On the Schur expansion of Hall-Littlewood and related polynomials via Yamanouchi words

This paper uses the theory of dual equivalence graphs to give explicit Schur expansions for several families of symmetric functions. We begin by giving a combinatorial definition of the modified Macdonald polynomials and modified Hall-Littlewood polynomials indexed by any diagram $\delta \subset {\mathbb Z} \times {\mathbb Z}$, written as $\widetilde H_{\delta}(X;q,t)$ and $\widetilde H_{\delta}(X;0,t)$, respectively. We then give an explicit Schur expansion of $\widetilde H_{\delta}(X;0,t)$ as a sum over a subset of the Yamanouchi words, as opposed to the expansion using the charge statistic given in 1978 by Lascoux and Sch\"{u}ztenberger. We further define the symmetric function $R_{\gamma,\delta}(X)$ as a refinement of $\widetilde H_{\delta}(X;0,t)$ and similarly describe its Schur expansion. We then analyze $R_{\gamma,\delta}(X)$ to determine the leading term of its Schur expansion. We also provide a conjecture towards the Schur expansion of $\widetilde H_{\delta}(X;q,t)$. To gain these results, we use a construction from the 2007 work of Sami Assaf to associate each Macdonald polynomial with a signed colored graph $\mathcal{H}_\delta$. In the case where a subgraph of $\mathcal{H}_\delta$ is a dual equivalence graph, we provide the Schur expansion of its associated symmetric function, yielding several corollaries.


Introduction
Adriano Garsia posed the question, when can the modified Hall-Littlewood polynomial H µ (X; 0, t) be expanded as in terms of Schur functions as a specific sum over Yamanouchi words, and is there a natural way to fix the expansion when it is not? At the time, he had already realized that the expansion he had in mind does not apply when H µ (X; 0, t) is indexed by the partition shape µ = (3, 3, 3). The results of this paper are in direct response to Garsia's question. In fact, the results we found proved to be more general than the question as originally posed.
In this paper, we will concentrate on three main families of polynomials, the modified Macdonald polynomials H µ (X; q, t), the modified Hall-Littlewood polynomials H µ (X; 0, t), and a refinement of the modified Hall-Littlewood polynomials, which we denote R γ,δ (X). Firstly, the Macdonald polynomials were introduced in [Macdonald, 1988] and are often defined as the set of q, t-symmetric functions that satisfy certain orthogonality and triangularity conditions. The modified Macdonald polynomials were shown to be Schur positive by Mark Haiman via representation-theoretic and geometric means in [Haiman, 2001]. Adhering to recent combinatorial work in [Haglund et al., 2005] and [Assaf, 2013], we choose to work with the modified Macdonald polynomials, though it is relatively straightforward to transition to other forms of Macdonald polynomials. For instance, the modified Macdonald polynomials may be converted to the transformed Macdonald polynomials J µ (X; q, t) by using J µ (X; q, t = t n(µ)H µ [X(1 − t); q, 1/t]. Here n(µ) is a constant given by the highest exponent of t inH µ (X; q, t), and X(1−t) is a plethystic substitution (see [Macdonald, 1995] for a definition of plethystic substitution). Macdonald polynomials also specialize to several well-known symmetric functions, including Hall-Littlewood polynomials and Jack polynomials. While combinatorial descriptions of the Schur expansions of specific families of Macdonald polynomials can be found in [Haglund et al., 2005] (which drew on the earlier work in [Carré and Leclerc, 1995] and [van Leeuwen, 2001]), [Fishel, 1995], [Zabrocki, 1998], [Zabrocki, 1999], [Lapointe and Morse, 2003], [Assaf, 2008/09], and [Roberts, 2014], an explicit combinatorial description remains elusive outside of these special cases. For details on the combinatorics of Macdonald polynomials, see [Haglund, 2008].
As just noted, Macdonald polynomials specialize to Hall-Littlewood polynomials, specifically by letting q = 0. Hall-Littlewood polynomials, in turn, specialize to the Schur functions, Schur Q-functions, and the monomial symmetric functions. They were first studied by Philip Hall in relation to the Hall algebra in [Hall, 1957] and later by D.E. Littlewood in [Littlewood, 1961]. It should be noted that the earliest known work on Hall-Littlewood polynomials actually dates back to the lectures of Ernst Steinitz in [Steinitz, 1901]. Similar to the case with Macdonald polynomials there are several easily related forms of Hall-Littlewood polynomials. In particular, a transformed H µ (X,0,t), is related to a modified Hall-Littlewood polynomial via H µ (X; 0, t) = t n(µ) H µ (X; 0, 1/t). Hall-Littlewood polynomials have proven to be a rich mathematical topic, with recent combinatorial work including (but certainly not limited to) [Nakayashiki and Yamada, 1997], [Carbonara, 1998], [Dalal and Morse, 2012],and [Loehr et al., 2013]. Expanding modified Hall-Littlewood polynomials into Schur functions can be achieved via the charge statistic, as found in [Lascoux and Schützenberger, 1978], though we will present a new expansion in this paper as a sum over a subset of the Yamanouchi words.
There is currently no simple method of deriving one expansion from the other -though the discovery of one would certainly be of interest. This new result is promising both because it allows for a greater use of the known machinery of Yamanouchi words to be applied to Hall-Littlewood polynomials, as in Part 1 of Remark 2.2 and Remark 2.5, and because it generalizes the usual setting of partition shapes to the more general δ ⊂ Z × Z. Equivalently, our result gives a new combinatorial rule for the coefficients of the Kostka-Foulkes polynomial in one variable t. For more background on the topic of Hall-Littlewood polynomials, see [Macdonald, 1995].
We use the statistics defined in [Haglund et al., 2005] to generalize the definition of the modified Macdonald polynomials H µ (X; q, t) and the modified Hall-Littlewood polynomials H µ (X; 0, t) to any diagram δ ⊂ Z × Z, giving the functions H δ (X; q, t) and H δ (X; 0, t). We may then write H δ (X; 0, t) in terms of the refinement polynomials R γ,δ (X), defined via row reading words of fillings of δ with a fixed descent set γ ⊂ δ. We will discuss these polynomials in the general context of diagrams, though the reader without concern for generality may freely replace the reader with a refined taste for the specific is free to replace δ with a partition shape in the statement of all major results. We may then write the main theorem of this paper as follows.
Theorem 1.1. If γ and δ are any diagrams such that γ ⊂ δ, then Here, Yam δ (λ) is the subset of the Yamanouchi words with content λ whose elements, when thought of as row reading words of a filling of the diagram δ, never have the j th from last i in the same pistol of δ as the j + 1 th from last i + 1. The above definitions and notation will be given a more thorough treatment in Section 2.
