijection in question is an involution on the disjoint union We could be a little less precise and write ϕ(x, (l 1 , l 2 )) = (x, (l 2 +1, l 1 −1)) without causing confusion.
The following generalizes the construction of Problem 1; indeed, for n = 2 the construction gives a sijection from [a Problem 2. Given a = (a 1 , . . . , a n−1 ) ∈ Z n−1 , b = (b 1 , . . . , b n−1 ) ∈ Z n−1 , x ∈ Z, construct a normal sijection Construction. The proof is by induction, with the case n = 1 being trivial and the case n = 2 was constructed in Problem 1. Now, for n ≥ 3, where for the last equivalence we have again used distributivity. Normality follows from the normality of all the sijections involved in the construction.
Problem 3. Given k = (k 1 , . . . , k n ) ∈ Z n and x ∈ Z, construct a normal sijection Construction. The proof is by induction with respect to n. The case n = 1 is trivial, and n = 2 is Problem 1. Now take n > 2. By the induction hypothesis (for (k 1 , . . . , k n−1 ) and x + 1), we have We use distributivity. We keep all terms except the one corresponding to i = n − 1 in the first part. Because we obtain the required Cartesian products for the first term on the right-hand side at i = n, the second term at i = n − 2, and the first term at i = n − 1. Again, normality follows from the fact that α is normal.

Gelfand-Tsetlin patterns
Using our definition of a disjoint union of signed sets, it is easy to define generalized Gelfand-Tsetlin patterns, or GT patterns for short (compare with [Fis05]).
Of course, one can think of an element of GT(k) in the usual way, as a triangular array A = (A i,j ) 1≤j≤i≤n of n+1 2 numbers, arranged as The sign of such an array is (−1) m , where m is the number of (i, j) with a i,j > a i,j+1 . Some crucial sijections for GT patterns are given by the following constructions.
Problem 4. Given a = (a 1 , . . . , a n−1 ) The result is important because while it adds a dimension to GT patterns, it (typically) greatly reduces the size of the indexing signed set. In fact, there is an analogy to the fundamental theorem of calculus: instead of extending the disjoint union over the entire signed box, it suffices to consider the boundary; x corresponds in a sense to the constant of integration.
In the following problem, the second sijection is necessary to construct the first.
the electronic journal of combinatorics 27(3) (2020), #3.35 By definition, the first signed set on the right-hand side is − GT(k 1 , . . . , k i−1 , k i+1 + 1, k i − 1, k i+2 , . . . , k n ). The other three disjoint unions all satisfy the condition needed for the existence of σ (for i, for i − 1 and for both, i − 1 and i, respectively), and hence we can siject them to ∅. If i = 1 or i = n − 1, the proof is similar but easier (as we only have to use α once, and we get only two factors after using distributivity). Details are left to the reader. Now take l = (l 1 , . . . , l n ) and l ′ = (l 1 , . . . , l i−1 , l i+1 + 1, l i − 1, l i+2 , . . . , l n ). The sijection σ can then be defined as .
It is easy to check that this is a sijection. Compare with Example 3.
All disjoint unions in the second term satisfy the conditions for the existence of σ from Problem 5, so we can siject them to ∅. This gives a sijection which is, by the definition of GT, a sijection GT(k 1 , . . . , k n ) ⇒ ⊔ n i=1 GT(k 1 , . . . , k i−1 , x + n − i, k i+1 , . . . , k n ).

Combinatorics of the monotone triangle recursion
One side of the operator formula: Monotone triangles Suppose that k = (k 1 , . . . , k n ) and l = (l 1 , . . . , l n−1 ) are two sequences of integers. We say that l interlaces k, l ≺ k, if the following holds: 1. for every i, 1 ≤ i ≤ n − 1, l i is in the closed interval between k i and k i+1 ; 2. if k i−1 ≤ k i ≤ k i+1 for some i, 2 ≤ i ≤ n − 1, then l i−1 and l i cannot both be k i ; For example, if k 1 < k 2 < ⋅ ⋅ ⋅ < k n , then l i ∈ [k i , k i+1 ] and l 1 < l 2 < ⋅ ⋅ ⋅ < l n−1 . A monotone triangle of size n is a map T ∶ {(i, j)∶ 1 ≤ j ≤ i ≤ n} → Z so that line i − 1 (i.e. the sequence T i−1,1 , . . . , T i−1,i−1 ) interlaces line i (i.e. the sequence T i,1 , . . . , T i,i ).
