Skew hook formula for $d$-complete posets

Peterson and Proctor obtained a formula which expresses the multivariate generating function for $P$-partitions on a $d$-complete poset $P$ as a product in terms of hooks in $P$. In this paper, we give a skew generalization of Peterson--Proctor's hook formula, i.e., a formula for the generating function for $(P \setminus F)$-partitions for a $d$-complete poset $P$ and its order filter $F$, by using the notion of excited diagrams. Our proof uses the Billey-type formula and the Chevalley-type formula in the equivariant $K$-theory of Kac--Moody partial flag varieties. This generalization provides an alternate proof of Peterson--Proctor's hook formula.


Introduction
One of the most elegant formulas in combinatorics is the Frame-Robinson-Thrall hook formula [2,Theorem 1] for the number of standard tableaux. Given a partition λ of n, a standard tableaux of shape λ is a filling of the cells of the Young diagram D(λ) of λ with numbers 1, 2, . . . , n such that each number appears once and the entries of each row and each column are increasing. The Frame-Robinson-Thrall hook formula asserts that the number f λ of standard tableaux of shape λ is given by where h D(λ) (v) denotes the hook length of the cell v in D(λ). Similar formulas hold for the number of shifted standard tableaux ( [13,5.1.4,Exercise 21], see also [30, § 40] and [35,Theorem 1]) and the number of increasing labeling of rooted trees ( [13,5.1.4,Exercise 20]). These tableaux and labelings can be regarded as linear extensions of certain posets.
Stanley [31] introduced the notion of P -partitions for a poset P , and found a relationship between the univariate generating function and the number of linear extensions of P . Given a poset P , a P -partition is an order-reversing map σ from P to N, the set of nonnegative integers. We denote by A(P ) the set of all P -partitions. For a P -partition σ, we write |σ| = v∈P σ(x). Then Stanley [31,Corollaries 5.3 and 5.4] proved that, for a poset P with n elements, there exists a polynomial W P (q) satisfying σ∈A(P ) and that W P (1) is equal to the number of linear extensions of P . Also in [32,Proposition 18.3] he proved that, if P is the Young diagram D(λ) of a partition λ, viewed as a poset, the generating function of D(λ)-partitions (also called reverse plane partitions of shape λ) is given by .
Proctor [27], [28] introduced a wide class of posets, called d-complete posets, enjoying "hook-length property", as a generalization of Young diagrams, shifted Young diagrams and rooted trees. d-Complete posets are defined by certain local structural conditions (see Section 2 for a precise definition). Peterson and Proctor obtained the following theorem, which is a far-reaching generalization of the hook formulas (1) and (3). Theorem 1.1. (Peterson-Proctor, see [29]) Let P be a d-complete poset. The multivariate generating function of P -partitions is given by (Refer to Section 2 for undefined notations.) However the original proof of this theorem is not yet published, though an outline of their proof is given in [29]. Different proofs are sketched by Ishikawa-Tagawa [8], [9] and Nakada [23], [24]. Our skew generalization (Theorem 1.2 below) provides an alternate proof of Theorem 1.1. In the univariate case, a full proof is given by Kim-Yoo [11].
Another direction of generalizing the Frame-Robinson-Thrall hook formula (1) is to consider skew shapes. For partitions λ ⊃ µ, a standard tableau of skew shape λ/µ is a filling of the cells of the skew Young diagram D(λ/µ) = D(λ)\D(µ) satisfying the same conditions as standard tableaux of straight shape. However one cannot expect a nice product formula for the number f λ/µ of standard tableaux of skew shape λ/µ in general. Naruse [25] presented and sketched a proof of a subtraction-free formula for f λ/µ : where n = |λ/µ|, and D runs over all excited diagrams of D(µ) in D(λ). Morales-Pak-Panova [21] gave a q-analogue of Naruse's skew hook formula for the univariate generating functions for P -partitions on P = D(λ/µ).
The main result of this paper is the following skew generalization of Peterson-Proctor's hook formula (Theorem 1.1): Theorem 1.2. Let P be a connected d-complete poset and F an order filter of P . Then the multivariate generating function of (P \F )-partitions, where P \F is viewed as an induced subposet of P , is given by where D runs over all excited diagrams of F in P . (See Sections 2 and 3 for undefined notations.) Taking an appropriate limit, we see that the number of linear extensions of P \ F is given by where n = #(P \ F ) and h P (v) is the hook length of v in P . (See Corollary 5.6(b).) If F = ∅, then our main theorem (Theorem 1.2) gives Peterson-Proctor's hook formula (Theorem 1.1). If P = D(λ) and F = D(µ) are the Young diagrams of partitions λ ⊃ µ, then (6) reduces Morales-Pak-Panova's q-hook formula [21, Corollary 6.17] after specializing z i = q for all i ∈ I, and (7) is nothing but Naruse's skew hook formula (5).
This paper is organized as follows. In Section 2, we review a definition and basic properties of d-complete posets. In Section 3, we introduce the notion of excited diagrams for d-complete posets, which is the key ingredient of the formulation of our main theorem, and study their properties. Our proof of the main theorem uses the Billey-type formula and the Chevalley-type formula for the equivariant K-theory of Kac-Moody partial flag varieties. In Section 4, we recall some properties of the equivariant K-theory and translate the Billey-type formula and the Chevalley-type formula in terms of combinatorics of d-complete posets. We will give a proof of our main theorem (Theorem 1.2) and derive some corollaries in Section 5.

d-Complete posets
In this section we review a definition and some properties of d-complete posets and explain their connections to Weyl groups. See [27], [28], [29] and [34] for details.