The main tool used in the proof of Theorem 1.1 is the theory of dual equivalence graphs. Dual equivalence has its roots in the work of Schützenberger in [Schützenberger, 1977], Mark Haiman in [Haiman, 1992], and Donald Knuth in [Knuth, 1970]. Sami Assaf introduced the theory of dual equivalence graphs in her Ph.D. dissertation [Assaf, 2007] and subsequent paper [Assaf, 2013]. The theory was further advanced by the author in [Roberts, 2014], from which we will derive the definition of dual equivalence graph used in this paper. In these papers, a dual equivalence graph is associated to a symmetric function so that each component of the graph corresponds to a single Schur function. Thus, the Schur expansion of said symmetric function is described by a sum over the set of components of the graph. Specifically, each component of a dual equivalence graph is isomorphic to a unique standard dual equivalence graph G λ , which in turn corresponds to the Schur function s λ . Variations of dual equivalence graphs have also been given for k-Schur functions in [Assaf and Billey, 2012], for the product of a Schubert polynomial with a Schur polynomial in , and for shifted tableaux in relation to the type B Lie group in [Billey et al.] and [Assaf, 2014]. Dual equivalence graphs were also connected to Kazhdan-Lusztig polynomials and W -graphs by Michael Chmutov in [Chmutov, 2013]. This paper will focus on dual equivalence graphs that emerge as components of a larger family of graphs. The involution D δ i : S n → S n was first introduced in [Assaf, 2007] and can be used to define the edge sets of a signed colored graph H δ with vertex set S n and vertices labeled by the signature function σ, which is defined via the inverse descent sets of permutations. We may then associate H µ (X; 0, t) and R γ,δ (X) to subgraphs of H δ . We show that these two subgraphs are dual equivalence graphs in Theorem 3.2. The main contribution of this paper to the theory of dual equivalence graphs can then be stated in the following theorem.
Theorem 1.2. Let δ be a diagram of size n, and let G = (V, σ, E) be a dual equivalence graph such that G is a component of H δ and G ∼ = G λ . Then there is a unique vertex of V in SYam δ (λ), and V ∩ SYam δ (µ) = ∅ for all µ = λ.
Here, SYam δ (λ) is the set of permutations resulting from standardizing the words in Yam δ (λ). This paper is organized as follows. We begin with the necessary material from the literature in Section 2, discussing tableaux, symmetric functions, and dual equivalence graphs. In Section 3, we give a classification of which connected components of H δ are dual equivalence graphs in Lemma 3.1, and use this to prove that the signed colored graphs associated to R γ,δ (X) and H δ (X; 0, t) are dual equivalence graphs. We then prove Theorem 1.2, followed by Theorem 1.1, as well as some related results. Next, Conjecture 3.8 gives a possible direction towards the Schur expansion of Macdonald polynomials. Section 4 is dedicated to further analysis of H µ (X; q, t) and H µ (X; 0, t). After classifying when H µ (X; q, t) and H µ (X; 0, t) expand via Yamanouchi words in Corollary 4.1 and Proposition 4.2, we then end by classifying when R γ,δ (X) = 0 in Proposition 4.4 and giving a description of the leading term in the Schur expansion of R γ,δ (X) in Proposition 4.6.

Tableaux and Permutations
By a diagram δ, we mean a finite subset of Z × Z. We let |δ| denote the size of this subset. By reflecting a diagram δ over the line x = y in the Cartesian plane, we may obtain the conjugate diagram, denoted δ ′ . A partition λ is a weakly decreasing finite sequence of nonnegative integers λ 1 ≥ . . . ≥ λ k ≥ 0. We write |λ| = n or λ ⊢ n if λ i = n. We will give the diagram of a partition in French notation by drawing left justified rows of boxes, where λ i is the number of boxes in the i th row, from bottom to top, with bottom left cell at the origin, as in the left diagram of Figure 1. Any diagram that arises from a partition in this fashion is said to have partition shape. Notice that conjugating a partition shape provides another partition shape. Figure 1: The diagrams for (4,3,2,2) and an arbitrary δ ⊂ Z × Z.
A filling is a function that takes each cell of a diagram δ to a positive integer. We express a filling visually by writing the value assigned to a cell inside of the cell. A standard filling uses each value in some [n] = {1, . . . , n} exactly once. Here, we say that T is a standard filling of δ, and we define SF(δ) as the set of standard fillings of δ. A standard Young tableau is a standard filling in which all values are required to be increasing up columns and across rows from left to right. The set of all standard Young tableaux on diagrams of partition shape λ is denoted by SYT(λ), and the union of SYT(λ) over all λ ⊢ n is denoted SYT(n). For more information, see [Fulton, 1997, Part I], [Sagan, 2001, Ch. 3], or [Stanley, 1999, Ch. 7].
Define the row reading word of a filling T , denoted rw(T ), by reading across rows from left to right, starting with the top row and working down, as in Figure 2. The row reading word of a standard filling is necessarily a permutation. For ease of reading, we will use π for permutations and use w in the more general context of words. Given a word w and diagram δ, both of size n, we let T δ (w) denote the filling of diagram δ with reading word w. By a pistol of a diagram δ, we mean a set of cells, in row reading order, between some cell c and the position one row below c, inclusive. By a pistol of a filling T , we mean a pistol of the diagram of T . If |δ| = n, we may associate each cell to a number in [n] in row reading order. In turn, this associates each pistol of δ with an interval I ⊂ [n]. In this case, we say that I is a pistol of δ and any collection of numbers in I is said to be δ-pistoled, as shown in Figure 2.1. In particular, given a word w of length |δ|, any collection of indices corresponding to cells in a pistol of T δ (w) is δ-pistoled. As an example, using the standard filling on the right of Figure 2, w = 3214, and the indices of 3 and 2 are δ-pistoled, as are the indices of 1 and 4.
Given a permutation π in one-line notation, the signature of π is a string of 1's and −1's, or +'s and −'s for short, where there is a + in the i th position if and only if i comes before i + 1 in π. Notice that a permutation is one entry longer than its signature. The signature of a standard filling T is defined as σ(T ) := σ(rw(T )). As an example, the signatures of the standard fillings in Figure 2 are + − − + − + −+ and − − +, respectively.
Each of these four sets is δ-pistoled, as is all of its subsets. same shape λ ⊢ n. Here, P is called the insertion tableau and Q is called the recording tableau. Let P : S n → SYT(n) be the function taking a permutation to its insertion tableau. One fact that will be use several times is that P (π| [n−1] ) is the result of removing the cell containing n from P (π). Given any T ∈ SYT(n), the set of permutations π such that P (π) = T is called a Knuth class. More generally, the R-S-K correspondence can be defined for all words w by considering each occurrence a value w i to be less than ay occurrence of the value later in the w. In this way, the P tableaux is allowed to have repeated entries, though this level generality will not be necessary for the proofs in this paper. We will assume a familiarity with the R-S-K correspondence, but the reader can find more details on the many properties of the R-S-K correspondence in Fulton [1997] or Sagan [2001].
