$(s,t)$-cores: a weighted version of Armstrong's conjecture

The study of core partitions has been very active in recent years, with the study of $(s,t)$-cores - partitions which are both $s$- and $t$-cores - playing a prominent role. A conjecture of Armstrong, proved recently by Johnson, says that the average size of an $(s,t)$-core, when $s$ and $t$ are coprime positive integers, is $\frac1{24}(s-1)(t-1)(s+t-1)$. Armstrong also conjectured that the same formula gives the average size of a self-conjugate $(s,t)$-core; this was proved by Chen, Huang and Wang. In the present paper, we develop the ideas from the author's paper [J. Combin. Theory Ser. A 118 (2011) 1525-1539] studying actions of affine symmetric groups on the set of $s$-cores in order to give variants of Armstrong's conjectures in which each $(s,t)$-core is weighted by the reciprocal of the order of its stabiliser under a certain group action. Informally, this weighted average gives the expected size of the $t$-core of a random $s$-core.


Introduction
The study of integer partitions continues to be a very active subject, with connections to representation theory, number theory and symmetric function theory. A particularly prominent theme is the study of s-core partitions, when s is a natural number: we say that a partition λ is an s-core if it does not have a rim hook of length s; if λ is any partition, then the s-core of λ is the partition obtained by repeatedly removing rim s-hooks. The set of all s-cores displays a geometric structure, with connections to Lie theory. In the case where s is a prime, s-cores play an important role in the s-modular representation theory of the symmetric group.
In the last few years, there has been considerable interest in the study of (s, t)-cores, i.e. partitions which are both s-and t-cores, for given natural numbers s and t. When s and t are coprime, there are only finitely many (s, t)-cores; the exact number was computed by Anderson [An], and in the particular case where t = s + 1 coincides with the sth Catalan number. The properties of (s, t)-cores have been studied from a variety of aspects: Fishel and Vazirani [FV] explored the connection with alcove geometry and the Shi arrangement, and several authors [K,OS,V,F1] have studied the properties of the unique largest (s, t)-core. The present author [F1] defined a level t action of the affine symmetric groupS s on the set of s-cores (generalising an action due to Lascoux [L] in the case t = 1) and showed that two s-cores have the same t-core if and only if they lie in the same orbit for this action.
Recently, Armstrong has examined the sizes of (s, t)-cores, conjecturing in [AHJ] that the average size of an (s, t)-core is given by 1 24 (s − 1)(t − 1)(s + t − 1); he made the same conjecture for (s, t)-cores which are self-conjugate, i.e. symmetric down the diagonal. The conjecture for self-conjugate (s, t)-cores was proved soon afterwards by Chen, Huang and Wang [CHW], but the original conjecture proved more difficult. The 'Catalan case' t = s + 1 was proved by Stanley and Zanello [SZ], and this was generalised to the case t ≡ 1 (mod s) by Aggarwal [Ag]. Very recently, the full conjecture was proved by Johnson [J] using Ehrhart theory.
In this paper, we connect Armstrong's conjectures to the level t action of the affine symmetric group on the set of s-cores, and present variants of these conjectures, in which the size of an (s, t)-core λ is weighted by the reciprocal of the order of the stabiliser of λ under this action. Surprisingly, these weighted averages are (apparently) given by simple formulae which are very similar to those in Armstrong's conjectures. We motivate our conjectures in terms of choosing an s-core at random and asking for the expected size of its t-core.
We now indicate the layout of this paper. In the next section we give some basic definitions and recall Armstrong's conjectures. In Section 3 we consider actions of the affine symmetric group on the set of s-cores and give our variant of Armstrong's conjecture. We show how to compute the stabiliser of an (s, t)-core, and connect this to Johnson's geometric approach. We then consider actions on certain finite sets of s-cores; this allows a rigorous interpretation of our conjecture in terms of the t-core of a randomly chosen s-core. Finally, we give (with proof) a formula for the denominator in our weighted average, i.e. the sum of the reciprocals of the orders of the stabilisers of the (s, t)-cores. In Section 4 we consider self-conjugate cores, introducing an action of the affine hyperoctahedral group on the set of self-conjugate (s, t)cores and giving a weighted variant of Armstrong's conjecture in this case. As in the non-selfconjugate case, we show how to compute the stabiliser of a self-conjugate (s, t)-core and explore the connections to Johnson's work, before studying actions on finite sets of self-conjugate scores.
This paper is mostly self-contained, although several results from the author's previous paper [F1] are used. We also use some standard results on Coxeter groups without proof.