Example. The following is a monotone triangle of size 5: This notion of (generalized) monotone triangle was introduced in [Rie13]. Other notions appeared in [Fis12].
The sign of a monotone triangle T is (−1) r , where r is the sum of: • the number of strict descents in the rows of T , i.e. the number of pairs (i, j) so that 1 ≤ j < i ≤ n and T i,j > T i,j+1 , and • the number of (i, j) so that 1 ≤ j ≤ i−2, i ≤ n and T i, The sign of our example is −1.
We denote the signed set of all monotone triangles with bottom row k by MT(k). Recall that monotone triangles form one side of the operator identity in Theorem 1.
It turns out that MT(k) satisfies a recursive "identity". Let us define the signed set of arrow rows of order n as AR n = ({↗, ↖}, {↖ ↗}) n .
Alternatively, we can think of them as rows of length n with elements ↖, ↗, ↖ ↗, where the positive elements are precisely those with an even number of ↖ ↗'s. The role of an arrow row µ of order n is that it induces a deformation of [k 1 , k 2 ] × [k 2 , k 3 ] × ⋅ ⋅ ⋅ × [k n−1 , k n ] as follows. Consider and if µ i ∈ {↖, ↖ ↗} (that is we have an arrow pointing towards [k i−1 , k i ]) then k i is decreased by 1 in [k i−1 , k i ], while there is no change for this k i if µ i =↗. If µ i ∈ {↗, ↖ ↗} (that is we have an arrow pointing towards [k i , k i+1 ]) then k i is increased by 1 in [k i , k i+1 ], while there is no change for this k i if µ i =↖.
Problem 7. Given k = (k 1 , . . . , k n ), construct a sijection Construction. The map we will construct maps all elements on the left to the right, and there will be quite a few cancellations on the right-hand side. More specifically, take a monotone triangle T with bottom row k. Then Ξ(T ) = ((T ′ , l), µ), where T ′ is the monotone triangle we obtain from T by deleting the last row, l is the bottom row of T ′ , and µ = (µ 1 , . . . , µ n ) is the arrow row defined as follows: • µ 1 =↖; • µ n =↗; • for 1 < i < n, µ i is determined as follows: 3. otherwise, take µ i =↖.
It is easy to check that l is indeed in e(k, µ). Note that in (1) and (2) of the third bullet point, µ i is forced if we require l ∈ e(k, µ). In (3), µ i =↖ ↗ would also be possible if and only if µ i =↗ would also be possible. On the other hand, for ((T ′ , l), µ), define Ξ((T ′ , l), µ) as follows. For the construction it is useful to keep in mind that l ∈ e(k, µ) implies that conditions (1) and (2) for l ≺ k are satisfied.
If no such i exists, we take Ξ((T ′ , l), µ) = T , where we obtain T from T ′ by adding k as the last row. It is easy to see that this is a well-defined sijection.
Remark. The previous construction could have been avoided by using alternative extensions of monotone triangles provided in [Fis12]. However, the advantage of the definition used in this paper is that it is more reduced than the others in the sense that it can obtained from these by canceling elements using certain sign-reversing involutions.
The other side of the operator formula: Arrow patterns and shifted GT patterns Alternatively, we can think of an arrow pattern of order n as a triangular array T = (t p,q ) 1≤p<q≤n arranged as with t p,q ∈ {↙, ↘, ↙ ↘}, and the sign of an arrow pattern is 1 if the number of ↙ ↘'s is even and −1 otherwise. The role of an arrow pattern of order n is that it induces a deformation of (k 1 , . . . , k n ), which can be thought of as follows. Add k 1 , . . . , k n as bottom row of T (i.e., t i,i = k i ), and for each ↙ or↙ ↘ which is in the same ↙-diagonal as k i add 1 to k i , while for each ↘ or ↙ ↘ which is in the same ↘-diagonal as k i subtract 1 from k i . More formally, letting δ ↙ (↙) = δ ↙ (↙ ↘) = δ ↘ (↘) = δ ↘ (↙ ↘) = 1 and δ ↙ (↘) = δ ↘ (↙) = 0, we set δ ↘ (t j,i ) and d(k, T ) = (k 1 + c 1 (T ), k 2 + c 2 (T ), . . . , k n + c n (T )) the electronic journal of combinatorics 27(3) (2020), #3.35 for k = (k 1 , . . . , k n ) and T ∈ AP n .