Combinatorics of d-complete posets
For an integer k ≥ 3, we denote by d k (1) the poset consisting of 2k − 2 elements u 1 , · · · , u k−2 , x, y, v k−2 , · · · , v 1 with covering relations Note that z and w are incomparable. The poset d k (1) is called the double-tailed diamond. The Hasse diagram of d k (1) is shown in Figure 1. (1). Then v and u are called the bottom and top of [v, u] respectively, and the two incomparable elements of [v, u] are called the sides. A subset I of P is called convex if x < y < z in P and x, z ∈ I imply y ∈ I. A convex subset I is called a d − k -convex set if it is isomorphic to the poset obtained by removing the top element from d k (1). Definition 2.1. A poset P is d-complete if it satisfies the following three conditions for every k ≥ 3: k -convex set, then there exists an element u such that u covers the maximal elements of I and I ∪ {u} is a d k -interval.
(D2) If I = [v, u] is a d k -interval and the top u covers u ′ in P , then u ′ ∈ I.
(D3) There are no d − k -convex sets which differ only in the minimal elements. It is clear that rooted trees, viewed as posets with their roots being the maximum elements, are d-complete posets.
If we regard P as an induced subposet of Z 2 with ordering given by (8), then P is a d-complete poset, called a swivel. See Figure 3 for the Hasse diagram of P .
A poset P is called connected if its Hasse diagram is a connected graph. It is easy to see that, if P is a d-complete poset, then each connected component of P is d-complete. Hence there is no harm in assuming that a d-complete poset is connected.
Let P be a d-complete poset. If P is connected, then P has a unique maximal element.
Let P be a poset with a unique maximal element. The top tree Γ 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. Let P be a connected d-complete poset and Γ its top tree. Let I be a set of colors whose cardinality is the same as Γ. Then a bijective labeling c : Γ → I can be uniquely extended to a map c : P → I satisfying the following three conditions: (C1) If x and y are incomparable, then c(x) = c(y).
Moreover this map c satisfies (C4) If x covers y, then the nodes labeled by c(x) and c(y) are adjacent in Γ.
(C5) If c(x) = c(y) or the nodes labeled by c(x) and c(y) are adjacent in Γ, then x and y are comparable.
Such a map c : P → I is called a d-complete coloring.
Let P be a connected d-complete poset and c : P → I a d-complete coloring. Let z = (z i ) i∈I be indeterminates. Given an order filter F of P , we regard P \ F as the induced subposet. For a (P \ F )-partition σ ∈ A(P \ F ), we put We are interested in the multivariate generating function σ∈A(P \F ) z σ of (P \ F )-partitions. For a subset D of P , we write Instead of giving a definition of hooks H P (u) ⊂ P for a general d-complete poset P , we define associated monomials z[H P (u)] directly by induction as follows: Definition 2.6. Let P be a connected d-complete poset with d-complete coloring c : P → I.
(i) If u is not the top of any d k -interval, then we define where x and y are the sides of [v, u].
Example 2.7. Let P = D(λ) be the shape corresponding to a partition λ. Then the top tree Γ of D(λ) is given by where λ ′ 1 is the number of cells in the first column of the Young diagram D(λ). A d-complete coloring c : Example 2.8. Let P = S(µ) be the shifted shape corresponding to a strict partition µ of length ≥ 2.. Then the top tree Γ of S(µ) is given by and a d-complete coloring c : S(µ) → I = {0, 0 ′ , 1, 2, . . . , µ 1 − 1} is given by

d-Complete posets and Weyl groups
Let P be a connected d-complete poset with top tree Γ. We regard Γ as a (simplylaced) Dynkin diagram with node set I and the d-complete coloring as a map c : P → I. Let A = (a ij ) i,j∈I be the generalized Cartan matrix of Γ given by We fix the following data associated to A: • a free Z module Λ, called the weight lattice, • a linearly independent subset Π = {α i : i ∈ I} of Λ, called the simple roots, • a subset Π ∨ = {α ∨ i : i ∈ I} of the dual lattice Λ * = Hom Z (Λ, Z), called the simple coroots, • a subset {λ i : i ∈ I} of Λ, called the fundamental weights, : Λ * × Λ → Z is the canonical pairing. Let W be the corresponding Weyl group generated by the simple reflections {s i : i ∈ I}, where s i acts on Λ and Λ * by the rule Then W is a Coxeter group, and we have the length function l and the Bruhat order < on W . The set of real roots Φ and the set of real coroots Φ ∨ are defined by Φ = W Π and Φ ∨ = W Π ∨ respectively. The set of simple roost Π (resp. the set of simple coroots Π ∨ ) determines the decomposition of Φ (resp. Φ ∨ ) into the positive system Φ + (resp. Φ ∨ + ) and the negative system Φ − (resp. Φ ∨ − ). We introduce the standard partial ordering on Φ + (resp. Φ ∨ + ) by setting α > β if α − β is a sum of simple roots For p ∈ P , we put Let α P and λ P be the simple root and the fundamental weight corresponding to the color i P of the maximum element of P . Take a linear extension and label the elements of P with p 1 , · · · , p N (N = #P ) so that p i < p j in P implies i < j. Then we construct an element w P ∈ W by putting w P = s(p 1 )s(p 2 ) · · · s(p N ).
A Weyl group element w ∈ W is called λ-minuscule if there exists a reduced expression w = s i 1 · · · s i l such that or equivalently A element w ∈ W is called fully commutative if any reduced expression of w can be obtained from any other by using only the Coxeter relations of the form st = ts.
If p = p k ∈ P , then we define It follows from Proposition 2.9 that, for each p ∈ P , the roots β(p), γ(p) and the coroot γ ∨ (p) are independent of the choices of linear extensions. For a Weyl group element w ∈ W , we put Then it is well-known (see [5, §5.6]) that Let W λ P be the stabilizer of λ P in W . Then W λ P is the maximal parabolic subgroup corresponding to I \ {i P }. Let W λ P be the set of minimum length coset representatives of W/W λ P . A subset F of P is called an order filter if x < y in P and x ∈ F imply y ∈ F . For a subset D = {p i 1 , · · · , p ir } (i 1 < · · · < i r ) of P , we define w D = s(p i 1 ) · · · s(p ir ).

