Abstract
We generalize multivariate hook product formulae for P -partitions. We use Macdonald symmetric functions to prove a (q, t)-deformation of Gansner's hook product formula for the generating functions of reverse (shifted) plane partitions. (The unshifted case has also been proved by Adachi.) For a d-complete poset, we present a conjectural (q, t)-deformation of Peterson-Proctor's hook product formula.
Introduction
R. Stanley [14] introduced the notion of P -partitions for a poset P , and studied univariate generating functions of them. A P -partition is an orderreversing map from P to the set of non-negative integers N, i.e., a map σ : P → N satisfying the condition:
if x ≤ y in P , then σ(x) ≥ σ(y).
Let A(P ) denote the set of all P -partitions. Typical examples of P -partitions are reverse plane partitions and reverse shifted plane partitions. If λ is a partition, then its diagram
can be viewed as a poset by defining (i, j) ≥ (k, l) if i ≤ k and j ≤ l, and the resulting poset is called a shape. A P -partition for this poset P = D(λ) is a where z k (k ∈ Z) are indeterminates. This weights σ by the sums of its diagonals. To state Gansner's hook product formulae, we introduce the notion of hooks for shapes and shifted shapes. For a partition λ and a cell (i, j) ∈ D(λ), the hook at (i, j) in D(λ) is defined by
For a strict partition µ and a cell (i, j) ∈ S(µ), the shifted hook at (i, j) in S(µ) is given by H S(µ) (i, j) = {(i, j)} ∪ {(i, l) ∈ S(µ) : l > j} ∪ {(k, j) ∈ S(µ) : k > i} ∪ {(j + 1, l) ∈ S(µ) : l > j}.
For a finite subset H ⊂ Z 2 , we write
Gansner [4] used the Hillman-Grassl correspondence to prove the following hook product formulae. (See also [13] .) Theorem 1.1. [4, Theorems 5.1 and 7.1] Let P be a shape D(λ) or a shifted shape S(µ). Then the multivariate generating function for A(P ) is given by
The first aim of this paper is to prove a (q, t) deformation of Gansner's hook product formulae. Let q and t be indeterminates and put f q,t (n; m) =
for non-negative integers n and m. And we use the notation F (x; q, t) = (tx; q) ∞ (x; q) ∞ ,
where (a; q) ∞ = i≥0 (1 − aq i ). If we take q = t, then f q,q (n; m) = 1 and F (x; q, q) = 1/(1 − x). Our main theorem is the following: Theorem 1.2. (a) Let λ be a partition. We define a weight W D(λ) (σ; q, t) of a reverse plane partition π ∈ A(D(λ)) by putting
where we use the convention that π k,l = 0 if k < 0 or l < 0. Then we have
(b) Let µ be a strict partition. We define a weight W S(µ) (σ; q, t) of a reverse shifted plane partition σ ∈ A(S(µ)) by putting
where σ k,l = 0 if k < 0. Then we have
Note that there are only finitely many terms different from 1 in the products in (7) and (9). If we put q = t, then f q,q (n; m) = 1 and all weights W P (σ; q, q) are equal to 1, so Theorem 1.2 reduces to Gansner's hook product formula (Theorem 1.1).
S. Adachi [1] proves the formula (8) by generalizing the argument of [15] . We give a similar but more transparent proof. If λ is a rectangular partition (r c ), then reverse plane partitions of shape (r c ) can be viewed as plane partitions by rotating 180
• , and the formula (8) gives Vuletić's generalization of MacMahon's formula [15, Theorem A] . O. Foda et al. [3, 2] uses fermion calculus to re-derive the Schur (q = t) and Hall-Littlewood (q = 0) case of Vuletić's generalization.