For I ⊂ [n] and π ∈ S n , let π| I be the word given by reading the values of I in the order they appear in π. For any such I, π| I is referred to as a subword of π. Given any standard filling T of size n, we let T | I denote the filling that results from removing all cells of T with values not in I, as shown in Figure 4. Given a set of permutations S ⊂ S n , we let S| I = {π| I : π ∈ S}. π = 483691257 T = We may standardize a word w of length n by replacing the values in w with the values in [n] as follows. If there are k 1's, replace the 1's in w with the values 1 through k from left to right. Then replace the 2's in w in a similar fashion, replacing the first 2 with the value k + 1. Repeat this recursively until w has been replaced by a permutation, which we will denote st(w). Notice st(π) = π for all π ∈ S n . It can further be shown that standardization preserves recording tableaux, in the context of R-S-K on words. We may unstandardize a permutation π by replacing each value i with 1 plus the number of −1's in σ(π| [i−1] ), resulting in a word that we will denote unst(π). That is, i and i + 1 are taken to the same value if i occurs before i + 1. Otherwise, i + 1 is taken to the value that is one larger than that of i. We may unstandardize a word w by letting unst(w) = unst(st(w)). Notice that st(unst(π)) = π and unst(unst(w)) = unst(w). See Figure 5 for an example.
For λ ⊢ n, let U λ be the standard Young tableau of shape λ given by placing the numbers in [n] in order across the first row of λ, then across the second row, and so on. Now define SYam(λ) := {π ∈ S n : P (π) = U λ }, and call this set of permutations the standardized Yamanouchi words of shape λ. Let Yam(λ) denote the set of all words w of length n such that there are never more i + 1's than i's while reading from right to left with the further requirement that i occurs λ i times in w, as demonstrated in Figure 5. Any such word is called a Yamanouchi word. It then follows from properties of the R-S-K correspondence that SYam(λ) = {st(w) : w ∈ Yam(λ)}. Similarly, Yam(λ) = {unst(π) : π ∈ SYam(λ)}. Finally, if π ∈ SYam(λ), where λ = (λ 1 , · · · , λ k ), then P (π| [n−1] ) is the result of removing n from U λ , and so π| [n−1] ∈ SYam((λ 1 , · · · , λ k−1 , λ k − 1)). Definition 2.1. Let δ be any diagram, and let w be any Yamanouchi word of length |δ|. We say that w jams δ if there exists some i and some j such that indices of the j th from last i in w and the j + 1 th from last i + 1 in w are δ-pistoled. A standardized Yamanouchi word is said to jam δ if unst(π) = w jams δ. In the context of having such a w, we refer to the index of w containing the j th from last i as jamming said pistol of δ.
We may then define, with examples of each set given in Figure 6.
1. One method for listing Yamanouchi words is to begin with the number 1 and add numbers to the left of it, as in the description of Yamanouchi words above. The further condition that a word not jam δ simply means that upon adding the j + 1 th i + 1, it needs to be checked that this i + 1 is not in a pistol with the j th i. That is, the process of generating Yam δ (λ) is readily integrated into the procedure for finding Yam(λ).
2. For the reader that prefers permutations, we may describe SYam δ (λ) as follows. Consider the result of right justifying U λ , and let A λ be the set of pairs of values in cells that are touching on a southeasterly diagonal. For any π ∈ SYam(λ), consider T δ (π). Then π ∈ SYam δ (λ) if and only if no pairs in A λ are in a pistol of T δ (π). In this way, we have encoded which pairs correspond to the j th from last i and the j + 1 th from last i + 1 in unst(π) into the sets of A λ , and then look for these pairs in the pistols of T δ (π). See Figure 7 for an example.  2,2) , followed by the result of right justifying, followed by A (5,2,2) , followed by a standard filling T of partition shape µ = (3, 3, 3) such that rw(T ) ∈ SYam((5, 2, 2)) and the pair (8, 7) ∈ A (5,2,2) is in a pistol of µ. Thus rw(T ) / ∈ SYam µ ((5, 2, 2)).
3. The set of permutations in SYam(λ) is a Knuth class. The set SYam δ (λ) is necessarily a subset of this class, and so can be expressed via some set of recording tableaux of a given partition shape. Finding a more explicit way of generating all such recording tableaux remains an interesting open problem.

Symmetric Functions
We briefly recall the interplay between standard fillings, quasisymmetric functions, Schur functions, Macdonald polynomials, and Hall-Littlewood polynomials. All three have rich connections to the theory of symmetric functions. The curious reader may also refer to [Stanley, 1999, Ch. 7], [Fulton, 1997, Part I], or [Sagan, 2001, Ch. 4].
We may now use the previous definition to define the Schur functions, relying on a result of Ira Gessel. While it is not the standard definition, it is the most functional for our purposes.
In order to define the modified Macdonald polynomials and the modified Hall-Littlewood polynomials, we will first need to define some statistics, relying on the results in [Haglund et al., 2005] for our definitions. Let T be a filling of a diagram δ.
where d is the cell directly below and adjacent to c. Here, c must have a cell directly below it in order to be a descent. We denote the set of descents of T as Des(T ). As an example, the filling on the left of Figure 2 has descents in the cells containing 3, 4, 6, 8, and 9, while the filling on the right has no descents. Given a cell c of δ, define leg(c) as the number of cells in δ strictly above and in the same column as c. Letting w = rw(T ), we may then define We are now able to define a combinatorial generalization of the modified Macdonald polynomials, We similarly define the modified Hall-Littlewood polynomials as, It follows immediately from these definitions that Using [Haglund et al., 2005], it is possible to write H δ (X; q, t) as a sum of Lascoux-Leclerc-Thibon polynomials, which are shown to be Schur positive in [Grojnowski and Haiman, 2007]. Hence, H δ (X; q, t) is both symmetric and Schur positive. While the coefficients of the Schur expansion of H δ (X; 0, t) and R γ,δ (X) are given by Theorem 1.1, finding a combinatorial description of this expansion for H δ (X; q, t) remains an open problem.
We should emphasize that our definition of H δ (X; q, t) and H δ (X; 0, t) are combinatorial generalizations, chosen to agree with definitions in [Haglund et al., 2005] and related definitions of LLT polynomials. Hence, they need not agree with any algebraic generalizations of Macdonald polynomials. Specifically, Garsia and Haiman conjectured a generalization of Macdonald polynomials to diagram indices in [Garsia and Haiman, 1995], with further results contributed by Jason Bandlow in his Ph.D. dissertation [Bandlow, 2007]. Their conjecture would require that H δ (X; q, t) = H δ ′ (X; t, q). While this is the case when δ is a partition, it fails for the diagram {(0, 0), (1, 1)}. Finding a way of recovering this symmetry, perhaps by modifying the maj statistic, is an important open problem.
Remark 2.5. In order to use Theorem 1.1 to expand H δ (X; 0, t) in terms of Schur functions, it is necessary to generate {w ∈ Yam δ (|δ|) : inv δ (w) = 0}. We may do this by making a tree: proceeding as mentioned in Part 1 of Remark 2.2 by filling δ in reverse row reading order and checking that there are no inversions, that we still have a Yamanouchi word, and that no pistol is jammed with the addition of each new entry. In the case of partition shapes, one can consider such fillings row by row to show that the bottom three rows must satisfy one of the three cases in Figure 9. Specifically, the bottom row must be all 1's, the second row starts with k 2's followed by all 1's, and the third row has j ≤ k 3's followed by one of three options. Either the rest of the third row is 1's, or there are k − j 1's followed by all 2's, or the rest may be all 2's if the result is still a Yamanouchi word. It is, in theory, possible to precompute more rows in this fashion at the expense of more complicated rules.
