{'fraction_non_alphanumeric': 0.05129356305067624, 'fraction_numerical': 0.01024187664852124, 'mean_word_length': 3.551596424010217, 'pattern_counts': {'":': 0, '<': 13, '<?xml version=': 0, '>': 9, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 21, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Littlewood-Richardson rules for symmetric skew quasisymmetric Schur functions

The classical Littlewood-Richardson rule is a rule for computing coefficients in many areas, and comes in many guises. In this paper we prove two Littlewood-Richardson rules for symmetric skew quasisymmetric Schur functions that are analogous to the famed version of the classical Littlewood-Richardson rule involving Yamanouchi words. Furthermore, both our rules contain this classical Littlewood-Richardson rule as a special case. We then apply our rules to combinatorially classify symmetric skew quasisymmetric Schur functions. This answers affirmatively a conjecture of Bessenrodt, Luoto and van Willigenburg.


Introduction
An important linear basis for the Hopf algebra of symmetric functions, Sym, is the basis of Schur functions, denoted by s λ where λ runs over all partitions. Since Sym has various interpretations such as the representation ring of the symmetric group, or the cohomology ring of the Grassmannian, it is natural to expect that the basis of Schur functions plays a significant role in these settings as well. Indeed, this basis captures a significant amount of the interplay between algebraic combinatorics and fields such as representation theory and algebraic geometry as is demonstrated aptly by the Littlewood-Richardson rule. This rule is a combinatorial algorithm that allows us to compute the structure coefficients c λ νµ occurring in the expansion of the skew Schur function s λ/µ in the Schur basis s λ/µ = ν c λ νµ s ν .
and have contributed to a rich theory that has strong links with other combinatorial constructions such as the Robinson-Schensted algorithm, jeu de taquin and the plactic monoid [6,17]. Arguably the most popular version involves Yamanouchi words (or lattice words) with content a partition, and this is the one that is relevant for our purposes. A Hopf algebra that contains Sym and is closely tied to it is the Hopf algebra of quasisymmetric functions, QSym. Quasisymmetric functions were introduced by Gessel in [9] as generating functions for P -partitions. Since then, they have become an indispensable tool in analyzing combinatorial structures as is evident from their applications to areas as diverse as enumerating permutations [10], card shuffling [23], combinatorial expansions of special functions such as Macdonald polynomials [11] and Kazhdan-Lusztig polynomials [4], representation theory of the Hecke algebra [14], discrete geometry [5], and combinatorial Hopf algebras [1].
Haglund, Luoto, Mason, and van Willigenburg [12] introduced a new basis for QSym using combinatorial objects called semistandard reverse composition tableaux. This is the basis of quasisymmetric Schur functions. These functions refine the Schur functions in a natural way and also refine many of their properties.
In [3], Bessenrodt, Luoto, and van Willigenburg defined skew quasisymmetric Schur functions and showed that these functions expand nonnegatively in the basis of quasisymmetric Schur functions.
The C α δβ are called noncommutative Littlewood-Richardson coefficients. Additionally, they possess an interpretation that is similar to an interpretation of the classical Littlewood-Richardson coefficients. What is lacking though is an interpretation involving Yamanouchi words, which can be attributed to the fact that constructing Yamanouchi words with content a composition (that is not a partition) is impossible. However this situation can be remedied if we restrict our attention to symmetric skew quasisymmetric Schur functions.
Our aim in this article is to give a Schur positive expansion for the symmetric skew quasisymmetric Schur functions, and to give a Yamanouchi interpretation for computing structure coefficients occurring in the expansion. We achieve this aim by giving a left Littlewood-Richardson rule in Theorem 3.2 and a right Littlewood-Richardson rule in Theorem 4.2. More specifically, we give a rule to compute the coefficient of S δ in S α/ /β in the case where δ is a partition or the reverse of a partition. As an application of these rules, we obtain a combinatorial classification of symmetric skew quasisymmetric Schur functions in Theorem 5.5. In so doing, we prove a conjecture of Bessenrodt, Luoto and van Willigenburg. More precisely, in [3,Corollary 5.2], they proved that if α/ /β is a uniform skew reverse composition shape then S α/ /β is symmetric, and further conjectured [3,Conjecture 6.1] that the reverse implication also holds. We establish that this is indeed the case and our two Littlewood-Richardson rules are crucial to this end.