Armstrong's conjectures
We assume throughout this paper that s and t are coprime natural numbers with s 2, and we define s•t = 1 2 (s − 1)(t − 1) and u = ⌊s/2⌋. In this paper, a partition means a weakly decreasing infinite sequence λ = (λ 1 , λ 2 , . . . ) of non-negative integers such that λ i = 0 for large i. If λ is a partition, we write |λ| = λ 1 + λ 2 + · · · , and refer to this as the size of λ. We write λ ′ for the conjugate partition, defined by λ ′ i = j 1 λ j i , and we say that λ is self-conjugate if λ = λ ′ .
The Young diagram of a partition λ is the set If (r, c) ∈ [λ], then the (r, c)-rim hook of λ is the set of all (s, d) ∈ [λ] such that s r, d c and (s + 1, d + 1) [λ]. The (r, c)-hook length of λ is the size of this rim hook, which equals 1 + (λ r − r) + (λ ′ c − c). We say that λ is an s-core if none of the hook lengths of λ equals s (or equivalently if none of them is divisible by s), and we let C s denote the set of all s-cores. We say that λ is an (s, t)-core if it is both an s-core and a t-core, i.e. it lies in C s ∩ C t . If λ is any partition, then the t-core of λ is the t-core obtained by repeatedly removing rim hooks of length t.
It is an easy exercise to show that (given the assumption that s and t are coprime) there are are only finitely many (s, t)-cores. More specifically, we have the following enumerative results.
1. [An,Theorems 1 & 3] The number of (s, t)-cores is 2. [FMS,Theorem 1] The number of self-conjugate (s, t)-cores is . (s, t)-cores have been intensively studied in the last few years. A very recent result is the following, which was conjectured by Armstrong [AHJ,Conjecture 1.6].
Theorem 2.2 [J, Theorem 1.7]. The average size of an (s, t)-core is Armstrong also conjectured the same statement for self-conjugate (s, t)-cores. This was proved (rather earlier than Theorem 2.2) by Chen, Huang and Wang.
Theorem 2.3 [CHW]. The average size of a self-conjugate (s, t)-core is The purpose of this paper is to present conjectured variants of these two statements, in which the sizes of the (s, t)-cores are weighted in a meaningful way. In Section 3 we give a weighted version of Theorem 2.2, and in Section 4 we do the same for Theorem 2.3.

Action of the affine symmetric group on s-cores
The weightings in our variant of Armstrong's conjecture are defined using an action of the affine symmetric group which first appeared in [F1]. LetS s denote the affine symmetric group of degree s; this can be defined as the set of all permutations w of Z satisfying the following conditions: We will say that a function w : Z → Z is s-periodic if it satisfies condition (1) above. We remark that if w : Z → Z is s-periodic and X ⊂ Z is any transversal of the congruence classes modulo s in Z, then w satisfies condition (2) (and hence lies inS s ) if and only if x∈X w(x) = x∈X x. S s has a well-known presentation by generators and relations. Before we give this, we establish some conventions of notation: if a ∈ Z, then we write a for the set a + sZ = { a + sm | m ∈ Z}. Then Z/sZ is the set a a ∈ Z , which is an abelian group under addition in the usual way. We let Z act additively and multiplicatively on Z/sZ in the natural way, i.e. via a + b = a + b and ab = ab, for a, b ∈ Z. Now for each i ∈ Z/sZ, let w i be the element ofS s defined by ThenS s is generated by { w i | i ∈ Z/sZ}, subject to defining relations ThusS s is the affine Coxeter group of typeÃ s−1 . We define the level t action ofS s on Z, denoted w →ẘ, bẙ Note that if t = 1 then s•t = 0, so this is just the natural action ofS s on Z.
We remark that the term −s•t is not really necessary in this section (and does not appear in the definition of the level t action given in [F1]); it can be removed with an easy modification of the results below. But using the term −s•t means that the action works well with self-conjugate partitions, which will be useful in Section 4. We can use the level t action ofS s on Z to describe an action on the set of s-cores, by using beta-sets. If λ is a partition, then the beta-set of λ is the set It is easy to check that if λ is a partition and w ∈S s thenẘB λ is also the beta-set of a (unique) partition, so we can define a level t action w →w ofS s on the set of partitions by B(wλ) =ẘB λ for every partition λ and every w ∈S s . This action was introduced by the author in [F2], where it was shown that the action preserves the set of s-cores. So we can restrict the level t action to give an action (which we also denote w →w) on C s . In the case t = 1, this action was introduced by Lascoux [L].