It should be noted that the tree described above may still have dead ends. In that respect, a key open problem is to find an algorithm that avoids any dead ends in order to maximize efficiency. Such an algorithm was provided for the Littlewood-Richardson coefficients in [Remmel and Whitney, 1984], suggesting that it may be possible in this case as well.

Dual Equivalence Graphs
The key tool used in this paper is the theory of dual equivalence graphs. We quickly lay out the necessary background on the subject in this section.
Definition 2.6 (Haiman [1992]). Given a permutation in S n expressed in one-line notation, define an elementary dual equivalence as an involution d i that interchanges the values i − 1, i, and i + 1 as and that acts as the identity if i occurs between i − 1 and i + 1. Two permutations are dual equivalent if one may be transformed into the other by successive elementary dual equivalences.
For example, 21345 is dual equivalent to 41235 because d 3 (d 2 (21345)) = d 3 (31245) = 41235. We may also let d i act on the entries of a standard Young tableau by applying them to the row reading word. It is not hard to check that the result of applying this action to a standard Young tableau is again a standard Young tableau. The transitivity of this action is described in the following theorem.
Theorem 2.7 ( [Haiman, 1992, Prop. 2.4]). Two standard Young tableaux on partition shapes are dual equivalent if and only if they have the same partition shape.
It follows from [Haiman, 1992, Lem. 2.3] that the action of d i on S n is further related to its action on SYT(n) by d i (P (π)) = P (d i (π)). (2.10) By definition, d i is an involution, and so we define a graph on standard Young tableaux by letting each nontrivial orbit of d i define an edge colored by i. By Theorem 2.7, the graph on SYT(n) with edges colored by 1 < i < n has connected components with vertices in SYT(λ) for each λ ⊢ n. We may further label each vertex with its signature to create a standard dual equivalence graph that we will denote G λ = (V, σ, E), where V = SYT(λ) is the vertex set, σ  Figure 10: The standard dual equivalence graphs on all λ ⊢ 5 up to conjugation.
is the signature function, and E is the colored edge set. Refer to Figure 10 for examples of G λ with λ ⊢ 5. Definition 2.4 and Theorem 2.7 determine the connection between Schur functions and dual equivalence graphs as highlighted in [Assaf, 2013, Cor. 3.10]. Given any standard dual (2.11) Here, G λ is an example of the following broader class of graphs. We denote a signed colored graph by G = (V, σ, E 2 ∪ · · · ∪ E n−1 ) or simply G = (V, σ, E).
In order to give an abstract definition of dual equivalence graphs, we will need definitions for isomorphisms and restrictions. Given two signed colored graphs G(V, σ, E) and H(V ′ , σ ′ , E ′ ), an isomorphism φ : G → H is a bijective map from V to V ′ such that both φ and φ −1 preserve colored edges and signatures. The definition of a restriction is a bit more technical. 2. E ′ i = E min(I)+i−1 when i ∈ {2, 3, . . . , |I| − 1} and E min(I)+i−1 is defined.
One useful example of Definition 2.9 arises by considering the restriction of G λ to an integer interval I ⊂ [|λ|], as in Figure 11. In this case, the resulting graph can be realized by removing cells with values not in I and then lowering the remaining values by |I| − 1. Edges are then given by the action of d i on these new vertices. We now proceed to the definition of a dual equivalence graph. Here, we use results in [Roberts, 2014] as our definition (see Remark 2.11 for details). For more general definitions, see [Assaf, 2013] and [Roberts, 2014]. Locally Standard Property: If I is any interval of integers with |I| = 6, then each component of G| I is isomorphic to some G λ .
Commuting Property: If {v, w} ∈ E i and {w, x} ∈ E j for some |i − j| > 2, then there exists y ∈ V such that {v, y} ∈ E j and {x, y} ∈ E i .
Remark 2.11. The definition of dual equivalence graph in [Assaf, 2013] requires that a signed colored graph satisfy six axioms. Theorem 3.17 of [Roberts, 2014] eliminates Axiom 6 by strengthening Axiom 4, replacing it with Axiom 4 + . The Commuting Property is a restatement of Axiom 5, while Part 2 of Remark 3.18 in [Roberts, 2014] states that Axioms 1, 2, 3, and 4 + are equivalent to our Locally Standard Property, though the names of these properties have not occurred before this paper. Next, we will associate to every Macdonald polynomial and Hall-Littlewood polynomial a signed colored graph. To do this, we need to define an involution D δ i to provide the edge sets of a signed colored graph, as defined originally in [Assaf, 2013]. First letd i : S n → S n be the involution that cyclically permutes the values i − 1, i, and i + 1 as 2.12) and that acts as the identity if i occurs between i − 1 and i + 1. For example,d 3 •d 2 (4123) = d 3 (4123) = 3142. We now define the desired involution. Given π ∈ S n and a diagram δ of size n, As an example, we may take π = 53482617 and δ as in Figure 12. Then D δ 3 (π) =d 3 (π) = 54283617 and D δ 5 (π) = d 5 (π) = 63482517. Figure 12: Three standard fillings of a diagram δ. At left, a standard filling with row word π = 53482617 followed by T δ (D δ 3 (π)) and then T δ (D δ 5 (π)).
Given some δ of size n, we may then define an Assaf Graph as the signed colored graph H δ = (V, σ, E) with vertex set V = S n , signature function σ given via inverse descents, and edge sets E i defined via the nontrivial orbits of D δ i . It is readily shown that the action of D δ i on π preserves inv δ (π), Des δ (π), and maj δ (π). Thus, these functions are all constant on components of H δ . We may study H δ (X; 0, t) by restricting our attention to components of H δ where inv δ is zero, as in the following definition.
Definition 2.13. Let γ and δ be diagrams such that γ ⊂ δ and |δ| = n. We define two subgraphs of H δ = (V, σ, E) induced by restricting the vertex set V as follows.
Notice that each subgraph is a union of connected components of H δ . Furthermore, (2.14) The Assaf graph H δ is the primary object of interest in the proof of the following lemma, which was originally stated in terms of Lascoux-Leclerc-Thibon polynomials but is easily translated using results in [Haglund et al., 2005].

.3]). Let δ be a diagram such that no pistol of δ contains more than three cells, then
Proof. For the sake of completeness, we briefly sketch how to translate [Roberts, 2014, Thm. 4.3] into the form used in Lemma 2.14. Every standard filling of δ can be taken to a unique standard filling of a tuple of not-necessarily connected ribbon tableaux ν, as described in [Haglund et al., 2005], taking the row reading word of a filling of δ to the content reading word of ν. In this way, H δ (X; q, t) can be expanded into a sum of LLT polynomials, where each term is multiplied by an appropriate power of q and t. The requirement that each pistol of δ contains no more than three cells is equivalent to requiring that the diameter of ν is at most 3, as defined in [Roberts, 2014]. If the diameter of ν is at most 3, then [Roberts, 2014, Thm. 4.3] states the the Schur expansion of the related LLT polynomial may be retrieved as a sum over fillings of ν whose reading words are Yamanouchi words. Lemma 2.14 is then an immediate result of sending fillings of δ to fillings of ν and applying this Schur expansion.