Moreover we have
Since w P is fully commutative (Proposition 2.9), we see that w D is independent of the choices of linear extensions of P . The map F → w F gives a poset isomorphism from the set of all order filters of P ordered by inclusion to the Bruhat interval [e, w P ] in W λ P .
Remark 2.12. Let W be an arbitrary symmetrizable Kac-Moody Weyl group corresponding to a Dynkin diagram Γ with node set I. Given a (not necessarily reduced) expression s i 1 s i 2 . . . s i N of an element w ∈ W in simple reflections, we can define a poset H, called the heap, as follows (see [33]). The poset H consists of the ground set {1, 2, . . . , N } and the partial ordering obtained by taking the transitive closure of the relations given by The heap H has a natural labeling (coloring) c : H → I given by c(a) = i a . If w ∈ W is fully commutative, then the haep defined by a reduced expression of w is independent of the choices of reduced expressions. In this case we denote the resulting heap by H(w). Every d-complete poset P is isomorphic to the simply-laced heap H(w P ). In general, if w ∈ W is dominant minuscule, i.e., λ-minuscule for some dominant weight λ, the corresponding heap H(w) is isomorphic (as a unlabeled poset) to a d-complete poset. See [34,Sections 3 and 4].
The propositions in this section hold literally for heaps H(w) of dominant minuscule elements, except for Proposition 2.5 and Proposition 2.10 (b). The latter half of Proposition 2.5 holds for heaps, i.e., the labeling c : H(w) → I satisfies (C4) and (C5). And we adopt Proposition 2.10 (b) as a definition of the hook monomial for H(w).
Example 2.13. Let µ = (µ 1 , . . . , µ l ) be a strict partition of length l. Then the shifted shape S(µ) can be regarded as a heap associated to a dominant minuscule element of the Weyl group of type B. Put m = µ 1 and let W be the Weyl group generated by s 0 , s 1 , . . . , s m−1 subject to the relations Then we define an element w µ ∈ W by putting Then it can be shown that w µ is λ 0 -minuscule, where λ 0 is the fundamental weight corresponding to s 0 , and that the map gives a poset isomorphism. We identify the ground set of H(w µ ) with S(µ) via the isomorphism (10). Then the natural labeling c ′ : which is also different from the shifted hook given in Example 2.8.

Excited diagrams
In this section we introduce the notion of excited and K-theoretical excited diagrams in a d-complete poset and study their properties.