Background
In this section, we will define the combinatorial objects that will be useful to us later. For more details on the classical notions discussed herein, the reader should refer to [22], while a detailed exposition on more modern constructs such as reverse composition diagrams, reverse composition tableaux, and quasisymmetric Schur functions can be found in [19].
2.1. Compositions and partitions. Given a positive integer n, a composition α = (α 1 , . . . , α k ) of n is defined to be an ordered list of positive integers whose sum is n. If α is a composition of n, then we denote this by α n. The α i for 1 ≤ i ≤ k are called the parts of α, while k is called the length of α and is denoted by ℓ(α). The size of a composition α, denoted by |α|, is the sum of its parts. We denote by α r the composition obtained by reversing α, that is, if α = (α 1 , . . . , α k ) then α r = (α k , . . . , α 1 ). Finally, for convenience we define the empty composition ∅ to be the unique composition of size and length equalling 0.
We depict a composition α = (α 1 , . . . , α k ) using its reverse composition diagram, also denoted by α, which is the array of left-justified cells with α i cells in row i from the top. A cell is said to be in row i and column j if it is in the i-th row from the top and j-th column from the left, and is referred to by the pair (i, j).
A partition λ = (λ 1 , . . . , λ k ) of n is a list of positive integers that sum to n and additionally satisfy λ 1 ≥ · · · ≥ λ k . The λ i for 1 ≤ i ≤ k are called the parts of λ, while k is called the length of λ, and is denoted by ℓ(λ). The size of λ, denoted by |λ|, is the sum of its parts. If λ is a partition of size n, then we denote this by λ ⊢ n. As with compositions, given a partition λ = (λ 1 , . . . , λ k ) we define λ r to be the ordered list of positive integers obtained by reversing λ, that is, λ r = (λ k , . . . , λ 1 ). Furthermore, we define the empty partition, denoted by ∅, to be the unique partition of size and length equalling 0. Finally, note that given any composition α, there is a corresponding partition, denoted by α, which is obtained by sorting the parts of α into weakly decreasing order.
We depict a partition using its Young diagram. Given a partition λ = (λ 1 , . . . , λ k ) ⊢ n, the Young diagram of λ, also denoted by λ, is the left-justified array of n cells, with λ i cells in the i-th row. We will be using the English convention for Young diagrams. That is, the rows are numbered from top to bottom and the columns from left to right. We refer to the cell in the i-th row and j-th column by the ordered pair (i, j). The transpose of a partition λ, denoted by λ t , is the partition whose Young diagram is the array of cells Example 2.2. Shown below on the left is the Young diagram of the partition λ = (4, 3, 2, 2) ⊢ 11. On the right is λ t .

2.2.
Tableaux. We will start by defining a poset structure on compositions that allows us to construct the main combinatorial object required to introduce quasisymmetric Schur functions. Let α = (α 1 , . . . , α ℓ(α) ) and β be compositions. Define a cover relation ⋖ c on compositions as follows.
The reverse composition poset L c is the poset on the set of compositions where the partial order < c is obtained by taking the transitive closure of the cover relation ⋖ c above. If β < c α and β is drawn in the bottom left corner of α, then the skew reverse composition shape α/ /β is defined to be the array of cells We refer to α as the outer shape and to β as the inner shape. If the inner shape is ∅, instead of writing α/ /∅, we just write α and refer to α as a straight shape. The size of the skew reverse composition shape α/ /β, denoted by |α/ /β|, is the number of cells in the skew reverse composition shape, that is, |α| − |β|.
Next we will define a semistandard reverse composition tableau. Definition 2.4. A semistandard reverse composition tableau (abbreviated to SSRCT) τ of shape α/ /β is a filling τ : α/ /β −→ Z + that satisfies the following conditions (1) the entries is each row decrease weakly from left to right, (2) the entries in the leftmost column increase strictly from top to bottom, (3) if i < j and (j, k + 1) ∈ α/ /β and either (i, k) ∈ β or τ (i, k) ≥ τ (j, k + 1) then either (i, k + 1) ∈ β or both (i, k + 1) ∈ α/ /β and τ (i, k + 1) > τ (j, k + 1). (This condition will be referred to as the triple condition.) We will denote the set of SSRCTs of shape α/ /β by SSRCT(α/ /β). Given an SSRCT τ , we define the content of τ , denoted by cont(τ ), to be the list of nonnegative integers where c i equals the number of times i appears in τ and max is the greatest integer appearing in τ .
