A generalized SXP rule proved by bijections and involutions

This paper proves a combinatorial rule expressing the product $s_\tau(s_{\lambda/\mu} \circ p_r)$ of a Schur function and the plethysm of a skew Schur function with a power sum symmetric function as an integral linear combination of Schur functions. This generalizes the SXP rule for the plethysm $s_\lambda \circ p_r$. Each step in the proof uses either an explicit bijection or a sign-reversing involution. The proof is inspired by an earlier proof of the SXP rule due to Remmel and Shimozono, A simple proof of the Littlewood--Richardson rule and applications, Discrete Mathematics 193 (1998) 257--266. The connections with two later combinatorial rules for special cases of this plethysm are discussed. Two open problems are raised. The paper is intended to be readable by non-experts.


Introduction
Let f • g denote the plethysm of the symmetric functions f and g. While it remains a hard problem to express an arbitrary plethysm as an integral linear combination of Schur functions, many results are known in special cases. In particular, the SXP rule, first proved in [9, page 351] and later, in a different way, in [2, pages 135-140], gives a surprisingly simple formula for the plethysm s λ • p r where s λ is the Schur function for the partition λ of n ∈ N and p r is the power sum symmetric function for r ∈ N. It states that s λ • p r = ν ν ν sgn r (ν ν ν )c λ ν ν ν s ν ν ν (1) where the sum is over all r-multipartitions ν ν ν = ν(0), . . . , ν(r − 1) of n, ν ν ν is the partition with empty r-core and r-quotient ν ν ν, sgn r (ν ν ν ) ∈ {+1, −1} is as defined in §2 below, and c λ ν ν ν = c λ (ν(0),...,ν(r−1)) is a generalized Littlewood-Richardson coefficient, as defined at the end of §3 below.
Each step in the proof uses either an explicit bijection or a sign-reversing involution on suitable sets of tableaux. The critical second step uses a special case of a rule for multiplying a Schur function by the plethysm h α • p r , where h α is the complete symmetric function for the composition α. This rule was first proved in [3, page 29] and is stated here as Proposition 2.3. A reader familiar with the basic results on symmetric functions and willing to assume this rule should find the proof largely self-contained. In particular, we do not assume the Littlewood-Richardson rule. We show in §6.1 that two versions of the Littlewood-Richardson rule follow from Theorem 1.1 by setting r = 1 and taking either τ or µ to be the empty partition. The penultimate step in our proof is (12), which restates Theorem 1.1 in a form free from explicit Littlewood-Richardson coefficients. In §6.2 we discuss the connections with other combinatorial rules for plethysms of the type in Theorem 1.1, including the domino tableaux rule for s τ (s λ •p 2 ) proved in [1].
An earlier proof of both the Littlewood-Richardson rule and the SXP rule, as stated in (1), was given by Remmel and Shimozono in [17], using a involution on semistandard skew tableaux defined by Lascoux and Schützenberger in [8]. The proof given here uses the generalization of this involution to tuples of semistandard tableaux of skew shape. We include full details to make the paper self-contained, while admitting that this generalization is implicit in [8] and [17], since, as illustrated after Example 3.3, a tuple of skew tableaux may be identified (in a slightly artificial way) with a single skew tableau. The significant departure from the proof in [17] is that we replace monomial symmetric functions with complete symmetric functions. This dualization requires different ideas. It appears to offer some simplifications, as well as leading to a more general result.
Outline. The necessary background results on quotients of skew partitions and ribbon tableaux are given in §2 below, where we also recall the plethystic Murnaghan-Nakayama rule and the Jacobi-Trudi formula. In §3 we give a generalization of the Lascoux-Schützenberger involution and define the generalized Littlewood-Richardson coefficients appearing in Theorem 1.1. The proof of Theorem 1.1 is then given in §4. An example is given in §5.
Further examples and connections with other combinatorial rules are given in §6. In particular we deduce the Littlewood-Richardson rule as stated in [6,Definition 16.1] and, originally, in [10,Theorem III]. In the appendix we prove a 'shape-content' involution that implies the version of the Littlewood-Richardson rule proved in [19]. We also prove a technical result motivating Conjecture 6.7.

Prerequisites on r-quotients, ribbons and tableaux
We assume the reader is familiar with partitions, skew partitions and border strips, as defined in [18,Chapter 7]. Fix r ∈ N throughout this section. We represent partitions using an r-runner abacus, as defined in [5, page 78], on which the number of beads is always a multiple of r; the r-quotient of a partition is then unambiguously defined by [5, 2.7.29]. (See §6.2 for a remark on this convention.) The further unnumbered definitions below are taken from [3, page 28], [4, §3] and [17, §3], and are included to make this note self-contained.