Excited diagrams
First we generalize the notion of exited diagrams for Young diagram and shifted Young diagram, which were introduced by Ikeda-Naruse [6] and Kreiman [14], [15] independently, to a general d-complete posets. And we give a generalization of backward movable positions or excited peaks introduced in [7], [12] and [21]. Let P be a connected d-complete poset with top tree Γ and d-complete coloring c : P → I. For a subset D ⊂ P and a color i ∈ I, we put For i ∈ I, let N i be the subset of P consisting of element x ∈ P whose color c(x) such that x is covered by u or covers v. Definition 3.1. Let P be a connected d-complete poset and let F be an order filter of P .
(a) Let D be a subset of P and u ∈ D. We say that u is D-active if there exists an (b) Let D be a subset of P and u ∈ D. If u is D-active, then we define α u (D) to be the subset of P obtained by replacing u ∈ D by the bottom element v of the d k -interval [v, u]. We call this replacement an (ordinary) elementary excitation.
(c) An excited diagram of F in P is a subset of P obtained from F after a sequence of elementary excitations on active elements. Let E P (F ) be the set of all excited diagrams of F in P .
If D is an excited diagram with an active element u, then we define If P is a shape or a shifted shape, our definition above coincides with the definitions of elementary excitations in [6], [7], or ladder moves in [14], [15], and backward movable positions in [7], [12] or excited peaks in [21] (only for a shape).
Example 3.2. If P = D(5, 4, 2, 1) is the shape corresponding to a partition (5, 4, 2, 1) and F = D(3, 1), then there are 6 excited diagrams in E P (F ) shown in Figure 4. In Figure 4 (and Figures 5, 6), the shaded cells form an exited diagram and a cell with × is an excited peak. And the arrow D → D ′ means that D ′ is obtained from D by an elementary excitation. Example 3.3. If P = S(5, 4, 2, 1) is the shifted shape corresponding to a strict partition (5, 4, 2, 1) and F = S(3, 1), then there are 5 excited diagrams in E P (F ) shown in Figure 5. Let W be the Weyl group corresponding to the top tree Γ viewed as a Dynkin diagram, and fix a labeling of the elements of P with p 1 , . . . , p N so that p i < p j in P implies i < j. Then we can associate to a subset D of P a well-defined element w D ∈ W as in (9). The following proposition gives a characterization of excited diagrams. To prove this proposition, we prepare two lemmas.
Lemma 3.6. Let D be a subset of P and u and v elements of P such that v < u and Proof. follows from Property (C5) in Proposition 2.5. Let v ∈ W be a fully commutative element and v = s i 1 · · · s ir = s j 1 · · · s jr be its reduced expressions. Then we have (a) Let i, j be adjacent nodes in the Dynkin diagram. Then the subsequence of (i 1 , · · · , i r ) consisting of i and j is identical with the subsequence of (j 1 , · · · , j r ) consisting of i and j.
(b) Let i be a node in the Dynkin diagram. Then the number of occurrence of i in (i 1 , · · · , i r ) is equal to the number of occurrence of i in (j 1 , · · · , j r ).
We use these lemmas to prove the characterization of excited diagrams.
Proof of Proposition 3.5. We follow the same idea used in the proof of [4, Proposition 4.8]. We denote by R P (F ) the set of all subsets D ⊂ P satisfying #D = #F and Then by using Lemma 3.6, we have Figure 6: Excited diagrams in a swivel Next we prove that R P (F ) ⊂ E P (F ). Since F is an order filter, we can take a linear extension of P such that F = {p n+1 , · · · , p N }, where n = #(P \ F ). We define the energy of any subset D ⊂ P with #D = #F by putting Then, by the assumption on our linear extension, we see that e(D) ≥ 0 and that e(D) = 0 if and only if D = F .
We proceed by induction on e(D) to prove D ∈ R P (F ) implies D ∈ E P (F ). Let D ∈ R P (F ). If e(D) = 0, then we have D = F ∈ E P (F ).
Suppose that e(D) > 0. Then there exists an element u ∈ F such that u ∈ D. Let u be the maximal element satisfying u ∈ F and u ∈ D. Since w D = w F , it follows from Lemma 3.7 (b) that there exists an element v ∈ D with the same color i as u. Let v be the maximal element satisfying v ∈ D and c(v) = c(u) = i. Then, by applying Lemma 3.7 (a), we see that [v, u] Lemma 3.6 and e(D ′ ) < e(D). Then by the induction hypothesis we have D ′ ∈ E P (F ). Since D is obtained from D ′ by a sequence of elementary excitations, we obtain D ∈ E P (F ).
Next we give a non-recursive description of the set of excited peaks B(D), which implies that B(D) is well-defined, i.e., it is independent of the choices of elementary excitations to reach D from F .
(a) The following are equivalent for x ∈ P : In the proof of this proposition, we utilize the following lemma, which will be used also in the sequel of this section.
Hence, by using Property (C5) in Proposition 2.5, we see that y < v < x < u and x ∈ [v, u] ∩ N j , which contradicts to the assumption.
(c) By an argument similar to (b), we can show that, if [y, x] ∩ D ∩ N i = ∅, then v < y < u < x, which contradicts to y ∈ [v, u] ∩ N j . We begin with considering the case where D = F . Let x and y be elements of F with the same color i satisfying y < x. Then it follows from Properties (C4) and (C5) in Proposition 2.5 that an element z covered by x or covers y belongs to We prove that B(α u (D)) = B ′ (α u (D)) and α u (D) ∩ B ′ (α u (D)) = ∅ for D ∈ E P (F ) and a D-active element u ∈ D. Let v be the element such that v < u, Next, in order to show B ′ (α u (D)) ⊂ B(α u (D)), we take an element x ∈ B ′ (α u (D)) such that x = u and prove . We consider the case where y = v. In this case, y ∈ D and it follows from Lemma 3.9 (c) that Hence, by using Property (C5) in Proposition 2.5, we have y < v < x < u and this contradicts to Finally we show that α u (D) ∩ B ′ (α u (D)) = ∅. Let x and y be elements of α u (D) with the same color i satisfying y < x. Since α u (D) = D \ {u} ∪ {v}, it is enough to consider the following three cases: In Case 1, assume that [y, x]∩α u (D)∩N i = ∅. Since [y, x]∩D∩N i = ∅ by the induction hypothesis, we have v ∈ [y, x] ∩ N i , i.e., y < v < x and c(u) = c(v) is adjacent to i in Γ. Since v ∈ α u (D), we have v ∈ [y, x] ∩ N i . Hence by using Property (C5) in Proposition 2.5 we see that v < y < u < x, which contradicts to the D-activity of u.
Hence by using Lemma 3.9 (a) we obtain [y, v] ∩ α u (D) ∩ N i = ∅. Therefore we see that any element satisfying the condition (ii) in (a) for α u (D) does not belong to α u (D), and α u (D) ∩ B ′ (α u (D)) = ∅. This completes the proof.