For k = (k 1 , . . . , k n ) define shifted Gelfand-Tsetlin patterns, or SGT patterns for short, as the following disjoint union of GT patterns over arrow patterns of order n: Shifted Gelfand-Tsetlin patterns form the other side of the operator identity in Theorem 1. We will prove Theorem 1 by showing that monotone triangles and shifted Gelfand-Tsetlin patterns satisfy the same recurrence.
Problem 8. Given n and i, 1 ≤ i ≤ n, construct a sijection Construction. For T ∈ AP n−1 , take Ψ(T ) = (t ′ p,q ) 1≤p<q≤n to be the arrow pattern defined by The relation between monotone triangles and shifted Gelfand-Tsetlin patterns The difficult part of this paper is to prove that SGT satisfies the same "recursion" as MT.
While the proof of the recursion was easy for monotone triangles, it is very involved for shifted GT patterns, and needs almost all the sijections we have constructed in this and previous sections.
Problem 9. Given k = (k 1 , . . . , k n ) ∈ Z n and x ∈ Z, construct a sijection Construction. To make the construction of Φ a little easier, we will define it as the composition of several sijections. The first one will reduce the indexing sets (from a signed box to its "corners") using Problem 4. The second one increases the order of the arrow patterns using the sijection from Problem 8. The third one further reduces the indexing set (from a signed set with 2 n−1 elements to [1, n]). The last one gets rid of the arrow row and then uses Problem 6.
If we switch disjoint unions one last time, we can use the sijection τ −1 (see Problem 6), and we get ⊔ T ∈AP n GT(d(k, T )) = SGT(k).
This completes the construction of Φ 4 and therefore of Φ.
Construction. The proof is by induction on n. For n = 1, both sides consist of one (positive) element, and the sijection is obvious. Once we have constructed Γ for all lists of length less than n, we can construct Γ k,x as the composition of sijections where ⊔ ⊔ Γ means ⊔ µ∈AR n ⊔ l∈e(k,µ) Γ l,x .
The main sijection Γ indeed depends on the choice x. As an example, take k = (1, 2, 3). In this case, MT(k) has 7 positive elements, and SGT(k) has 10 positive and 3 negative elements. For x = 0, the sijection is given by • The operator formula was used in [Fis07] to show that n × n ASMs are counted by (1) and, more generally, to count ASMs with respect to the position of the unique 1 in the top row.
• While working on this project, we actually realized that the final calculation in [Fis07] also implies that ASMs are equinumerous with DPPs without having to use Andrews' result [And79] on the number of DPPs; more generally, we can even prove that the refined count of n × n ASMs with respect to the position of the unique 1 in the top row agrees with the refined count of DPPs with parts no greater than n with respect to the number of parts equal to n. This was conjectured in [MRR83] and first proved in [BDFZJ12].
• In [Fis19b], the operator formula was used to show that ASTs with n rows are equinumerous with TSSCPPs in a 2n × 2n × 2n-box. Again we do not rely on Andrews' result [And94] on the number of TSSCPPs and we were actually able to deal with a refined count again (which has also the same distribution as the position of the unique 1 in the top row of an ASM).
• In [Fis19a], we have considered alternating sign trapezoids (which generalize ASTs) and, using the operator formula, we have shown that they are equinumerous with objects generalizing DPPs. These objects were already known to Andrews and he actually enumerated them in [And79]. Later Krattenthaler [Kra06] realized that these more general objects are (almost trivially) equivalent to cyclically symmetric lozenge tilings of a hexagon with a triangular hole in the center. Again we do not rely on Andrews' enumeration of these generalized DPPs, and in this case we were able to include three statistics.
We plan to work on converting the proofs just mentioned into bijective proofs. For those mentioned in the first and second bullet point, this has already been worked out. The attentive reader will have noticed that working out all of them will link all four known classes of objects that are enumerated by (1).