A particular SRCT that will be of importance to us is the canonical reverse composition tableau. Given a composition α = (α 1 , . . . , α k ), the canonical reverse composition tableau of shape α, denoted by τ α , is constructed as follows. In the reverse composition diagram of α, the i-th row for 1 ≤ i ≤ k is filled with the consecutive positive integers from i j=1 α j down to 1 + i−1 j=1 α j in that order from left to right.
µ ⋖ Y λ ⇐⇒ λ = (µ 1 , . . . , µ ℓ(µ) , 1) or λ = (µ 1 , . . . , µ k + 1, . . . , µ ℓ(µ) ) and µ i = µ k for all i < k The poset on the set of partitions obtained by taking the transitive closure of the cover relation ⋖ Y is well-known as Young's lattice, and we will denote it by L Y . We will denote the order relation on L Y by < Y . If µ < Y λ, the skew shape λ/µ is defined to be the array of cells where µ is drawn in the top left corner of λ. We refer to λ as the outer shape and to µ as the inner shape. If the inner shape is ∅, instead of writing λ/∅, we just write λ and call λ a straight shape. The size of a skew shape λ/µ, denoted by |λ/µ|, is the number of cells in the skew shape, which is |λ| − |µ|.
Definition 2.6. A semistandard reverse tableau (abbreviated to SSRT) T of shape λ/µ is a filling

that satisfies the following conditions
(1) the entries in each row decrease weakly from left to right, (2) the entries in each column decrease strictly from top to bottom.
We will denote the set of SSRTs of shape λ/µ by SSRT(λ/µ). Exactly as we did with SSRCTs, we define the content of an SSRT T , denoted by cont(T ), to be the list of nonnegative integers where c i equals the number of times i appears in τ and max is the greatest integer appearing in τ .
As in the case of SRCTs, there is a distinguished tableau associated with any partition. Given a partition λ = (λ 1 , . . . , λ k ), the canonical reverse tableau of shape λ, denoted by T λ , is constructed as follows. In the Young diagram of λ, the i-th row for 1 ≤ i ≤ k is filled with the consecutive positive integers from k j=i λ j down to 1 + k j=i+1 λ j in that order from left to right. Example 2.7. Shown below, on the left, is an SSRT of shape (4, 3, 2, 2)/(2, 1) and, on the right, is T (4,3,3,2) .
Let SSRCT(−/ /β) denote the set of all SSRCTs with inner shape β, and SSRT(−/ β) denote the set of all SSRTs with inner shape β. Then is defined as follows [19,Chapter 4]. Given τ ∈ SSRCT(−/ /β), obtain ρ β (τ ) by writing the entries in each column in decreasing order from top to bottom and top-justifying these new columns on the inner shape β (that might be empty).
The inverse map take the set of i entries in the leftmost column of T and write them in increasing order in rows 1, 2, . . . , i above the inner shape with a cell in position (i + 1, 1) but not (i, 1) to form the leftmost column of τ , (2) take the set of entries in column 2 in decreasing order and place them in the row with the smallest index so that either • the cell to the immediate left of the number being placed is filled and the row entries weakly decrease when read from left to right • the cell to the immediate left of the number being placed belongs to the inner shape, (3) repeat the previous step with the set of entries in column k for k = 3, . . . , β 1 .
Another tool we will need is the rectification of an SRCT, which is inserting the entries of a column of the SRCT taken in increasing order (and columns taken from left to right) into an empty SSRCT using the following insertion process [20, Procedure 3.3]. Suppose we start with an SSRCT τ whose longest row has length r. To insert a positive integer k 1 , the result being denoted k 1 → τ , scan column positions starting with the top position in column j = r + 1.
(1) If the current position is empty and at the end of a row of length j − 1, and k 1 is weakly less than the last entry in the row, then place k 1 in this empty position and stop. Otherwise, if the position is nonempty and contains k 2 < k 1 and k 1 is weakly less than the entry to the immediate left of k 2 , let k 1 bump k 2 , that is, swap k 2 and k 1 . Given an SRCT τ , its rectification is denoted by rect(τ ).
Example 2.11. Let τ be the SRCT shown below.
Then rect(τ ) is computed by inserting the integers 3, 1, 2, 5, 4, 6 in that order using the insertion process outlined above, starting from the empty SSRCT. We obtain the following sequence of SSRCTs, with the final tableau being rect(τ ).