K-theoretical excited diagrams
We define K-theoretical excited diagrams and study their properties. For shapes and shifted shapes, these diagrams were introduced in [4]. (a) Let D be a subset of P and u ∈ D. If u is D-active and [v, u] is a d k -interval, then we define α * u (D) to be the subset of P obtained by adding v to D. We call this operation a K-theoretical elementary excitation.
(b) A K-theoretical excited diagram of F in P is a subset of P obtained from F after a sequence of ordinary and K-theoretical elementary excitations on active elements. Let E * P (F ) be the set of all K-theoretical excited diagrams of F in P . For a fixed linear extension of P and a subset D = {p i 1 , · · · , p ir } (i 1 < · · · < i r ) of P , we define an element w * D ∈ W by putting where * : W × W → W is the associative product, called the Demazure product, defined by Since w P is fully commutative (Proposition 2.9), the element w * D is independent of the choices of linear extensions of P The following proposition is a key to rephrase the Billey-type formula for equivariant K-theory in terms of combinatorics of d-complete posets (see Proposition 4.7). Let v ∈ W and v = s i 1 * · · · * s ir with i 1 , . . . , i r ∈ I.
(a) There is a increasing sequence 1 ≤ k 1 < k 2 < · · · < k l ≤ r such that v = s i k 1 s i k 2 . . . s i k l is a reduced expression of v. In particular, we have l(v) ≤ r.
(c) If v is fully commutative and l(v) < r, then there exist a < b such that s ia = s i b and commutes with s ic for every a < c < b.
Since u is D-active, it follows from Lemma 3.6 and s(p k ) * s(p l ) = s(p l ) * s(p l ) = s(p l ) that w * D ′ = w * D in both cases. Next we prove R * P (F ) ⊂ E * P (F ). We proceed by induction on #D to prove D ∈ R * P (F ) implies D ∈ E * P (F ). Since w * D = w F , we have #D ≥ l(w F ) = #F by Lemma 3.12 (a). If #D = #F , then w * D = w D by Lemma 3.12 (b), thus we have D ∈ E P (F ) ⊂ E * P (F ) by Proposition 3.5. If #D > #F , then it follows from Lemma 3.12 (c) that there exist u, v ∈ D with v < u such that c(u) = c(v) and s(u) = s(v) commutes with every elements between u and v in the expression of w * D . If we put D ′ = D \ {v}, then we see that w * D ′ = w * D . Hence by the induction hypothesis we have D ′ ∈ E * P (F ). Since D ′ is obtained from D by a sequence of ordinary and K-theoretical elementary excitations, we obtain D ′ ∈ E * P (F ).
The following proposition plays a crucial role in the proof of our main theorem (see the proof of Theorem 5.3).
In order to prove this proposition, we prepare several lemmas.
Hence by using Property (C5) in Proposition 2.5 we see that w < y < z < x, which contracts to For E ∈ E * P (F ), we define a subset S(E) of E by putting It follows from Property (C4) in Proposition 2.5 that S(F ) = ∅ for an order filter F of P .
Lemma 3.15. Let E ∈ E * P (F ) and u ∈ E an E-active element. Then we have where [v, u] is the d k -interval with top element u. In particular, we have Proof. Since α u (E) = E \ {u} ∪ {v}, the equality (13) follows from the following three claims: To prove (i), we take x ∈ S(α u (E)) such that x = v. Then, by the definition (12), there exists y ∈ (α u (E)) c(x) such that y < x and [y, . We consider the case where y = v. In this case, y ∈ E and it follows from [v, u] ∩ E ∩ N c(u) = ∅ that y ∈ [v, y] ∩ N c(u) . Now we can use Lemma 3.9 (c) to obtain [y, x] ∩ E ∩ N c(x) = ∅, hence x ∈ S(E).
Next we prove (ii). Since u ∈ S(E), there exists z ∈ E c(u) such that z < u and To prove (iii), we take x ∈ S(E) such that x = u. By the definition (12), there exists y ∈ E c(x) such that y < x and [y, x] ∩ E ∩ N c(x) = ∅. Then we have y = u or y ∈ α u (E). If y = u, then [u, x] ∩ E ∩ N c(x) = ∅ and [v, u] ∩ α u (E) ∩ N c(u) = ∅ by the E-activity of u. Hence by using Lemma 3.9 (a) we have [v, x] ∩ α u (E) ∩ N c(x) = ∅ and x ∈ S(α u (E)). We consider the case where y ∈ α u (E). Since [v, u] Since α * u (E) = E ∪ {v}, the equality (14) follows from the following three claims: ). To prove (iv), we take x ∈ S(α * u (E)) such that x = u, v. Then there exists y ∈ (α * u (E)) c(x) such that y < x and [y, Next we prove (v). Since u ∈ S(E), there exists z ∈ E c(u) such that z < u and (u) . Hence by using Lemma 3.9 (b), we see [y, x]∩α * u (E)∩N c(x) = ∅ and x ∈ S(α * u (E)). Proof. First we proceed by induction on #S to prove D ⊔ S ∈ E * P (F ). If S = ∅, then we have D ∈ E P (F ) ⊂ E * P (F ). If S = ∅, we take an element x ∈ S and put S ′ = S \ {x}. By the induction hypothesis and Proposition 3.11, we have D ⊔ S ′ ∈ E * P (F ) and w * D⊔S ′ = w F . Using Proposition 3.8 (a), we see that there exists y ∈ D c(x) such that y < x and [y, x] ∩ D ∩ N c(x) = ∅. Then it follows from Lemma 3.14 that By using Lemma 3.6 and s(p k ) * s(p l ) = s(p k ) * s(p k ) = s(p k ), we obtain w * D⊔S = w * D⊔S ′ = w F . Hence by Proposition 3.11 we have D ⊔ S ∈ E * P (F ). Next we put E = D ⊔ S and prove that S = S(E). In order to show the inclusion S ⊂ S(E), we take x ∈ S ⊂ B(D). Then by Proposition 3.8 (a), there exists y ∈ D c(x) such that [y, x] ∩ D ∩ N c(x) = ∅. Hence by using Lemma 3.14 we see that [y, x] ∩ E ∩ N c(x) = ∅ and x ∈ S(E). In order to show the reverse inclusion S(E) ⊂ S, we take x ∈ S(E) and prove x ∈ B(D). By the definition (12), there exists y ∈ E c(x) such that y < x and [y, x]∩E ∩N c(x) = ∅. Since D ⊂ E, we have [y, x]∩D∩N c(x) = ∅. If y ∈ D, then by Proposition 3.8 (a) we have x ∈ B(D). If y ∈ S, then there exists z ∈ D c(y) such that z < y and [z, y] ∩ D ∩ N c(y) = ∅. Then by using Lemma 3.9 (a) we have [z, x] ∩ D ∩ N c(x) = ∅ and x ∈ B(D).
Proof. By the definition (12), there exists w ∈ E c(z) such that w < z and [w, z] ∩ E ∩ N c(z) = ∅.
Next we prove that S(E) \ {z} ⊂ S(E ′ ). We take an element x ∈ S(E) such that x = z. Then there exists y ∈ E c(x) such that y < x and [y, Now we are in position to give a proof of Proposition 3.13.
Proof of Proposition 3.13. By using Lemma 3.16, it is enough to show that any E ∈ E * P (F ) can be written as E = D ⊔ S with D ∈ E P (F ) and S ⊂ B(D). Given E ∈ E * P (F ), we put D = E \ S(E) and prove that D ∈ E P (F ) and S(E) ⊂ B(D). We proceed by induction on #S(E). If S(E) = ∅, then E ∈ E P (F ) by Lemma 3.15. We consider the case where S(E) = ∅. Then we take z ∈ S(E) and put By the induction hypothesis, D ∈ E P (F ) and S(E ′ ) ⊂ B(D). It remains to show that z ∈ B(D). By the definition (12), there exists w ∈ E c(z) such that w < z and [w, z] ∩ E ∩ N c(z) = ∅. Let w be the minimal such element. If w ∈ D, i.e., w ∈ S(E), then by definition there exists w ′ ∈ E c(w) such that [w ′ , w] ∩ E ∩ N c(w) = ∅. Then it follows from Lemma 3.9 (a) that [w ′ , z] ∩ E ∩ N c(z) = ∅, which contradicts to the minimality of w. Therefore we have w ∈ D and z ∈ B(D).    (c) If u = (i, j) ∈ D is D-active, then we define This notion of excited diagrams is the same as Ikeda-Naruse's excited diagrams of type I introduced in [6]. For example, if P = S(5, 4, 2, 1) and F = S(3, 1), then there are 10 excited diagrams of F in P as a heap for the type B Weyl group. See Figure 7.