The graphs P δ and R γ,δ
Next we give a classification for when a component of H δ is a dual equivalence graph. To do this, we will need the following definition. Given permutations p ∈ S m and π ∈ S n with m ≤ n, we say that p is a strict pattern of π if there exists some sequence i 1 < i 2 < . . . < i m such that π i j = p j + k for some fixed integer k and all 1 ≤ j ≤ m. Furthermore, we say that p is a δ-strict pattern of π if the indices i 1 , i 2 , . . . , i m of π are δ-pistoled. As an example, on the left side of Figure 12, p = 231 is a strict pattern of π = 534826179 occurring in π at indices 2, 3, and 5. These indices correspond to the second, third, and fifth cell in row reading order of the given diagram δ. These cells are contained in a pistol δ. Thus, p = 231 is a δ-strict pattern of π.  Proof. If G contains a vertex with one of the patterns described above, then we may consider the smallest interval I such that restricting to π| I gives a word that still contains one of the strict patterns. Here, |I| = 4 or 5 depending on whether the pattern is in the first or second case above, respectively. We may then consider the component of π in G| I . There are four possibilities corresponding to the four strict patterns described above. Direct inspection shows that each such component does not satisfy the Locally Standard Property in Definition 2.10, as demonstrated in Figure 13. If, on the other hand, G does not contain a vertex with one of the above patterns, we need only show that G satisfies Definition 2.10. The proof of the Commuting Property follows from the fact that D δ i fixes all values except i − 1, i, and i + 1. To demonstrate the Locally Standard Property, it suffices to check H δ with |δ| = 6. While it is possible to meticulously do this check by hand, it is more straight forward to verify the finitely many cases by computer. More specifically, it suffices to consider the possible graphs on S 6 . Here, we must consider the different ways of grouping the six indices into pistols separately, since the action of D i is dependent on the choice of these pistols. The code for said verification can be found at <http://www.math.washington.edu/∼austinis/Proof DEG by strict patterns.pdf>, cited as [Roberts, 2013].
Theorem 3.2. If γ and δ are any diagrams such that γ ⊂ δ, then R γ,δ and P δ are dual equivalence graphs.
Proof. Let π be some arbitrary vertex of R γ,δ or P δ . It suffices to show that if the hypotheses of Lemma 3.1 are not satisfied by a permutation π and diagram δ, then inv δ (π) = 0. Let T = T δ (π) such that π contains one of the strict patterns mentioned in Lemma 3.1. Let π| I be this pattern. We will use the location of π| I in T to show that inv δ (π) ≥ 1.
First notice that as a word, π| I ends in a descent. If the last two values of π| I occur in the same row of T , they must be part of an inversion triple or an inversion pair, since the cell completing said triple cannot have a value in between the last two values in π| I , by the definition of a strict pattern. We may thus restrict our attention to the case where the last value of π| I does not share its row in T with any other value of I.
We will now demonstrate an inversion triple or inversion pair by ignoring any rows and columns that do not contain the last four values of π| I . By the assumption that the last four values of π| I are contained in a pistol, we have restricted to a diagram with exactly two rows. We may then demonstrate an inversion triple or an inversion pair in all possible cases, as shown in Figure 14. Thus inv δ (π) ≥ 1, completing our proof.

The proofs of Theorems 1.1 and 1.2
We begin by giving several lemmas necessary for the proof of Theorem 1.2. These lemmas, however, may be safely skipped without hindering the understanding of later results. In the proof of Lemmas 3.3-3.5, we will use the variables s, t, u, and v to denote vertices of graphs. Moreover, the vertices of graphs in the proofs of these lemmas will always be permutations.
Lemma 3.3. Let γ be a diagram such that |γ| = n, and let λ ⊢ n have at most two rows. Let π ∈ SYam γ (λ) and u ∈ S n−1 such that u = π| [n−1] . Let δ be the diagram of T γ (π) [n−1] . Then u is connected by a path p in H δ to some vertex v such that: Furthermore, the sequence of edge colors in p is not dependent on the choice of π ∈ SYam γ (λ).
Proof. We proceed by induction on the size of λ 2 . If λ 2 ≤ 1, then u must be the identity permutation 123 · · · n − 1, since π ∈ SYam(λ). We may then let u = v in order to satisfy the result. For the inductive step, suppose that λ 2 > 1 and that the result holds for all two row partitions whose second row is smaller than λ 2 . We will apply a sequence of edges to find v, as portrayed in Figures 15 and 16. Here, it may be helpful to notice that P (π) = U λ , by the definition of SYam(λ), and that P (u) is the result of removing n from U λ . It may also be helpful to recall that the action of d i on P (u) can be understood via (2.10). It follows from the fact that π ∈ SYam(λ) and λ 2 ≥ 2 that u n−2 = λ 1 − 1 and u n−1 = λ 1 . To apply the inductive hypotheses, it suffices to notice that u satisfies the requirements of π in our hypothesis. That is, if we let π ′ = u, then we may satisfy the hypotheses by noting that restricting u to values in [n − 2] yields a permutation π ′ | [n −2] , where π ′ ∈ SYam δ ′ (µ), µ = (λ 1 , λ 2 −1), and δ ′ is the diagram of T δ (π ′ )| [n −2] . Specifically, π ′ = u. We may then apply induction to move n − 2 into position n − 1 of u. Similarly restricting to values in [n − 3], we may then move n − 3 into position n − 2 via some path q.
We have not changed the index of n − 1, and so n − 1 must now occur before n − 3, which occurs before the n − 2 in the last index. We may thus apply an n − 2-edge to move n − 1 into the last index. Since the last index of π does not jam a pistol of γ, the last index of u and the index of n − 1 in u cannot be δ-pistoled. In particular, the n − 2-edge must be defined via d i . Finally, we may consider the restriction to values in [n − 3] again to apply the edges of q in reverse order, ensuring u i = v i whenever u i < λ 1 . The result is the desired v, as given by applying a sequence of edges that was not dependent on the choice of π ∈ SYam(λ). Proof. Let s be as in the statement of the lemma. We begin with the case where λ has at most two rows, and proceed by induction on the size of λ 2 . If λ 2 ≤ 1, then it follows from the hypothesis that s ∈ SYam(λ) and the fact that the isomorphism φ preserves signatures that unst(φ(s)) has at most one 2. Hence, φ(s) cannot jam δ, by Definition 2.1. Next assume that λ has exactly two rows and that λ 2 ≥ 2. Applying induction, we further assume the result for all two row partitions whose second row has fewer than λ 2 cells. We wish to show that φ(s) does not jam δ. Because λ has two rows, all values of s weakly less than λ 1 are taken to 1 in unst(φ(s)) and all values greater than λ 1 are taken to 2 in unst(φ(s)).
Applying Definition 2.1, it thus suffices to show that the indices of φ(s) with values weakly less than λ 1 do not jam δ.