Motivation: the t-core of an s-core
Here we recall some results which will give some meaning to the weighted average in Conjecture 3.1. We begin with a result of Olsson.

Theorem 3.2 [O, Theorem 1].
If λ is an s-core, then the t-core of λ is also an s-core.
Taking the t-core of an s-core therefore gives a map from the set of s-cores to the set of (s, t)-cores. The next result says that the fibres of this map are determined by the level t action ofS s . We can informally interpret the weighted average in Conjecture 3.1 as weighting each (s, t)core λ 'by the size of the orbit containing λ'. In fact, all the orbits are infinite, so this does not strictly make sense, though we will make it rigorous below by working with finite sets of s-cores. Thus, where Armstrong's conjecture addresses the question 'given a random (s, t)-core, what is its expected size?', our weighted version addresses the question 'given a random s-core, what is the expected size of its t-core?' It is surprising that the apparent answer is so simple and so similar to Armstrong's conjecture.
Before looking at finite sets of s-cores, we define s-sets, and examine a connection to Johnson's geometric proof of Armstrong's conjecture.

s-sets
It will be useful to encode an s-core as a set of s integers. To do this, we use the fact (first observed by Robinson [R,2.8]) that a partition λ is an s-core if and only if for every b ∈ B λ we have b − s ∈ B λ . With this in mind, we define the s-set of an s-core λ to be the set S(λ) = { a i | i ∈ Z/sZ}, where a i is the smallest integer in i but not in B λ , for each i. S(λ) is then a set of s integers which are pairwise incongruent modulo s, and which sum to s 2 . In general, we refer to any set of s integers with these two properties as an s-set; it is shown in [F1] that any s-set is the s-set of a unique s-core.
This bijection between s-cores and s-sets is used in [F1] to describe a geometric structure on the set of s-cores. Later we will see a different version of this structure which was used by Johnson in the proof of Armstrong's conjecture.
Note that we can describe the level t action ofS s on C s using s-sets: we have for any λ ∈ C s and w ∈S s . This will allow us to give a formula for |Stab s,t (λ)| in terms of S(λ) below. First we need to examine the level t action ofS s in more detail.

Basic results on the level t action
In this section we make some simple observations about the level t action ofS s on Z. The definition given in Section 3.1 specifiesẘ i for each i ∈ Z/sZ, but it is useful to have an explicit expression forẘ when w is any element ofS s . This is given by the following easy lemma. Hence we can explicitly determine the image of the level t action.
Proposition 3.5. The level t action ofS s on Z is faithful, and its image is the set Proof. To show that the level t action is faithful, observe that for w ∈S s we haveẘ(it − s•t) = tw(i) − s•t for all i ∈ Z, by Lemma 3.4. Hence ifẘ is the identity permutation, then so is w. Now we consider the image of the level t action. Take w ∈S s ; then it is clear from Lemma 3.4 thatẘ(m) ≡ m (mod t) for all m and thatẘ is s-periodic. For each m ∈ {0, . . . , s − 1} let i m be the element of {0, . . . , s − 1} such that i m t ≡ m + s•t (mod s). Since s and t are coprime, the map m → i m is bijective. So by Lemma 3.4 . The (s, t)-cores: a weighted version of Armstrong's conjecture we have m n (mod s) (since s and t are coprime) and hence So w is an s-periodic permutation of Z; one can show (by essentially a reverse of the argument in the first part of the proof) that w(0) + · · · + w(s − 1) = s 2 , so w ∈S s . By construction x =ẘ, and we are done.