Equivariant K-theory and localization
Let A = (a ij ) i,j∈I be a symmetrizable generalized Cartan matrix, and Γ the corresponding Dynkin diagram with node set I. Then the associated Kac-Moody group over C is constructed from the following data: the weight lattice Z-module Λ, the simple roots Π = {α i : i ∈ I}, the simple coroots Π ∨ = {α ∨ i : i ∈ I}, and the fundamental weights {λ i : i ∈ I} (see the beginning of Subsection 2.2).
In what follows, we fix a subset J of I. Let B be a Borel subgroup corresponding to the positive system Φ + and T ⊂ B a maximal torus. Let P − be the opposite parabolic subgroup corresponding to the subset J, which contains the opposite Borel subgroup B − such that B ∩ B − = T . Then we can introduce the Kashiwara thick partial flag variety X = G/P − . (We refer the readers to [10] for a construction of X .) Let W J be the parabolic subgroup of W corresponding to J and W J be the set of minimum length coset representatives of W/W J . For each element v ∈ W J , we put X • v = BvP − /P − and X v = X • v , the Zariski closure of X • v , which are called the Schubert cell and the Schubert variety respectively. Then X v has codimension l(v) in X and Let K T (X ) be the T -equivariant K-theory of X . Then K T (X ) has a commutative associative K T (pt)-algebra structure. Here the T -equivalent K-theory K T (pt) of a point is isomorphic to the group algebra Z[Λ] with basis {e λ : λ ∈ Λ}, and to the representation ring R(T ) of T . For each v ∈ W J , let [O v ] be the class of the structure sheaf O v of X v in K T (X ) and call it the equivariant Schubert class. Then we have Any elements of K T (X ) is a (possibly infinite) K T (pt)-linear combination of the equivariant Schubert classes.
Each w ∈ W J gives a T -fixed point e w = wP − /P − ∈ X . Then the inclusion map ι w : {e w } → X induces the pull-back ring homomorphism, called the localization map at w, ι * w : K T (X ) → K T (e w ) ∼ = Z[Λ]. If L λ is the line bundle on X corresponding to a weight λ ∈ Λ, then the image of the class [L λ ] under the localization map is given by ι * w ([L λ ]) = e wλ . For two elements v, w ∈ W J , we denote by ξ v | w the image of the T -equivariant Schubert class ξ v = [O v ] ∈ K T (X ) under the localization map ι * w : Then the Billey-type formula for the equivariant K-theory can be stated as follows: where the summation is taken over all sequences (k 1 , . . . , k r ) such that 1 ≤ k 1 < k 2 < · · · < k r ≤ N and s i k 1 * · · · * s i kr = v (with respect to the Demazure product), and β (k) is given by β (k) = s i 1 . . . s i k−1 (α i k ) for 1 ≤ k ≤ N . By using Lemma 3.12 (a), we can deduce the following corollary from Proposition 4.1.
(b) Let v, w ∈ W J . If ξ v | w = 0, then we have v ≤ w in the Bruhat order.