It follows from (2.13) that a permutation π is only contained in an i/i + 1-double edge of H δ if the indices of i and i + 1 in π are not δ-pistoled. Specifically, D i and D i+1 fix i + 2 and i − 1, respectively, and so an i/i + 1-double edge must act be switching the locations of i and i + 1, as is only the case when i and i + 1 are not δ-pistoled. We will use this fact to show that the index of λ 1 in φ(s) does not jam δ by demonstrating a particular double edge in G, as shown in Figure 17. By considering the restriction to values in [n − 2], we may apply Lemma 3.3 to move n − 2 into the index of λ 1 in s. Call this vertex t. Next we may consider the restriction to values in [n − 3] to similarly move n − 3 into the index of λ 1 − 1 in s. Call the resulting vertex u. Notice that s is connected to u by a sequence of edges defined via the action of d i and whose labels are less than n − 2. In particular, n − 1 and n are not moved. Thus the indices of n − 2 and n − 1 in u are not γ-pistoled, since the index of λ 1 in s does not jam γ. Furthermore, n and n − 3 are between n − 2 and n − 1 in u, so u must admit an n − 2/n − 1-double edge.
Now consider the effect of the same sequence of edges on φ(s). Notice that φ(s)| [n−2] also satisfies the hypotheses of Lemma 3.3. Thus, φ(s) is connected to φ(u) by a sequence of edges that are defined via d i and do not move n − 1 or n. Because isomorphisms preserve edges, φ(u) must also admit an n − 2/n − 1-double edge. In particular, the indices of n − 2 and n − 1 in φ(u) are not δ-pistoled. Thus, the indices of λ 1 and n − 1 in s are not δ-pistoled. Figure 17: The relationships between the standard Young tableaux P (s), P (t), P (u), P (v), and P (v| I ).
We still need to show that no index of φ(s) with value less than λ 1 can jam δ. Let v be the vertex connected to t by an n − 1-edge, as in Figure 17. Applying the above analysis, s is connected to v by a sequence of edges that are all defined via d i , and similarly for φ(s) and φ(v). Notice that v| [n−2] ∈ SYam((λ 1 − 1, λ 2 − 1)) and that v| [n−2] does not jam γ ′ , where γ ′ is the diagram of T γ (v)| [n−2] . By our inductive hypothesis, φ(v)| [n −2] does not jam δ ′ , where δ ′ is the diagram of T δ (φ(v))| [n−2] . Comparing φ(s) to φ(v), it follows that no index of φ(s) with value less than λ 1 can jam δ. Hence, φ(s) does not jam δ. We have thus completed our inductive argument for λ with at most two rows.
Finally, consider the case where λ has more than two rows. Suppose, for the sake of contradiction, that φ(s) jams δ. In particular, there are some values i and i + 1 in unst(s) that satisfy the definition of jamming. We wish to restrict our attention to those values in s that are sent to i and i + 1 in unst(s) and then force a contradiction by applying the two row case above. Specifically, there must exist some interval of integers I such that st(s| I ) ∈ SYam(µ), where µ has exactly two rows, and st(φ(s)| I ) jams δ ′ , where δ ′ is the diagram of T δ (φ(s))| I . Lemma 3.5. Let G and H be connected components of H γ and H δ , respectively. Further suppose that there exists an isomorphism φ : G → H. If s is a vertex of G such that s ∈ SYam γ (λ), then φ(s) ∈ SYam δ (λ).
Proof. Let s be as in the statement of the lemma. We begin with the case where λ has at most two rows and proceed by induction on the size of λ 2 . If λ 2 ≤ 1, then the fact that isomorphisms preserve signatures guarantees that unst(φ(s)) has at most one 2. Hence, φ(s) cannot jam δ by Definition 2.1. Also via signature considerations, it follows that φ(s) ∈ SYam(λ). Now assume that λ has exactly two rows and λ 2 ≥ 2. Applying our induction, we further assume the result for all two row partitions whose second row has fewer than λ 2 cells. Applying Lemma 3.4, φ(s) does not jam δ. To complete the argument, we need to show that φ(s) ∈ SYam(λ). By induction, φ(s)| [n−1] ∈ SYam(λ)| [n−1] . Since P (φ(s)) can be obtained by adjoining an a cell containing n to P (φ(s)| [n−1] ), we need only show that n is in the second row of P (φ(s)). Because σ(φ(s)) n−1 = σ(s) n−1 = +, n cannot be in the third row. It thus suffices to show that n is not in the first row of P (φ(s)).
To show that n is not in the first row of P (φ(s)), consider the vertex v as defined in the proof of Lemma 3.4 and depicted in Figure 17. Recall that φ(s) is connected φ(t) by a sequence of edges defined via d i with i < n − 1. Vertex φ(t) is then connected to φ(v) by an n − 1-edge defined via d i . Also recall that v| [n−2] ∈ SYam((λ 1 − 1, λ 2 − 1)). Thus, φ(v)| [n−2] ∈ SYam((λ 1 − 1, λ 2 − 1)), by our inductive hypothesis. Because φ(s) is connected to φ(v) by a sequence of edges defined via d i , we may use (2.10) to consider the relationship between P (φ(s)) and P (φ(v)). Suppose that n is in the bottom row of P (φ(s)). Then it must also be in the same location of P (φ(t)), as shown in Figure 18, since φ(s) and φ(t) are connected by edges with labels less than n − 1. Applying an d n−1 must then leave n − 2 in the bottom row of φ(v). In particular, φ(v)| [n−2] / ∈ SYam((λ 1 − 1, λ 2 − 1)). This contradiction completes our inductive argument for λ with at most two rows.
Finally, consider the case where λ has more than two rows. If |λ| = 1, then φ(s) clearly does not jam δ. Inducting on |λ|, we may assume Lemma 3.5 holds for |λ| = n − 1. The  Figure 18: The standard Young tableaux P (φ(t)), P (φ(v)), and P (φ(v)| [n−2] ) if n were in the bottom row of P (s). statement of Lemma 3.4 is then the inductive step necessary to prove that φ(s) does not jam δ. Now suppose, for the sake of contradiction, that φ(s) is not a standardized Yamanouchi word. Then there must be some minimal integer r such that there is a value in row r of P (s) occurring in a lower row of P (φ(s)). We may let I be the set of values in row r − 1 and r of P (s). Thus, st(s| I ) ∈ SYam(µ) for some partition µ with exactly two rows, and st(φ(s)| I ) is not a standardized Yamanouchi word. As in the end of the proof of Lemma 3.4, there must exist an isomorphism from the component of st(s| I ) in some H γ ′ to the component of st(φ(s)| I ) in some H δ ′ . However, this contradicts the conclusion of the two row case above. Thus φ(s) ∈ SYam δ (λ).
Proof of Theorem 1.2. Recall that δ is a diagram and that G = (V, σ, E) is a dual equivalence graph such that G is a component of H δ . By Theorem 2.12, we may further suppose that G ∼ = G λ , where λ ⊢ n. We need to show that |V ∩ SYam δ (λ)| = 1 and that |V ∩ SYam δ (µ)| = 0 if λ = µ.