Corollary 3.6. Suppose X and Y are s-sets and
Proof. Since X contains exactly one integer in each congruence class modulo s, there is a unique s-periodic function v : Z → Z such that v| X = φ; this function satisfies v(m) ≡ m (mod t) for every m since φ does, so by Proposition 3.5 it suffices to show that v ∈S s . To see that v is a bijection, it suffices to show that v(m) v(n) (mod s) when m n (mod s); since v is s-periodic we may as well take m, n ∈ X, in which case the result follows because the elements of Y are pairwise incongruent modulo s and φ is injective.
So v is an s-periodic permutation of Z. Since in addition with X a transversal of the congruence classes modulo s, we have v ∈S s .

s-sets and stabilisers
Now recall that Stab s,t (λ) denotes the stabiliser of an s-core λ under the level t action ofS s . The next result shows how to compute Stab s,t (λ) from S(λ).
Proof. The description of the level t action on C s in terms of s-sets given in Section 3.3 means that |Stab s,t (λ)| equals the number of elements ofS s fixing S(λ) setwise under the level t action on Z. The case t = 1 of Corollary 3.6 implies that for every permutation v of S(λ) there is a unique element ofS s extending v. By Proposition 3.5, this element lies in the image of the level t action if and only if it fixes every integer modulo t, which happens if and only if v fixes every element of S(λ) modulo t. Since the level t action ofS s is faithful, different permutations of S(λ) correspond to different elements ofS s , so the size of the stabiliser is just the number of permutations of S(λ) that fix every element modulo t. Now we show how to interpret this formula geometrically. We work in Euclidean space R s , with coordinates labelled using the set Z/sZ. Following [F1] we define the affine subspace Given an s-core λ, define a point x λ ∈ P s by defining (x λ ) i to be the unique element of S(λ) ∩ i for each i ∈ Z/sZ. The one-to-one correspondence between s-cores and s-sets then gives Note that this set is a lattice (or rather, an affine lattice), which we denote Λ s . This lattice was introduced (with different conventions) by Johnson [J], who calls Λ s the lattice of s-cores.
Johnson's construction is central to his proof of Armstrong's conjecture via Ehrhart theory; indeed, Johnson makes the legitimate claim that his paper 'establishes lattice point geometry as a foundation for the study of simultaneous core partitions'. Note that this construction is different from that in [FV,F1], where an s-core λ is represented a point p λ in the dominant region of P s ; this construction yields a bijection between C s and the set of dominant alcoves in P s , but (if s 2) does not yield a lattice.
The advantage of Johnson's construction is the easy identification of the set of (s, t)-cores as the set of points of Λ s lying inside a certain simplex. Define a hyperplane H i in P s for each i ∈ Z/sZ by Let SC s (t) denote the simplex bounded by the hyperplanes H i ; that is, Then we have the following.