Gansner's hook product formulae are generalized to other posets than shapes and shifted shapes. R. Proctor [10, 11] introduced a wide class of posets, called d-complete posets, enjoying "hook-length property" and "jeude-taquin property," as a generalization of shapes and shifted shapes. And he announced [12] that, in collaboration with D. Peterson, he obtained the hook product formula for d-complete posets, but their formulation and proof are still unpublished. K. Nakada [7] gives a purely algebraic proof to a hook product formula equivalent to Peterson-Proctor's formula. The second aim of this paper is to present a conjectural (q, t)-deformation of Peterson-Proctor's multivariate hook formula. This paper is organized as follows. In Section 2, we use the theory of Macdonald symmetric functions to give a generating function for reverse shifted plane partitions with prescribed shape and profile. Section 3 is devoted to the proof of our main theorem by using the result in Section 2. In Section 4, we consider d-complete posets and give a conjectural (q, t)-deformation of Peterson-Proctor's hook formula.
2
Reverse shifted plane partitions of given shape and profile
In this section, we give a generating function for reverse shifted plane partitions of given shape and profile, from which our main theorem (Theorem 1.2) follows. Our approach is based on an observation made in A. Okounkov et al. [8, 9] . Let µ be a strict partition of length r and σ a reverse shifted plane partition of shape µ. We put
and call it the profile of σ. And we associate to σ a weight V S(µ) (σ; q, t) defined by
Here we use the convention that σ k,l = 0 if k ≤ 0 or k > l, so only finitely many terms in the product over m are different from 1. For a partition τ , we denote by A(S(µ), τ ) the set of reverse shifted plane partitions of shape µ and profile τ .
Theorem 2.1. Given strict partition µ of length r, let N be such that
. Let τ be a partition of length ≤ r. Then we have
where the product is taken over all pairs (k, l) satisfying µ c k < µ l , and
And Q τ (x; q, t) is the Macdonald symmetric function.
Here we recall the definition of Macdonald symmetric functions. (See [6, Chap. VI] for details.) Let Λ be the ring of symmetric functions in x = (x 1 , x 2 , · · · ) with coefficients in a field of characteristic 0, which contains q, t and formal power series in z k 's. Define a bilinear form on Λ by
where p λ is the power-sum symmetric function. Then [6, Chap. VI, (4.7)] there exists a unique family of symmetric functions P λ (x; q.t) ∈ Λ, indexed by partitions, satisfying the following two conditions:
(i) P λ is a linear combination of monomial symmetric functions of the form
where < denotes the dominance order on partitions.
(ii) If λ = µ, then P λ , P µ = 0.
Then the P λ 's form a basis of Λ. The Macdonald Q functions Q λ (x; q, t) are the dual basis of P λ , i.e.,
We use operator calculus on the ring of symmetric functions Λ to show Theorem 2.1. We denote by g n the Macdonald symmetric function Q (n) corresponding to a one-row partition (n). Let g + n and g − n be the multiplication and skewing operators on Λ associated to g n respectively. They satisfy
We consider the generating functions
Also we introduce the degree operator D(y) defined by
For a strict partition µ and a partition τ , we put
Fix a positive integer N satisfying N ≥ µ 1 and define a sequence ε = (ε 1 , · · · , ε N ) of + and − by putting
The first step of the proof of Theorem 2.1 is the following Lemma.
Lemma 2.2. In the ring of symmetric functions Λ, we have
where τ runs over all partitions of length at most the length of µ.