This algorithm is analogous to the rectification of an SRT [19, Section 2.6], which is inserting the entries of a column of the SRT taken in increasing order (and columns taken from left to right) into an empty SSRT. The following insertion process is used, which inserts a positive integer k 1 into an SSRT T and is denoted by T ← k 1 .
(1) If k 1 is less than or equal to the last entry in row 1, place it at the end of the row, else (2) find the leftmost entry in that row strictly smaller than k 1 , say k 2 , then (3) replace k 2 by k 1 , that is, k 1 bumps k 2 . (4) Repeat the previous steps with k 2 and row 2, k 3 and row 3, etc.
Example 2.12. Given an SRT T , we denote its rectification by rect(T ).
Example 2.13. Let T be the SRT shown below.
To compute the rectification of T , we insert the integers 3, 1, 2, 5, 4, 6 in that order into an empty SSRT. We get the following sequence of SSRTs.
The final SRT obtained above is rect(T ).

Symmetric functions and Schur functions. Consider
, the graded Hopf algebra of formal power series in the commuting indeterminates x 1 , x 2 , . . . graded by total monomial degree. An important Hopf subalgebra of it is the Hopf algebra of symmetric functions. We will define it by giving a basis for it.
We can now define the Hopf algebra of symmetric functions, Sym, which is a graded Hopf Sym n = span{m λ | λ ⊢ n}. Arguably the most important basis of Sym is the basis of Schur functions for the reasons given in the introduction. To describe this basis, we need to associate monomials with tableaux. Given T ∈ SSRT(λ/µ), associate a monomial x T to it as follows.
This given, we define the skew Schur function indexed by the skew shape λ/µ, denoted by s λ/µ , to be If µ = ∅ in the above definition, instead of writing s λ/∅ , we write s λ and call it a Schur function. For an empty diagram, we set s ∅ = 1. Though not evident from the above definition, s λ is a symmetric function. Furthermore, the elements of the set {s λ | λ ⊢ n} form a basis of Sym n for any nonnegative integer n. This implies that we can expand a skew Schur function as a C-linear combination of Schur functions. In fact, any skew Schur function is a linear combination of Schur functions with nonnegative integral coefficients: The structure coefficients c λ νµ above are called Littlewood-Richardson coefficients, and there are many combinatorial algorithms for computing them, two of which we will encounter later.
The skew Schur functions belong to a special class of symmetric functions, called Schur positive functions, which are defined to be those symmetric functions that expand as a nonnegative linear combination of Schur functions.

Quasisymmetric functions and quasisymmetric Schur functions.
In this subsection we will define another Hopf subalgebra of C[[x 1 , x 2 , x 3 , . . .]] that contains Sym as a Hopf subalgebra. We will then proceed to describe a distinguished basis of this Hopf subalgebra, called the basis of quasisymmetric Schur functions. This basis will be central to our investigations later on.
We can now define the Hopf algebra of quasisymmetric functions, QSym, which is a graded QSym n = span{M α | α n}.
Just as we did in the case of SSRTs, we can associate a monomial with an SSRCT. Given τ ∈ SSRCT(α/ /β), we associate to it a monomial x τ as follows.
Then we can define the skew quasisymmetric Schur function indexed by the skew reverse composition shape α/ /β, denoted by S α/ /β , to be If β = ∅, instead of writing S α/ /∅ , we write S α and call this the quasisymmetric Schur function indexed by α. For an empty diagram, we set S ∅ = 1. An important property of quasisymmetric Schur functions is that The original motivation for naming these functions quasisymmetric Schur functions is the following equality that shows that quasisymmetric Schur functions refine Schur functions in a natural way.
s λ = α=λ S α Example 2.18. We compute the following expansion the monomials associated with the following SSRCTs. In [3], it was shown that skew quasisymmetric Schur functions expand as a nonnegative integer linear combination of quasisymmetric Schur functions, and the expansion was given precisely as a generalization of the classical Littlewood-Richardson rule. We now recall this main result [3, Theorem 3.5, Equation (3.5)], which gives a noncommutative Littlewood-Richardson rule.
We also say a function is quasisymmetric Schur positive if it can be written as a nonnegative sum of quasisymmetric Schur functions.