In order to apply Lemma 3.5, we first need to show that G λ is isomorphic to a component of a well-behaved H γ . By conflating standard Young tableaux in G λ with their row reading words, we may consider G λ as a component C of H γ , where γ is the subset of the vertical axis {(0, i) ∈ Z × Z : 0 ≤ i < n}. Notice that the unique standardized Yamanouchi word in C is rw(U λ ) and that SYam γ (λ) = SYam(λ). By composing isomorphisms, we may find an isomorphism φ : G → C. By Lemma 3.5, φ acts as a bijection between the set of standardized Yamanouchi words in G that do not jam δ and the set of standardized Yamanouchi words in C that do not jam γ. Thus, there is a unique permutation in V ∩ SYam δ (λ) corresponding to φ −1 (rw(U λ )). Hence, |V ∩ SYam δ (λ)| = 1. Since SYam(λ) and SYam(µ) are disjoint when λ = µ, it follows that V ∩ SYam δ (µ) = ∅ when λ = µ.
Proof. This is an immediate consequence of Theorem 1.2 and (2.11).
Proof of Theorem 1.1. Expressing R γ,δ (X) and H δ (X; 0, t) as in (2.14), the result follows from Theorems 3.2, Corollary 3.6, and the fact that the maj δ statistic is constant on components of P δ .
1. For each partition λ and diagram δ, there exists a set Y δ (λ) defined as the intersection of SYam δ (λ) with the set of permutations of length |δ| whose component in H δ is a dual equivalence graph. That is, if G = (V, σ, E) is a component of H δ , then |V ∩ Y δ (λ)| = 1 if G ∼ = G λ , and |V ∩ S δ (λ)| = 0 otherwise. Finding a more direct way to generate Y δ , however, is an open problem.
2. We may use Theorem 3.2 to find two related families of dual equivalence graphs in H δ . Consider the graphs obtained by reversing the values in the vertex sets of P δ and R γ,δ , sending i to |δ| + 1 − i. If we similarly reverse the edge labels and multiply the signatures by −1 so that the edges and signatures are given by d i and σ, respectively. This process necessarily sends dual equivalence graphs in H δ to other dual equivalence graphs in H δ , as described in [Roberts, 2014, Cor. 3.8]. We may then apply Corollary 3.6 to describe the Schur expansions of the related symmetric functions. Combinatorially, we are restricting our attention to permutations π that achieve the maximal inv δ (π), denoted m(δ). The associated symmetric function may also be computed by applying the function ω, which sends s λ to s λ ′ , and replacing maj δ with comaj δ , which is the result of subtracting maj from the maximal value of maj δ (π). That is, 3. At this point it is appropriate to briefly discuss a connection to the Lascoux-Leclerc-Thibon (LLT) polynomials. The involution D δ i was originally defined in [Assaf, 2013] in order to assign a signed colored graph to the LLT polynomial G ν (X; q), which is the symmetric function generated as the sums over SYT(ν), the set of standard Young fillings of the tuples of skew tableaux ν = (ν (0) , . . . , ν (k−1) ). In this case, we use the involution D ν i , and the resulting signed colored graphs are called LLT graphs. In G ν (X; q), the exponent of q is defined by the inv statistic on T ∈ SYT(ν), which is closely related to our earlier definition of inv δ . The action of D ν i on T ∈ SYT(ν) preserves inv(T), where this action is defined via a reading word rw(T). For the moment, assume that ν is a tuple of ribbon tableaux. Using the relationship between Macdonald polynomials and LLT polynomials described in [Haglund et al., 2005], we may send each ν (j) to a column of some δ and then treat the graph associated to G ν (X; 0) as a collection of components of some R γ,δ . In particular, the graph associated to G ν (X; 0) is a dual equivalence graph, by Theorem 3.2. As in Theorem 1.1, we may further conclude that For the more general case where ν is a tuple of skew shapes, we need only separate each skew tableau in ν into the separate ribbons given by its columns. Replacing each ν ∈ ν, in order, by its columns provides some ν that is a tuple of all such columns. It should be noted that each ribbon may need to be translated vertically to ensure that the reading word is preserved. The result is that ν will have the same inv statistic and signature as ν, so we may use ν to obtain the analogous result.

Related Conjectures
The above analysis lends itself to an interesting conjecture about the Schur expansion of the quasisymmetric function associated to any graph comprised of components of H δ .
Conjecture 3.8. Given any diagram δ, there exists an injective function f δ : SYam(n) ֒→ S n fixing SYam δ (|δ|) and preserving σ such that for any component Conjecture 3.9 (Corollary of Conjecture 3.8). Given any diagram δ and function f δ as in Conjecture 3.8, Conjecture 3.9 has been explicitly checked when δ is a partition shape of size at most seven. It should be mentioned, however, that f δ was defined in an ad hoc fashion for each new δ.
4 Further Applications to Symmetric Functions 4.1 More analysis of H µ (X; 0, t) and H µ (X; q, t) We can now explicitly answer the question of Garsia mentioned in Section 1. We also provide the analogous result for Macdonald polynomials.
Corollary 4.1. Given a partition µ, the following equality holds if and only if µ does not contain (3, 3, 3) as a subdiagram.
Proof. First assume that (3, 3, 3) is a subdiagram of µ. In light of Theorem 1.1, it suffices to show that there exists w ∈ Yam(λ) for some λ ⊢ |µ| such that inv µ (w) = 0 and w jams µ. We may explicitly demonstrate the desired w by placing 1's in all cells of µ except the first cell of the second row and the first three cells of the third row. Now fill the four remaining cells with 3232 in row reading order, as in Figure 19, and then define w as the row reading word of this filling. Thus, inv µ (w) = 0, and the index of the last 2 of w jams the pistol starting at the first cell of the third row from the bottom. Now suppose that (3, 3, 3) is not a subdiagram of µ. We need to show that Yam µ (λ) = Yam(λ) to apply Theorem 1.1. Let w ∈ Yam(λ) be the row reading word of a filling T of µ such that inv(T ) = 0. Since a Yamanouchi word must end in a 1, and the bottom row of T must be weakly increasing to avoid inversion pairs, the bottom row must be all 1's. Similarly focusing on the construction of Yamanouchi words and the description of inversion triples in Figure 8, it is readily shown that the second row starts with some number of 2's followed by all 1's. Thus, w cannot jam any of the pistols contained in the bottom two rows. Now consider any pistol that ends before the bottom row. Since µ does not contain (3, 3, 3), any such pistol has at most three cells. In a Yamanouchi word, the index of the j th i differs by at least three from the index of the j + 1 th i + 1, and so cannot share a pistol containing less than four cells. Hence, no pistol that ends before the bottom row can be jammed by an index of w. That is, Yam µ (λ) = Yam(λ).  Figure 19: A filling with row reading word w ∈ SYam(λ) such that inv µ (w) = 0 and w jams µ.