Signs and quotients of skew partitions. Let n ∈ N 0 and let ν/τ be a skew partition of rn. We say that ν/τ is r-decomposable if there exist partitions is a border strip of size r (also called an r-border strip) for each j ∈ {1, . . . , n}. In this case we define the r-sign of ν/τ by sgn r (ν/τ ) = n j=1 (−1) ht(σ (j) /σ (j−1) ) .
Ribbons. Let ν/σ be a border strip in the partition ν. If row a is the least numbered row of ν meeting ν/σ then we say that ν/σ has row number a and write R(ν/σ) = a. Let r ∈ N and q ∈ N 0 . A skew partition ν/τ of rq is a horizontal r-ribbon strip if there exist partitions such that σ (j) /σ (j−1) is an r-border strip for each j ∈ {1, . . . , q} and For examples see Figure 1 above and Figure 3 in §5.
Proof. Let A be an abacus representing τ . If β is a position in A containing a bead then the row-number of the r-border strip corresponding to a singlestep downward move of this bead is one more than the number of beads in the positions {β + r + j : j ∈ N} of A. Thus a sequence of single-step downward bead moves, moving beads in positions β 1 , . . . , β c in that order, adds r-border strips in decreasing order of their row number, as required by (3), if and only if β 1 ≤ . . . ≤ β c . It follows that (i) and (ii) are equivalent. It is easily seen that (ii) is equivalent to (iii) and (iv).
Ribbon tableaux. Let n ∈ N 0 . Let ν/τ be a skew partition of rn and let α be a composition of n with exactly parts. An r-ribbon tableau of shape ν/τ and weight α is a sequence of partitions such that ρ (j) /ρ (j−1) is a horizontal r-ribbon strip of size rα j for each j ∈ {1, . . . , }. We say that ρ (j) /ρ (j−1) has label j. We denote the set of all rribbon tableaux of shape ν/τ and weight α by r-RT(ν/τ, α). For an example see §5 below.
A plethystic Murnaghan-Nakayama rule. In the second step of the proof of Theorem 1.1 we need the following combinatorial rule. Recall that h α denotes the complete symmetric function for the composition α.
Proposition 2.3. Let n ∈ N 0 . If α is a composition of n and τ is a partition then where the sum is over all partitions ν such that ν/τ is a skew partition of rn.
This rule was first proved in [3, page 29], using Muir's rule [15]. For an involutive proof of Muir's rule see [11,Theorem 6.1]. The special case when τ = ∅ and α has a single part is proved in [14, I.8.7]. In this case the result also follows from Chen's algorithm, as presented in [2, page 130]. The special case when α has a single part was proved by the author in [20] using a sign-reversing involution. The general case then follows easily by induction, using that h (α 1 ,...,α ) The Jacobi-Trudi formula. Let ∈ N. The symmetric group Sym acts on Z by place permutation. Given α ∈ Z and g ∈ Sym , we define where the entries in the middle are in positions k and k + 1.
The Jacobi-Trudi formula states that if λ is a partition with exactly parts and λ/µ is a skew partition then where if α has a strictly negative entry then we set h α = 0. A proof of the formula is given in [18, page 342] by a beautiful involution on certain tuples of paths in Z 2 .

A generalized Lascoux-Schützenberger involution
We begin by presenting the coplactic maps in [13, §5.5]. For further background see [8]. Let w be a word with entries in N and let k ∈ N. Following the exposition in [17], we replace each k in w with a right-parenthesis ')' and each k + 1 with a left-parenthesis '('. An entry k or k + 1 is k-paired if its parenthesis has a pair, according to the usual rules of bracketing, and otherwise k-unpaired. Equivalently, reading w from left to right, an entry k is k-unpaired if and only if it sets a new record for the excess of ks over (k + 1)s; dually, reading from right to left, an entry k + 1 is k-unpaired if and only if it sets a new record for the excess of (k + 1)s over ks. We may omit the 'k-' if it will be clear from the context.