Next, we will establish some small but useful results relating symmetric and quasisymmetric functions. Proof. This follows since s λ = α=λ S α . Proof. This follows immediately from Theorem 2.19, which shows that every skew quasisymmetric Schur function is quasisymmetric Schur positive.
The above corollary immediately begs the question of whether a combinatorial formula exists for the coefficients appearing in the Schur function expansion of a symmetric skew quasisymmetric Schur function. In answer, we will give explicit combinatorial interpretations for these, not only positive but also integral, coefficients in Theorem 3.2 and Theorem 4.2.
Lemma 2.23. Let β be a composition such that β = µ, for some partition µ. Let λ be a partition. Then Proof. This is a consequence of [3, Proposition 2.17].

A left Littlewood-Richardson rule for symmetric skew quasisymmetric Schur functions
Our aim in this section is to give a left Littlewood-Richardson rule for symmetric skew quasisymmetric Schur functions. We start by reminding the reader of a version of the Littlewood-Richardson rule before we state our own rule that resembles it. For this we need to define left Littlewood-Richardson reverse tableaux. Given a skew shape λ/µ, let LRRT l (λ/µ) denote the set of all SSRTs of shape λ/µ that have been constructed as follows. Given a partition ν = (ν 1 , . . . , ν ℓ(ν) ), using the cover relations of L Y to place cells on the inner shape µ, place ν ℓ(ν) cells containing ℓ(ν) from left to right, ν ℓ(ν)−1 cells containing ℓ(ν) − 1 from left to right, . . ., ν 1 cells containing 1 from left to right such that the k-th i from the left is weakly left of the k-th i + 1 from the left. Whenever the resulting SSRT is of shape λ/µ, this is an element in our set LRRT l (λ/µ). The elements of LRRT l (λ/µ) are referred to as left Littlewood-Richardson reverse tableaux of shape λ/µ. Now recall the following classical Littlewood-Richardson rule for reverse tableaux (for example, see [16,Theorem 2.4

]).
Theorem 3.1. Given λ/µ and a partition ν, the coefficient of s ν in s λ/µ is equal to the cardinality of the set LRRT l ν (λ/µ) = {T ∈ LRRT l (λ/µ) | cont(T ) = ν}. An analogous rule holds for reverse composition tableaux, which we will state after establishing some more notation. Given a skew reverse composition shape α/ /β, let LRRCT l (α/ /β) denote the set of all SSRCTs of shape α/ /β that have been constructed as follows. Given a partition ν = (ν 1 , . . . , ν ℓ(ν) ), using the cover relations of L c to place cells on the inner shape β, place ν ℓ(ν) cells containing ℓ(ν) from left to right, ν ℓ(ν)−1 cells containing ℓ(ν) − 1 from left to right,. . ., ν 1 cells containing 1 from left to right such that the k-th i from the left is weakly left of the k-th i + 1 from the left. Whenever the resulting SSRCT is of shape α/ /β, this is an element in our set LRRCT l (α/ /β). We will refer to the elements of LRRCT l (α/ /β) as left Littlewood-Richardson reverse composition tableaux of shape α/ /β. Now we can state the left Littlewood-Richardson rule for symmetric skew quasisymmetric Schur functions; we will prove this at the end of the section. The reader should compare the statement in the following theorem with that of Theorem 3.1.
Theorem 3.2. Given α/ /β such that S α/ /β is symmetric, and a partition ν, the coefficient of s ν in S α/ /β is equal to the cardinality of the set LRRCT l ν (α/ /β) = {τ ∈ LRRCT l (α/ /β) | cont(τ ) = ν}. Remark 3.3. Notice that if α and β are partitions of the same length then α/β = α/ /β and we recover the classical Littlewood-Richardson rule, as the cover relations on L Y and L c are identical since nothing is added to the leftmost column.
The two examples that follow illustrate the expansion into Schur functions of two symmetric skew quasisymmetric Schur functions using Theorem 3.2.
We will spend the remainder of this section establishing Theorem 3.2. Henceforth in this section, fix compositions α and β such that β < c α. Suppose further that α = λ, β = µ. Then observe that the bijection ρ −1 β restricts to a bijection ρ −1 β : LRRT l (λ/µ) → γ=λ LRRCT l (γ/ /β) and hence we have a set partition We will establish that only the elements of LRRCT l (α/ /β) contribute to the quasisymmetric Schur function expansion of S α/ /β in a sense that will become precise once we prove the following lemma.