Equivariant K-theoretical Littlewood-Richardson coefficients
We consider the structure constants for the multiplication in K T (X ) with respect to the equivariant Schubert classes. For u, v, w ∈ W J , we denote by c w u,v ∈ K T (pt) the structure constant determined by Proof. We use the induction on l(w) to prove that, if u ≤ w or v ≤ w, then c w u,v = 0. Assume that u ≤ w or v ≤ w. By apply the localization map ι * w to and then by using Corollary 4.2 (b), we have If there exists an element x ∈ W J satisfying x < w and c x u,v = 0, then we have u ≤ x and v ≤ x by the induction hypothesis, and hence u ≤ w and v ≤ w, which contradicts to the assumption. Hence, by using Corollary 4.2 (b) and the assumption, we have 0 = c w u,v ξ w | w . Since ξ w w = 0 (Corollary 4.2 (a)), we obtain c w u,v = 0.
Proof. By apply the localization map ι and then by using Corollary 4.2 (b), we have By Lemma 4.3, we see that c x v,w = 0 unless w ≤ x. By Corollary 4.2 (b), we see that ξ x | w = 0 unless x ≤ w. Hence we have Since ξ w w = 0 (Corollary 4.2 (a)), we obtain the desired equality. The following lemma gives a recurrence of the equivariant K-theoretical Littlewood-Richardson coefficients c w u,v . We use the same idea as [20,Corollary 6.5] and [26, Proposition 3.1].
Lemma 4.5. Let u, v, w ∈ W J and s ∈ W J a simple reflection. If c w s,w = c u s,u , then we have In particular, we have Proof. Consider the associativity Taking the coefficients of [O w ] in the both hand sides and using Lemma 4.3, we have from which we get the conclusion.
The Chevalley formula give a combinatorial expression of c w s,v for a simple reflection s. To state the Chevalley formula of [19] we need several notations. For a dominant weight λ ∈ Λ, we put Fix a total order on I so that I = {i 1 < · · · < i r }, and define a map ι : Then it is known that ι is injective. We define a total ordering < on H λ by where < lex is the lexicographical ordering on Q r+1 . For h = (γ ∨ , k), we define affine transformations r h and r h on Λ by Note that r h = s γ . Now we can state the Chevalley formula for the equivariant K-theory of the partial flag variety X .
Proposition 4.6. ( [19,Theorem 4.8 (4.12) and (4.13)], see also [17,Corollary 7.1]) Let s be a simple reflection such that s ∈ W J and v, w ∈ W J . If s = s i and λ = λ i is the corresponding fundamental weight, then we have where the summation is taken over all sequences (h 1 , · · · , h r ) of length r ≥ 1 satisfying the following two conditions:

Connection to d-complete posets
In this subsection we rephrase the Billey-type formula and the Chevalley-type formula in terms of combinatorics of d-complete posets. Let P be a connected d-complete poset with top tree Γ. We regard Γ as a simplylaced Dynkin diagram with node set I. Let α P and λ P be the simple root and the fundamental weight corresponding to the color i P of the maximum element of P . We apply the above argument to the Kashiwara thick partial flag variety X = G/P − , where P − is the maximal parabolic subgroup corresponding to J = I \ {i P }. In this case, the parabolic subgroup W J coincides with the stabilizer W λ P of λ P in W , and the minimum length coset representatives W J is denoted by W λ P .
By using a labeling of the elements of P with p 1 , · · · , p N (N = #P ) so that p i < p j in P implies i < j, we can associate to each subset D = {i 1 , . . . , i r } (i 1 < · · · < i r ) of P a well-defined element w D = s(p i 1 ) · · · s(p ir ) ∈ W . Then the following formula is obtained from the Billey-type formula.
Proposition 4.7. Let P be a connected d-complete poset and F an order filter of P . Then we have under the identification z i = e α i (i ∈ I).
Proof. Follows from Proposition 4.1 by using Proposition 2.10 (b) and Proposition 3.11.
Also the following explicit expression is obtained from the Chevalley-type formula.
Proposition 4.8. Let P be a connected d-complete poset and put s = s i P . For two order filters F and F ′ of P , we have under the identification z i = e α i (i ∈ I).
First we consider the case r = 1 in Proposition 4.6.
Lemma 4.9. Let F be an order filter of P and h = (γ ∨ , k) ∈ H λ P . If w F r h ∈ W λ P and w F ⋖ w F r h ≤ w P , then there exists p ∈ P such that F ⊔ {p} is an order filter of P , w F r h = w F ′ and γ = γ(p). In this case k = 0 and r h λ P = λ P .
Proof. Since the interval [e, w P ] in W λ P is isomorphic to the poset of order filters of P (Proposition 2.11 (a)), there exists a unique order filter F ′ of P such that F ′ ⊃ F , #F ′ = #F + 1 and w F ′ = w F r h . Hence we have p ∈ P such that F ′ = F ⊔ {p} and w F ′ = s(p)w F . We take a linear extension of P such that F = {p n+1 , · · · , p N } with N = #P and n = #(P \ F ). If p = p m , then p is incomparable with p m+1 , · · · , p n , hence s(p) is commutative with s(p m+1 ), · · · , s(p n ) by Property (C5) in Proposition 2.5. Hence we have By Proposition 2.10 (c), we see that k = 0 and r h λ P = λ P . Now we deduce Proposition 4.8 from the Chevalley-type formula.
Proof of Proposition 4.8. It follows from Proposition 2.11 (b) that Suppose that there exists a sequence (h 1 , · · · , h r ) of elements in H λ P satisfying Conditions (H1) and (H2) in Proposition 4.6. Then by Lemma 4.9, we have a sequence (q 1 , · · · , q r ) of elements of P such that F i = F ⊔ {q 1 , · · · , q i } is an order filter of P , h i = (γ ∨ (q i ), 0) for 1 ≤ i ≤ r and r h 1 · · · r hr λ P = λ P . Now we show that {q 1 , · · · , q r } is an antichain. Assume to the contrary that there exist i and j such that i < j and q i and q j are comparable. Since q i is maximal in P \ (F ⊔ {q 1 , · · · , q i−1 }) and q j ∈ P \ (F ⊔ {q 1 , · · · , q i−1 }), we have q i > q j . Then by Proposition 2.10 (a), we see that γ ∨ (q i ) < γ ∨ (q j ). Hence by the definition of the total order on H λ P , we have h i < h j , which contradicts to Condition (H1). Moreover it follows from Proposition 2.11 (b) that Conversely, suppose that F ′ F and F ′ \ F is an antichain. For q ∈ F ′ \ F , we put h(q) = (γ ∨ (q), 0) ∈ H λ P . Since F ′ \ F is an antichain, we can label the elements of F ′ \ F so that h(q 1 ) > · · · > h(q r ). Then (h(q 1 ), · · · , h(q r )) is the unique sequence satisfying Conditions (H1) and (H2) in Proposition 4.6.