For example, if w = 3422243312311 then the 2-unpaired entries are shown in bold and the corresponding parenthesised word is (4)))4((1)(11 while keeping all other positions the same, gives a new word which has kunpaired entries in exactly the same positions as w. Proof. It is clear that any k to the right of the rightmost unpaired k + 1 in w is paired. Dually, any k + 1 to the left of the leftmost unpaired k in w is paired. Hence the subword of w formed from its unpaired entries has the claimed form. When d ≥ 1, changing the unpaired subword from k c (k + 1) d to k c+1 (k + 1) d−1 replaces the first unpaired k + 1, in position i say, with a k; since every k + 1 to the left of position i is paired, the new k is unpaired. The dual result holds when c ≥ 1; together these imply the lemma. Definition 3.2. Let w be a word with entries from N. Suppose that the k-unpaired subword of w is k c (k + 1) d . If d > 0, let E k (w) be defined by changing the subword to k c+1 (k+1) d−1 , and if c > 0, let F k (w) be defined by changing the subword to k c−1 (k + 1) d+1 . Let S k (w) be defined by changing the subword to k d (k + 1) c .
The shape of t is (3, 2), (3, 2, 1)/(1), (2) and the 2-unpaired entries are shown in bold. By Definition 3.2, E 2 (t) is obtained from t by changing the leftmost unpaired 3 to a 2, and F 2 (t) is obtained from t by changing the rightmost unpaired 2 to a 3. It follows that As mentioned in the introduction, one may identify a skew m-multitableau with a single skew tableau of larger shape. For example, the semistandard skew 3-multitableau t above corresponds to This identification may be used to reduce the next two results to Proposition 4 and the argument in §3 of [17]. We avoid it in this paper, since it has an artificial flavour, and loses combinatorial data: for instance, the skew tableau above may also be identified with two different semistandard skew 2-multitableaux.
Proof. Let t = t(1), . . . , t(m) ∈ SSYT(σ σ σ/τ τ τ , α). The main work comes in showing that E k (t), F k (t) are semistandard (when defined). Suppose that E k (t) = t (1), . . . , t (m) and that the first unpaired k + 1 in w(t) corresponds to the entry in row a and column b of tableau t(j). Thus t (j) is obtained from t(j) by changing this entry to an unpaired k and t (i ) for the entry of a tableau u in row a and column b. If t fails to be semistandard then a > 1, This k is to the right of the unpaired k + 1 in w(t), so by Lemma 3.1 it is paired, necessarily with a k + 1 in row a and some column b > b of t. Since Thus t (a,e) = k + 1 and t (a−1,e) = k for every e ∈ {b, . . . , b }. Since t (a−1,b) is paired with t (a,b ) under the k-pairing, we see that t (a−1,b+j) is paired with t (a,b −j) for each j ∈ {0, . . . , b − b}. In particular, the k + 1 in position (a, b) of t is paired, a contradiction. Hence E k (t) is semistandard. The proof is similar for F k in the case when t has an unpaired k.
It is now routine to check that E k F k and F k E k are the identity maps on their respective domains, so E k and F k are bijective. If the unpaired subword Hence S k is an involution. A similar argument shows that S k E k is an involution. By (5) at the end of §2, the image of S k E k is as claimed.
We are ready to define our key involution. Say that a semistandard skew multitableau t is latticed if w(t) has no k-unpaired (k + 1)s, for any k. Let λ be a partition of n ∈ N 0 with exactly parts, let σ σ σ/τ τ τ be a skew m-multipartition of n and let T = g∈Sym SSYT(σ σ σ/τ τ τ , g · λ).
Observe that if g = id Sym then g · λ is not a partition, and so no element of SSYT(σ σ σ/τ τ τ , g · λ) is latticed. Therefore the set is precisely the latticed elements of T . Let t ∈ T . If t is latticed then define G(t) = t. Otherwise consider the k-unpaired (k +1)s in w(t) for each k ∈ N.
If the rightmost such entry is a k-unpaired k +1 then define G(t) = S k E k (t). For instance, in Example 3.3 we have k = 2 and G(t) = S 2 E 2 (t).
Proof. This follows immediately from Lemma 3.4.
This is a convenient place to define our generalized Littlewood-Richardson coefficients. In §6.1 we show these specialize to the original definition.
We now give an explicit bijection or involution establishing each step. For an illustrative example see §5 below.
Proof of (9). Apply the Jacobi-Trudi formula for skew Schur functions, as stated in §2.