Proposition 4.2. Given a partition µ, the following equality holds if and only if µ does not contain (4) or (3, 3) as a subdiagram. (4.2) Proof. If we assume that µ does not contain (4) or (3, 3) as a subdiagram, the result is given by Lemma 2.14. It then suffices to assume that µ contains (4) or (3, 3) and show that Equation (4.2) does not hold. We proceed by considering the coefficients of q 2 t 0 . Focusing on the right hand side of Equation (4.2), consider a Yamanouchi word w such that inv µ (w) = 2 and maj µ (w) = 0. In particular, maj µ (w) = 0 forces the columns of T µ (w) to be weakly increasing when read downward. The bottom row of T µ (w) must contribute at most two inversions, so w must end in 1. . . 111, 1. . . 121, or 1. . . 1211. That is, the bottom row can have at most one 2. Applying the fact that all columns are weakly increasing, all but one column must be all 1's.
We now consider each of these possibilities for the bottom row separately. If the bottom row is all 1's, then every column must be all 1's, and so there cannot be any inversions. If the bottom row is 1 . . . 121 or 1 . . . 1211 and there is an x > 1 above the 2 in the bottom row, then there are at least two inversion triples containing x in the first case and at least one in the second case. Both situations force inv µ (w) > 2. Hence, any values greater than one must occur in the bottom row. Because inv µ (w) = 2, we are left with only the filling containing all 1's except for a bottom row filled by 1 . . . 1211. The conclusion of this analysis is that λ⊢|µ| w∈Yam(λ) q invµ(w) t maj µ (w) s λ q 2 t 0 = s (n−1,1) . (4.3) Now consider the coefficient of q 2 t 0 in H µ (X; q, t), as described in (2.5), when µ contains (4) or (3, 3). By letting π be the standardization of w = 1 . . . 1 1 2 1 3 2, we have maj µ (π) = 0 and inv µ (π) = 2 (see Figure 20). Thus, q 2 t 0 F σ(π) has a positive coefficient in the expansion of H µ (X; q, t) into fundamental quasisymmetric functions. However, F σ(π) has coefficient 0 in the expansion of s (n−1,1) . This is clear because σ(π) = + · · · + + − +−, but all fundamental quasisymmetric functions that contribute to s (n−1,1) have exactly one minus sign. Hence, there must be some term of H µ (X; q, t) of the form q 2 t 0 s λ , where λ = (n − 1, 1). Therefore, (4.2) cannot hold if µ contains (4) or (3,3) as a subdiagram, completing our proof.

Further Analysis of R γ,δ (X)
In this section we give a method for quickly finding the leading term of the expansion of R γ,δ (X). We begin by using the relationship between γ and δ to give a characterization of when R γ,δ (X) = 0. To do so, we will need the following definition.
Definition 4.3. Given any diagrams γ and δ such that γ ⊂ δ, we refer to γ as a realizable descent set of δ if the following hold. Proposition 4.4. Given any two diagrams γ and δ such that γ ⊂ δ, then R γ,δ (X) = 0 if and only if γ is a realizable descent set of δ.
The proof of Proposition 4.4 is postponed until the end of this section.  Figure 22: At left, a diagram δ with bullets in cells of a realizable subset γ. At right, a filling whose row reading word is the leading Yamanouchi word w γ,δ = 12131221111.
Definition 4.5. Given diagrams γ and δ such that γ is a realizable descent set of δ, define the leading Yamanouchi word of R γ,δ (X), denoted w γ,δ , as the row reading word of the filling of δ achieved by placing a 1 in every cell of δ \ γ and then placing values in the rows of γ from bottom to top, filling each cell with one plus the value in the cell immediately below it in δ.
See Figure 22 for an example of Definition 4.5. Notice that w γ,δ is indeed a Yamanouchi word. We can then use w γ,δ to provide the leading term in the expansion of R γ,δ (X) into Schur functions.
Proposition 4.6. Given diagrams γ and δ such that γ is a realizable descent set of δ, let R γ,δ (X) = c λ s λ for some nonzero integers c λ , and let w γ,δ ∈ Yam(µ), then Proof. Let w γ,δ be as described in the proposition. We begin by showing that T = T δ (w γ,δ ) has Des(T ) = γ and inv(T ) = 0. The fact that Des(T ) = γ is a result of the definition of w γ,δ . We thus need to show that inv(T ) = 0. First consider a triple of T consisting of cells c, d, and e, in row reading order. Notice that if c is a descent of T , then the value in c is one greater than the value in e. It then follows that cells c, d, and e cannot form an inversion triple, regardless of the value in d. Now suppose that c is not a descent of T . Then the value in c is a 1, so it suffices to show that the value in d is weakly less than the value in e. If the value in d is 1, we are done. We may then assume that d is a descent. By the definition of w γ,δ , it follows that the value in d is one greater than the number of consecutive descents of T occurring in the cells weakly below d. Appealing to the definition of realizable descent set, there must be at least as many consecutive descents of T starting at e and continuing down. Thus, the value in d is weakly less than the value in e. That is, the cells c, d, and e do not form an inversion triple.
Next consider two cells c and d, in row reading order, that could form an inversion pair. If d is in the row below c, then the same reasoning as above shows that d must be weakly greater than c. If c and d are in the same row, then the cell below c cannot be in δ. Thus c is not a descent of T and so has value 1 in T . In particular, c and d do not form an inversion pair. Therefore, inv(T ) = 0.
Notice that the values in each cell of T are as small as possible while respecting Des(T ) = γ. Thus, there cannot be any other filling with row reading word w ∈ SYam(λ), inv δ (w) = 0, and Des δ (w) = γ, where λ ≥ µ in lexicographic order. Appealing to Theorem 1.1 completes the proof.
Proof of Proposition 4.4. We wish to show that γ is a realizable descent set of δ if and only if R γ,δ (X) = 0. First assume that γ is not a realizable descent set of δ. Notice that if γ does not satisfy Part 1 of Definition 4.3, then there are no fillings of δ with descent set γ. Thus, R γ,δ (X) = 0. Next, suppose that T is a filling of δ with Des(T ) = γ such that γ satisfies Part 1 but not Part 2 of Definition 4.3. It suffices to show that inv(T ) = 0. We suppose, for the sake of contradiction, that inv(T ) = 0.
Choose {x 1 , x 2 } × I violating Part 2 of Definition 4.3. Label the values of T in this rectangle by a, b, c, and so on, in row reading order. Here, we will call the upper left value a, whether or not the value exists in T . Regardless of existence in δ, the cell containing a is not in γ. See Figure 23 for an illustration.
Because a is not in a descent of T , and T has no inversion triples or inversion pairs, it follows that a < b < c. Here, we just have b < c if a does not exist. Also, b > d because b is in a descent of T , and so c > d. Assuming that c is a descent of T , it then follows that c > e > d > f , since T has no inversions and d is in a descent of T . In particular, notice that e > f . Continuing in this fashion recursively, it follows that the value in the bottom left corner is greater than the value in the bottom right corner. However, the bottom left corner is not a descent of T , and so the fact that its value is greater than the value to its right guarantees an inversion triple or inversion pair in T . Hence, inv(T ) = 0, as desired. Therefore, R γ,δ (X) = 0 whenever γ is not a realizable descent set of δ.