Note that the first number we will insert is ν 1 , which will be placed in the leftmost column. Now assume the result is true for m insertions. Let the m + 1-th insertion be of some number j. Then there are two cases to consider.
If j ∈ {ν 1 , ν 1 +ν 2 , . . .}, then since τ ∈ LRRCT l (α/ /β) this implies that the pre-image under standardization of j is the leftmost occurrence of some number i, and hence all numbers inserted before j are smaller than j. Thus the insertion algorithm will place j in the leftmost column.
If j ∈ {ν 1 , ν 1 + ν 2 , . . .}, then since τ ∈ LRRCT l (α/ /β) this implies that the pre-image under standardization of j is the k-th occurrence from the left of some number i, where k > 1, which is weakly left of the k-th i + 1 from the left, which in turn is weakly left of the k-th i + 2 from the left, etc. Thus by our induction hypothesis no row whose rightmost cell contains a number larger than j is longer than the row containing j + 1, which has already been inserted since its pre-image is the (k − 1)-th occurrence of i from the left. Thus the insertion algorithm will place j to the immediate right of j + 1 as desired.
Consider now a partition ν ⊢ |α/ /β|. Recall Theorem 2.19, which implies that the coefficient of S ν in S α/ /β equals the number of SRCTs of shape α/ /β that rectify to τ ν . From Lemma 3.6, we obtain the following inclusion.
The proposition above immediately implies the following corollary, after which we give a proof for Theorem 3.2.
Proof. (of Theorem 3.2) If S α/ /β is symmetric, then the coefficient of s ν in S α/ /β is the coefficient of S ν in S α/ /β since s ν = S ν + γ=ν γ =ν S γ , and the supports of the Schur functions in the basis of quasisymmetric Schur functions are disjoint. The result now follows by Proposition 3.7.

A right Littlewood-Richardson rule for symmetric skew quasisymmetric Schur functions
In this section, we will give a right Littlewood-Richardson rule for symmetric skew quasisymmetric Schur functions. The classical version that our rule will resemble is different from the one stated in the previous section. We will first define right Littlewood-Richardson reverse tableaux.
Given a skew shape λ/µ, let LRRT r (λ/µ) denote the set of all SSRTs of shape λ/µ that have been constructed as follows. Given a partition ν = (ν 1 , . . . , ν ℓ(ν) ), using the cover relations of L Y to place cells on the inner shape µ, place ν 1 cells containing ℓ(ν) from left to right, ν 2 cells containing ℓ(ν) − 1 from left to right,. . ., ν ℓ(ν) cells containing 1 from left to right, such that the k-th i + 1 from the right is weakly right of the k-th i from the right. Whenever the resulting SSRT is of shape λ/µ, this is an element in our set LRRT r (λ/µ). Note that the aforementioned procedure implies that once such an SSRT has been constructed, as we read the entries of each column taken from right to left (where within a column we read the entries from largest to smallest) the number of i + 1s we have read is always weakly greater than the number of is we have read. Note the converse also holds. The elements of LRRT r (λ/µ) will be referred to as right Littlewood-Richardson reverse tableaux of shape λ/µ.
Recall now the following classical Littlewood-Richardson rule for reverse tableaux (for example, see [7, Section 5.2]). Theorem 4.1. Given λ/µ and a partition ν, the coefficient of s ν in s λ/µ is equal to the cardinality of the set LRRT r ν r (λ/µ) = {T ∈ LRRT r (λ/µ) | cont(T ) = ν r }. An analogous rule holds for reverse composition tableaux, and to state it we need some notation. Given a skew reverse composition shape α/ /β, let LRRCT r (α/ /β) denote the set of all SSRCTs of shape α/ /β that have been constructed as follows. Given a partition ν = (ν 1 , . . . , ν ℓ(ν) ), using the cover relations of L c to place cells on the inner shape β, place ν 1 cells containing ℓ(ν) from left to right, ν 2 cells containing ℓ(ν) − 1 from left to right,. . ., ν ℓ(ν) cells containing 1 from left to right, such that the k-th i + 1 from the right is weakly right of the k-th i from the right. Whenever the resulting SSRCT is of shape α/ /β, this is an element in our set LRRCT r (α/ /β). Again, we have that once such an SSRCT has been constructed, as we read the entries of each column taken from right to left (where within a column we read the entries from largest to smallest) the number of i + 1s we have read is always weakly greater than the number of is we have read. Note that the converse also holds. We will refer to the elements of LRRCT r (α/ /β) as right Littlewood-Richardson reverse composition tableaux of shape α/ /β. Armed with this notation, we are ready to state our analogue of Theorem 4.1 for symmetric skew quasisymmetric Schur functions.