Proof and corollaries of Main Theorem
In this section, we give a proof of the main theorem (Theorem 1.2 in Introduction) and derive several consequences.

Proof of the Main Theorem
Recall the main theorem of this paper: Theorem 5.1. Let P be a connected d-complete poset and F an order filter. Then the multivariate generating function of (P \ F )-partitions, where P \ F is viewed as an induced subposet of P , is given by where D runs over all excited diagrams of F in P .
Theorem 5.1 is a direct consequence of the following two theorems, which describe the ratio ξ w F | w P ξ w P | w P of the localizations of elements in the equivariant K-theory K T (X ) in two ways.
Theorem 5.2. For a connected d-complete poset P and its order filter F , we have under the identification z i = e α i (i ∈ I).
Theorem 5.3. For a connected d-complete poset P and its order filter F , we have under the identification z i = e α i (i ∈ I).
First we prove Theorem 5.2 by using the Chevalley-type formula (Proposition 4.8).
Proof of Theorem 5.2. For an order filter F of P , we put It is clear that Z P/P (z) = G P/P (z) = 1. Hence it is enough to show that Z P/F (z) and G P/F (z) satisfy the same recurrences: where F ′ runs over all order filters such that F F ′ ⊂ P and F ′ \ F is an antichain. First we prove (25). Under the isomorphism of posets given in Proposition 2.11 (a), the interval (w F , w P ] = {z ∈ W λ P : w F < z ≤ w P } corresponds to {F ′ : F ′ is an order filter of P and F F ′ ⊂ P }. Then by using the recurrence (18) and Proposition 4.8, we see that Next we prove (26). Let M be the set of maximal elements of P \ F . Then we have Then we have G P/(F ⊔I) (z) = σ∈A(P \F ) I z σ .
By the Inclusion-Exclusion Principle, we have where we put Given σ ∈ A(P \ F ), let m = min{σ(x) : x ∈ P \ F } and define σ ′ ∈ A(P \ F ) by Then the map σ → (m, σ ′ ) gives a bijection from A(P \ F ) to N × A ′ (P \ F ), and Hence we have σ∈A(P \F ) This completes the proof.
Next we derive Theorem 5.3 from the Billey-type formula (Proposition 4.7). By dividing the both sides by ξ w P | w P = p∈P (1 − z[H P (p)]), we obtain the desired identity (24).

Corollaries of the Main Theorem
First we derive the equivariant cohomology version of Theorem 5.1. This corollary is a skew generalization of Nakada's colored hook formula [22,Corollary 7.2].
Corollary 5.4. Let P be a connected d-complete poset with d-complete coloring c : P → I and F its order filter. Let a = (a i ) i∈I be indeterminates. We put a(p) = a c(p) (p ∈ P ) and define a linear polynomial a H P (u) as follows: Then we have (q 1 ,...,qn) 1 a(q 1 )(a(q 1 ) + a(q 2 )) . . . (a(q 1 ) + · · · + a(q n )) where the summation is taken over all linear extensions of P \ F , i.e., all labelings of the elements of P \ F with q 1 , . . . , q n so that q i < q j in P \ F implies i < j.
Proof. Any (P \ F )-partition σ ∈ A(P \ F ) is determined by a nonnegative integer r, an increasing sequence i 1 < · · · < i r of positive integers and an increasing sequence F ⊂ F 0 F 1 · · · F r = P of order filters of P , by the condition Hence we have . Now by using Theorem 5.1, we have . By substituting z i = t a i (i ∈ I) and multiplying the both sides by (1 − t) n with n = #(P \ F ), and then by taking the limit t → 1, we obtain where the summation on the left hand side is taken over all increasing sequences F = F 0 F 1 · · · F n = P of order filters of length n, and a D = p∈D a c(p) for a subset D ⊂ P . Such increasing sequences of order filters are in one-to-one correspondence with linear extensions (q 1 , · · · , q n ) of P \ F by the relation F k = F ∪ {q n , · · · , q n−k+1 } (0 ≤ k ≤ n).
Hence we obtain the desired result. By specializing z i = q for all i ∈ I in (22), and a i = 1 for all i ∈ I in (27), we obtain Corollary 5.6. Let P be a connected d-complete poset and F an order filter of P . We define the hook length h P (u) at u ∈ P as follows: (i) If u is not the top of any d k -interval, then we define h P (u) = #{w ∈ P : w ≤ u}. Then we have (a) The univariate generating function of (P \ F )-partitions is given by (b) The number of linear extensions of P \ F is given by where n = #(P \ F ).
If P = D(λ) and F = D(µ) are shapes corresponding to partitions λ ⊃ µ, Equations (28) and (29) reduce to Morares-Pak-Panova's q-hook formula [21, Corollary 6.17] and Naruse's hook formula [25] respectively. The trace generating function of revers plane partitions of skew shape [21, Corollary 6.20] is obtained from Theorem 5.1 by specializing z i = tq if i is the color of the maximum element of D(λ), q otherwise.
Remark 5.7. Theorem 5.1 and its corollaries hold for heaps H(w) associated to dominant minuscule elements w in any symmetrizable Kac-Moody Weyl groups, after suitable modifications are made. See Remarks 2.12 and 3.18.