For example, if µ = (6, 5, 2) and N = 6, then ε = (−, +, −, −, +, +) and
Proof. For a map σ : S(µ) → N (or an array of non-negative integers of shape µ) and an integer k (0 ≤ k ≤ N), we define the kth trace σ[k] to be the sequence (· · · , σ 2,k+2 , σ 1,k+1 ) obtained by reading the k-th diagonal from SE to NW. For example, a reverse shifted plane partition 0 1 2 3 3  1 2 3 3 3  2 4 has traces
For two partitions α and β, we write 
In the above example, we have
A key ingredient is the Pieri rule for Macdonald symmetric functions. For two partitions α and β satisfying α ≻ β, we put
Then the Pieri rule [6, Chap. VI, (6,24)] can be stated as follows:
where α in the first summation (resp. β in the second summation) runs over all partition satisfying α ≻ β (resp. β ≺ α). (In [6, Chap. VI, (6.24)], the coefficients ϕ + α,β = ϕ α,β and ϕ − β,α = ψ α,β are given in terms of arm and leg lengths, but it is not hard to rewrite them in the form (12) by using f q,t (m, n) defined by (5) . See also [6, Chap. VI, 6, Ex. 2].) And the weight function V S(µ) (σ; q, t) is expressed in terms of ϕ ± α,β (q, t) as
Now the claim of Lemma easily follows by induction on µ 1 . The second step is to rewrite the composite operators on the right-hand side of Lemma 2.2 by using some commutation relations. 
where the first product is taken over all pairs (k, l) satisfying µ c k < µ l .
Proof. By the Pieri rule (13), we have
Also we have
By using these relations, we move D(z i )'s to the right and see that
where F (x; q, t) is defined by (6) . (Details are left to the reader.) It follows from this commutation relation that
Now we are in position to finish the proof of Theorem 2.1.
Proof of Theorem 2.1. It follows from Lemmas 2.2 and 2.3 that
By definition, we have D(z)1 = 1 and G − (u)1 = 1, so we see that
Since the generating function of g n (x; q, t), where
It follows from the Cauchy identity [6, Chap. VI, (4.13)] that
Hence we see that
Equating the coefficients of P τ (x; q, t) completes the proof.
Proof of Theorem 1.2
In this section, we derive Theorem 1.2 from Theorem 2.1. If we put
then we have (see [6, Chap. VI, (4,12) and (6.19)]) Q τ (x; q, t) = b τ (q, t)P τ (x; q, t).
Proof of Theorem 1.2 (b).
First we prove our (q, t) deformation for shifted shapes. By comparing (9) and (11), for a reverse shifted plane partition σ of shifted shape µ, we have
where τ = σ[0] is the profile of σ and
.
with the product taken over all i and j such that i ≤ j and j − i is even. Hence it follows from Theorem 2.1 that
t).
By applying the Schur-Littlewood type formula [6, Chap. VI, 7, Ex.
we see that
On the other hand, we can compute the monomial z[H S(µ) (i, j)] associated to the shifted hook H S(µ) (i, j) at (i, j) ∈ S(µ). By using the fact that the length of column j (j > r) of the shifted diagram S(µ) is equal to j − µ c N −j+1 , we have
Noticing that (i, j) ∈ S(µ) if and only if µ i > µ c N −j+1 for 1 ≤ i ≤ r < j ≤ N, the desired product has been obtained.
Proof of Theorem 1.2 (a).
Next we show the (q, t)-deformation for shapes. For a given partition λ, let r = #{i : λ i ≥ i} be the number of cells on the main diagonal of D(λ) and define two strict partitions µ = (µ 1 , · · · , µ r ) and ν = (ν 1 , · · · , ν r ) by
where t λ is the conjugate partition of λ. Also we put
Then a reverse plane partition π ∈ A(D(λ)) is obtained by gluing two reverse shifted plane partitions σ ∈ A(S(µ)) and ρ ∈ A(S(ν)) with the same profile
, and
Hence it follows from Theorem 2.1 that
Now applying the Cauchy identity [6, Chap. VI, (4.13)], we obtain
F (x µ iỹ ν j ; q, t).
On the other hand, the monomial z[H D(λ) (i, j)] associated to the hook
This completes the proof of Theorem 1.2. isomorphic to the shape D((2, 2)), while d 4 (1) is isomorphic to the shifted shape S ((3, 2, 1) ). The poset d k (1) is called the double-tailed diamond poset. [10, 11] .)
A poset P is d-complete if it satisfies the following three conditions for every k ≥ 3: (b) For each v ∈ P , every saturated chain from v to the maximum element v 0 has the same length. Hence P admits a rank function r : P → N such that r(x) = r(y) + 1 if x covers y.