Theorem 4.2. Given α/ /β such that S α/ /β is symmetric, and a partition ν, the coefficient of s ν in S α/ /β is equal to the cardinality of the set As before, notice that if α and β are partitions of the same length then we recover the classical Littlewood-Richardson rule, as the cover relations on L Y and L c are identical since nothing is added to the leftmost column.
Next we consider the same examples as we did in the previous section, but use Theorem 4.2 instead.
For the remainder of this section, fix compositions α and β such that β < c α. Suppose further that α = λ, β = µ. Similar to the previous section, we observe that we have the following set partition.
As in the previous section, we will establish that only the elements of LRRCT r (α/ /β) contribute to the quasisymmetric Schur function expansion of S α/ /β . Towards this end, we need the following lemma.
LRRT r ν r (λ/µ) Our strategy now is similar to the one we employed in the previous section. Consider a partition ν of |α/ /β|. By Theorem 2.19, we have that the coefficient of S ν r in S α/ /β is the number of SRCTs of shape α/ /β that rectify to τ ν r . From Lemma 4.6, we obtain the following inclusion.
This gives us the following corollary.
Proof. (of Theorem 4.2) If S α/ /β is symmetric, then the coefficient of s ν in S α/ /β is the coefficient of S ν r in S α/ /β since s ν = S ν r + γ=ν γ =ν r S γ , and the supports of the Schur functions in the basis of quasisymmetric Schur functions are disjoint. The result now follows by Proposition 4.7.

The classification of symmetric skew quasisymmetric Schur functions
The aim of this section is to use the left and the right Littlewood-Richardson rules obtained earlier to combinatorially classify symmetric skew quasisymmetric Schur functions. To this end, we need three lemmas and a proposition. For the remainder of the section fix compositions α and β where β < c α, ℓ(α) = k + ℓ and ℓ(β) = k. Our next lemma is a result about the shape of the tableaux obtained during the process of rectification of an SRCT.
Lemma 5.1. When rectifying τ ∈ SRCT(α/ /β), after the insertion of the j-th column, to form τ j , no row of τ j contains more than j cells. Proof. We proceed by induction on the number of columns. The result is trivially true for the first column. Now assume that the result is true for j − 1 columns. On the insertion of the j-th column into τ j−1 observe that since we insert entries from smallest to largest, if i → τ j−1 results in i being placed in column j, the subsequent insertions of larger numbers cannot result in a number being placed in column j + 1 since row entries must weakly decrease when read from left to right. So no row of τ j contains more than j cells.
We will now use the lemma above to prove that S α/ /β contains a distinguished summand in its quasisymmetric Schur function expansion. Not only is this summand easy to compute, it is also crucial for our desired classification. Proof. Observe that the proposition will follow by Theorem 2.19 if we can show that there exists a τ ∈ SRCT(α/ /β) whose rectification is τ δ . Let the upper shape of α/ /β be denoted by α/ /β | u and the lower shape of α/ /β be denoted by α/ /β | l . Furthermore, let δ u = (δ 1 , . . . , δ ℓ ) and δ l = (δ ℓ+1 , . . . , δ m ).
Let r 1 , . . . , r t where t = m − ℓ, be the rows with a nonzero number of cells in α/ /β | l that we will fill in the following construction. We will now order the rows according to how far to the right their leftmost empty cell lies. In particular, we say r i > r j if either the leftmost empty cell in row r i lies to the right of the the leftmost empty cell in row r j , or if the leftmost empty cell in both rows are in the same column but i < j. Thus we have a linear order on the rows with a nonzero number of cells in α/ /β | l , say, r it > r i t−1 > · · · > r i 1 .
We will now fill the cells of α/ /β | l to create τ ′ as follows.