Let G be the involution on T defined in §3. Let u(µ) be the semistandard µ-tableau having all its entries in its j-th row equal to j for each relevant j. Note that u(µ) is the unique latticed semistandard µ-tableau. Thus if then SSYTL(ν ν ν/τ τ τ : µ, λ) ⊆ T µ . Let t ∈ T µ . The final |µ| positions of w(t) correspond to the entries of u(µ). Every entry k + 1 in these positions is k-paired. If an entry k in one of these positions is k-unpaired then there is no k-unpaired k + 1 to its left, so every k + 1 in w(t) is k-paired. It follows that the final semistandard tableau in G(t) is u(µ) and so G restricts to an involution on T µ . By Proposition 3.5, the fixed-point set of G, acting on either T or T µ , is SSYTL(ν ν ν/τ τ τ : µ, λ).
Note that we obtain only a numerical equality: even cyclic permutations of skew r-multitableaux, do not, in general preserve the lattice property. For example, changing the abaci in Figure 2 in §5 so that 7 beads are used to represent (3, 2) and (6, 5, 5, 5, 2) induces a rightward cyclic shift of the skew tableaux forming the skew 3-multitableaux t 1 , t 2 , t 3 , t 4 . After one or two such shifts, the unique latticed skew 3-multitableaux are the shifts of t 3 and t 2 , respectively; t 4 remains unlatticed after any number of shifts. The identification of t 1 as the unique skew 3-multitableau contributing to the coefficient of s (6,5,5,5,2) in s (3,2) (s (3,3) • p 3 ) is therefore canonical, but not entirely natural.
The author is aware of two combinatorial rules in the literature for special cases of the product s τ (s λ • p r ) that avoid this undesirable feature of the SXP rule. To state the first, which is due to Carré and Leclerc, we need a definition from [1]. Let T be an r-ribbon tableau of shape ν/τ and weight λ. Represent T , as in Figure 3, by a tableau of shape ν/τ in which the boxes of the α j disjoint r-border strips forming the horizontal r-ribbon in T labelled j all contain j. The column word of T is the word of length n obtained by reading the columns of this tableau from bottom to top, starting at the the leftmost (lowest numbered) column, and recording the label of each r-border strip when it is first seen, in its leftmost column.
Theorem 6.2 ([1, Corollary 4.3]). Let r ∈ N and let n ∈ N 0 . Let ν/τ be a skew partition of rn and let λ be a partition of n. Up to the sign sgn 2 (ν/τ ), the multiplicity s τ (s λ • p 2 ), s ν is equal to the number of 2-ribbon tableaux T of shape ν/τ and weight λ whose column word is latticed.
Proof. By Theorem 6.3 in [4], up to the sign sgn r (ν/τ ), the multiplicity s τ (s (a,1 b ) • p r ), s ν is the number of (a, 1 b )-like border-strip r-diagrams of shape ν/τ , as defined in [4,Definition 6.2]. (The required translation from character theory to symmetric functions is outlined in [4, §7].) To relate these objects to r-ribbon tableaux, we define a skew partition ρ/τ to be a vertical r-ribbon strip if ρ /τ is a horizontal r-ribbon strip.
Let T be an r-ribbon tableau of shape ν/τ and weight (a, 1 b ). There is a unique partition ρ such that ρ/τ is the horizontal a-ribbon strip in T and ν/ρ is a vertical b-ribbon strip, formed from the border strips labelled 2, . . . , b + 1. Suppose RNT(T ) is latticed. Then the row numbers of these border strips are increasing. Moreover, the rightmost border strip in either of the ribbons ρ/τ and ν/ρ lies in the Young diagram of ρ/τ , and the skew partition formed from this border strip and ν/ρ is a vertical (b + 1)-ribbon strip. Therefore T corresponds to an (a, 1 b )-like border-strip r-diagram of shape ν/τ , and the column word of T is as claimed. Conversely, each such r-ribbon tableau arises in this way.
For general weights we have the following result.
Proposition 6.5. Let r ∈ N. Let T be an r-ribbon tableau. If the column word of T is latticed then the row-number tableau of T is latticed.
The proof is given in the appendix. The converse of Proposition 6.5 is false. For example s (2,2) • p 3 , s (3,3,3,3) = 1. The two 3-ribbon tableaux in r-RT (3, 3, 3, 3), (2, 2) are shown below. Both have a latticed row-number tableau, with word 2211. The column words are 2112 and 2121 respectively; only the second is latticed. In both Theorem 6.2 and Corollary 6.4 there is a lattice condition that refers directly to certain sets of r-ribbon tableaux, without making use of r-quotients. In the following problem, which the author believes is open under its intended interpretation (except when r = 2 or λ = (a, 1 b ) for some a ∈ N and b ∈ N 0 ) we say that such conditions are global. Problem 6.6. Find a combinatorial rule, simultaneously generalizing Theorem 6.2 and Corollary 6.4, that expresses s τ (s λ • p r ), s ν as the product of sgn r (ν/τ ) and the size of a set of r-ribbon tableaux of shape ν/τ satisfying a global lattice condition.