Let P be a poset with a unique maximal element. The top tree T of P is the subgraph of the Hasse diagram of P , whose vertex set consists of all elements x ∈ P such that every y ≥ x is covered by at most one other element. (a) Let λ be a partition. Then the top tree of the shape D(λ) is given by
and the content function c :
is a d-complete coloring.
(b) If µ is a strict partition with length ≥ 2, then the top tree of the shifted shape S(µ) is given by 4, 3, 1) ) and the shifted shape S( (7, 6, 3, 1) ). The top tree consists of the nodes denoted by •.
Let P be a connected d-complete poset and c : P → I a d-complete coloring. Let z i (i ∈ I) be indeterminates. For a P -partition σ ∈ A(P ), we put
For example, if we use the d-complete colorings given in Example 4.3, the monomial z tr(σ) is the same as z σ for a reverse plane partition σ ∈ A(D(λ)), while, for a reverse shifted plane partition σ ∈ A(S(µ)), z tr(σ) is obtained from z σ by putting z 0 = z 0 ′ . Instead of giving a definition of hooks H P (v) for a general d-complete poset P , we define associated monomials z[H P (v)] directly by induction as follows: 
, where x and y are the sides of [w, v] .
It is easy to see that, for shapes, this definition of z[H D(λ) (v)] is consistent with the definition (1) and (3) given in Introduction. And, for shifted shapes, the monomial z[H S(µ) (v)] defined above reduces to the monomial z[H S(µ) (v)] defined by (2) and (3), if we put z 0 = z 0 ′ . Now we are ready to state our conjectural (q, t)-deformation of PetersonProctor's hook formula. Let P be a connected d-complete poset with the maximum element v 0 , and the rank function r : P → N. Let T be the top tree of P . Take T as a set of colors and c : P → T be the d-complete coloring such that c(v) = v for all v ∈ T . Given a P -partition σ ∈ A(P ), we define a weight W P (σ; q, t) by putting
where c(x) ∼ c(y) means that c(x) and c(y) are adjacent in T , and
Note that, if c(x) ∼ c(y), then r(y) − r(x) is odd, and, if c(x) = c(y), then r(y) − r(x) is even, hence d(x, y) and e(x, y) are integers. If we consider the extended posetP = P ⊔ {1}, and its top treeT = T ⊔ {1}, where1 is the new maximum element ofP and1 is adjacent to v 0 inT , then the weight W P (σ; q, t) can be expressed in the following form:
whereĉ :P →T andσ :P → N are the extensions of c and σ defined bŷ c(1) =1 andσ(1) = 0 respectively. Conjecture 4.4. Let P be a connected d-complete poset. Using the notations defined above, we have
If we put q = t, then W P (σ; q, q) = 1 and F (x; q, q) = 1/(1 − x), hence Conjecture 4.4 reduces to Peterson-Proctor's hook formula (see [7] ).
Our main theorem (Theorem 1.2) supports Conjecture 4.4. We use these identities with a = z 0 ′ /z 0 together with Theorem 2.1 to derive (16) for P = S(µ), by the argument similar to the proof of Theorem 1.2 (b). We proceed by induction on the number of vertices in the tree T . Since it is obvious for #T = 1, we may assume #T > 1. Let v 1 , · · · , v r be the children of v 0 , and T 1 , · · · , T r be the rooted subtrees of T with roots v 1 , · · · , v r respectively. Then there is a natural bijection S : A(T 1 ) × · · · × A(T r ) × N −→ A(T ), which associates to (π 1 , · · · , π r , k) ∈ A(T 1 ) × · · · × A(T r ) × N, the T -partition π ∈ A(T ) defined by
Under this bijection S, we have W T (π; q, t) = f q,t (k; 0)
Hence we have
By using the binomial theorem we can complete the proof.

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