(1) Place the integers from t down to 1 in the leftmost empty cells of rows taken in the order r it down to r i 1 . (2) Repeat until all the cells are filled. By construction τ ′ ∈ LRRCT l (α/ /β | l ) once we confirm that τ ′ is an SSRCT. First note that after every iteration, the linear order on the rows is preserved, but the chain may decrease in length. Hence in every application of the first step, the entries in the rows constructed will weakly decrease when read from left to right. By construction there are no entries in the first column, thus we only need confirm that the triple condition always holds once all the cells are filled.
Consider rows r i and r j where r i is a row higher in α/ /β | l than r j . Let r j have a cell in column c containing entry τ ′ (r j , c). There are two cases to consider. If r i > r j then the leftmost cell of r i in the outer shape is weakly right of the leftmost cell of r j in the outer shape. Thus by the cover relations in L c if r i has a cell in column c − 1, then r i has a cell in column c. Let their respective entries be τ ′ (r i , c − 1) and τ ′ (r i , c). By construction we have τ ′ (r j , c) < τ ′ (r i , c). If r i < r j then the leftmost cell of r i in the outer shape is strictly left of the leftmost cell of r j in the outer shape, and hence by construction we have τ ′ (r j , c) > τ ′ (r i , c − 1), if there exists a cell in row r i and column c − 1 with entry τ ′ (r i , c − 1). Thus τ ′ is an SSRCT and hence τ ′ ∈ LRRCT l (α/ /β | l ). So by [3,Corollary 5.2] and Theorem 3.2, S δ l and hence S δ l is a summand of S α/ /β| l . In particular, by Theorem 2.19 this implies that there exists a τ ′′ ∈ SRCT(α/ /β | l ) that rectifies to τ δ l . Now consider τ ∈ SRCT(α/ /β) whose upper shape α/ /β | u consisting of the first ℓ rows is filled as the canonical reverse composition tableau of shape δ u , and the remaining nonempty rows are those of τ ′′ with |δ u | added to each entry. We claim that the rectification of τ is τ δ .
Observe the entries in the top ℓ rows trivially rectify to τ δ u , since by our insertion procedure and Lemma 5.1 the leftmost column when inserted will be δ u 1 , δ u 1 + δ u 2 , . . . and, thereafter, if i is in column j of τ to the immediate right of i + 1 then i will be placed in column j to the immediate right of i + 1 during rectification. Since every entry in the remaining rows of τ is larger than those in the first ℓ rows, they can never be placed in the same row as any number appearing in the first ℓ rows of τ during rectification. Hence since τ ′′ rectifies to τ δ l , the remaining rows will rectify to τ δ l with |δ u | added to each entry.
Hence τ rectifies to an SRCT whose top ℓ rows are τ δ u and whose remaining rows are τ δ l with |δ u | added to each entry, that is, τ δ .
Our next lemma shows that Theorem 3.2 places a constraint on what the upper shape of α/ /β can be. Lemma 5.3. If S α/ /β is symmetric then the upper shape of α/ /β is a partition, that is, α 1 ≥ · · · ≥ α ℓ .
Proof. Assume, for the sake of contradiction, that S α/ /β is symmetric but the upper shape of α/ /β is not a partition.
We will now try to construct τ ∈ LRRCT l ν (α/ /β) for some partition ν. If we have such a τ , then by construction we know the leftmost column read from top to bottom must read 1, . . . , ℓ. Then the second left must contain entries 1, . . . , ℓ 2 for some ℓ 2 ≤ ℓ, the third left must contain entries 1, . . . , ℓ 3 for some ℓ 3 ≤ ℓ 2 and so on. By the definition of SSRCTs it follows that the entries in row i for 1 ≤ i ≤ ℓ will be equal to i. Furthermore, since the upper shape of α/ /β is not a partition, there exists some α j > α j−1 (j minimal) for 2 ≤ j ≤ ℓ. Note an (α j−1 + 1)-th j − 1 from the left cannot appear weakly left of the (α j−1 + 1)-th j from the left, as the definition of SSRCTs guarantees that this j − 1 cannot belong to the lower shape in column α j−1 + 1.
Thus, if the upper shape of α/ /β is not a partition then it is not possible to construct τ ∈ LRRCT l ν (α/ /β) for some partition ν. Hence by Theorem 3.2 the coefficient of s ν in S α/ /β is zero for all partitions ν, and hence S α/ /β = 0. However, by Proposition 5.2 S α/ /β = 0, a contradiction.
On the other hand, as the following lemma shows, using Theorem 4.2 we get a different constraint on what the upper shape of α/ /β can be.