Conjecture 6.7. Let r ∈ N, let n ∈ N 0 , let ν be a partition of rn and let (a, b) be a partition of n. The number of r-ribbon tableaux T of shape ν and weight (a, b) such that the row-number tableau RNT(T ) is latticed is an upper bound for the absolute value of s (a,b) • p r , s ν .
Appendix: the shape-content involution and proof of Proposition 6.5 In the proof of (19) we used the following proposition. This proposition follows immediately from Lemma 7.2(iii) below, by setting α = ν and β = ∅. The 'shape-content involution' given in this lemma is surely well known to experts, but the author has not found it in the literature in this generality. The lemma may also be used to show that the final corollary in Stembridge's involutive proof of the Littlewood-Richardson rule [19] is equivalent to (17); this is left as a 'not-too-difficult exercise' in [19].
Given t ∈ RSYT(λ/µ, α/β), let SC(t) be the row-standard tableau of shape α/β defined by putting a k in row a of SC(t) for every a in row k of t.
Proof. (i) is obvious. For (ii) observe that if t ∈ RSYTL(λ/µ, α/β) then t, u(β) is not latticed if and only if there exists k ∈ N and an entry k + 1 in row a of t and position i of w(t) such that {j : w(t) j = k + 1, j ≥ i} + β k+1 = {j : w(t) j = k : j > i} + β k + 1.
Let b be the common value. The first b − 1 − β k entries in row k of SC(t) are at most a − 1, and the next entry is the number of a row a with a ≥ a. The entry below is the (b − β k+1 )-th entry in row k + 1 of SC(t + 1), namely a. Therefore SC(t) is not semistandard. The converse may be proved by reversing this argument. It follows from (ii) that SC restricts to involutions SSYT(λ/µ, α/β) → RSYTL(α/β, λ/µ) and RSYTL(λ/µ, α/β) → SSYT(α/β, λ/µ); taking the common domain and codomain of these involutions we get (iii).
We end with the proof of Proposition 6.5. One final definition will be useful. Let D be a subset of the boxes of a Young diagram of a partition ν. If column b is the least numbered column of ν meeting D, then we say that D has column number b, and write C(D) = b. (Thus if D is a border strip in ν then D has column number b if and only if the conjugate border strip D in ν has row number b.) For an example see Figure 5 overleaf.
Proof of Proposition 6.5. Suppose that the labels of the r-ribbons in T are {1, . . . , }. Fix k < . Let D 1 , . . . , D q be the subsets of the Young diagram of ν/τ that form the r-border strips lying in the r-ribbon strips of T labelled k and k +1, written in the order corresponding to the column word of T . Thus be the subword of the column word of T formed from the entries k and k +1. By hypothesis, w has no k-unpaired k + 1.
Let v be the subword of the word of the row-number tableau RNT(T ) formed from the entries k and k + 1. We may obtain v by reading the rows of T from left to right, starting at the highest numbered row, and writing down the label N (D j ) of D j on the final occasion when we see a box of  Figure 5. Border strips D 1 , . . . , D 12 labelled k (grey) or k + 1 (white) forming the 3-ribbons in a 3-ribbon tableau T are shown. Numbers are as in the proof of Proposition 6.5. For example, R(D 9 ) = 4 and C(D 9 ) = 9. The subword of the column word with entries k and k+1 is k + kk + kk + k + k + kk + kkk, where k + denotes k +1. The inversions are 2, 4 and 8. The subword of the row word of the row-number tableau of T with entries k and k + is obtained by sorting the entries in positions 2, 3, 4, 5 and 8, 9 into decreasing order, giving k + k + k + kkk + k + k + kkkk. and R(D j ) < R(D j+1 ). We say that such j are inversions. If there are no inversions, then v and w are equal. Otherwise, let j be minimal such that j is an inversion, and let s be maximal such that R(D j ) < R(D j+s ); note that N (D j+s ) = k + 1, by (3). The word v is obtained from w sorting its entries in positions j, j + 1, . . . , j + s into decreasing order, and then continuing inductively with the later positions. It is clear that this procedure does not create a new k-unpaired k + 1. Hence v has no k-unpaired k + 1.