Lemma 3.8 [J, Lemma 3.1].
Suppose λ is an s-core. Then λ is also a t-core if and only if x λ ∈ SC s (t).
Example. Suppose s = 3 and t = 4. We illustrate part of the lattice of 3-cores in Figure 1, where we label each point x of Λ s by its coordinates x 0 , x 1 , x 2 and also by the corresponding 3-core. The three lines drawn are the hyperplanes H 0 , H 1 , H 2 , and the triangle bounded by these three lines is SC 3 (4). The 3-cores corresponding to points of Λ 3 inside this triangle are precisely the (3, 4)-cores.
The lattice of s-cores is also relevant to our study of the level t action ofS s on s-cores. For j ∈ Z/sZ let r j : P s → P s denote the reflection (with respect to the usual inner product on R s ) in the hyperplane H j . Then, as is well known in the theory of reflection groups, the group W s := r j j ∈ Z/sZ is isomorphic toS s , and an isomorphism θ :S s → W s may be given by mapping for each i ∈ Z/sZ. Moreover, this isomorphism connects the level t action ofS s on C s to the action of W s on the lattice Λ s , via the following lemma.
Lemma 3.9. If λ ∈ C s and w ∈S s then Proof. In the case where w = w i for i ∈ Z/sZ, this follows directly from the formula for a reflection in R s and the definition of the level t action on C s . The case for arbitrary w then follows from the fact that θ is a homomorphism.
With this geometric interpretation of the level t action, we can realise the stabiliser Stab s,t (λ) geometrically. First we show that Stab s,t (λ) is a parabolic subgroup ofS s . Lemma 3.10. Suppose λ ∈ C s ∩ C t , and let I be the set of i ∈ Z/sZ such that Proof. Recall from the proof of Proposition 3.7 the correspondence between Stab s,t (λ) and the group of permutations of S(λ) that fix every element modulo t. It follows from the proof of [F1,Proposition 4.1] that since λ is an (s, t)-core the elements of S(λ) lying in a given congruence class modulo t form an arithmetic progression with common difference t, say a, a + t, . . . , a + rt. The group of permutations of these integers is generated by the transpositions (a + (k − 1)t, a + kt) So Stab s,t (λ) is generated by those w i for which S(λ) contains integers m, m − t with m ∈ it − s•t. This is exactly the condition that x λ ∈ H it−s•t .
From Lemmas 3.9 and 3.10 we deduce the following, which enables us to calculate |Stab s,t (λ)| purely geometrically.
Corollary 3.11. If λ is an (s, t)-core, then Stab s,t (λ) is isomorphic to the group generated by r j x λ ∈ H j .
Example. Continuing from the last example, we see that x ∅ does not lie on any of the hyperplanes H 0 , H 1 , H 2 , so Stab 3,4 (∅) is trivial. The 3-cores (1), (2) and (1 2 ) each lie on only one of the three hyperplanes, so the stabiliser of each of these 3-cores has order 2. (3, 1 2 ) lies on H 1 and H 2 , so its stabiliser is isomorphic to the group generated by r 0 and r 1 , which has order 6.
So the weighted average in Conjecture 3.1 is and Conjecture 3.1 is verified in the case (s, t) = (3, 4).

Actions on finite sets of cores
In this section we define a family of finite sets of s-cores on whichS s acts. This will enable us to give rigorous meaning to our interpretation of Conjecture 3.1 in terms of random cores.
Choose N ∈ N, and let C (N) s denote the set of all s-cores λ such that k − l < Ns for all k, l ∈ S(λ). We begin by enumerating these cores.
Proof. Choosing an element of C (N) s amounts to choosing an s-set whose elements differ by less than Ns. Define a shifted s-set to be a set of s integers with exactly one in each congruence class modulo s, and with smallest element 0. Given an s-set X, there is a unique shifted s-set arising as a translation of X, and this shifted s-set will be contained in the interval [0, Ns − 1] if and only if the elements of X differ by less than Ns. Conversely, given a shifted s-set Y, we have x∈Y x ≡ s 2 (mod s), so there is a unique s-set arising as a translation of Y, i.e. the translation by 1 s s 2 − x∈Y x . So it suffices to count the shifted s-sets contained in [0, Ns − 1], and clearly there are N s−1 of these: for each 1 i s − 1, we choose exactly one of the integers i, i + s, i + 2s, . . . , i + (N − 1)s to be in the set.
To define an action ofS s on C (N) s , we will show that C (N) s is a transversal of the equivalence classes for an equivalence relation on C s which is fixed by the action ofS s on C s . Given λ, µ ∈ C s , set λ ≡ N µ if there is a bijection φ : S(λ) → S(µ) such that φ(k) ≡ k (mod Ns) for all k ∈ S(λ). Then obviously ≡ N is an equivalence relation on C s , and we have the following two results. with λ ≡ N µ; then we must show that λ = µ. Let φ : S(λ) → S(µ) be the bijection such that φ(k) ≡ k (mod Ns) for all k. Since S(λ) lies within an interval of length Ns and so does S(µ), the only possibility is that for every k, l ∈ S(λ) with k > l, But this gives So a = bs, and therefore a = b = 0. So S(λ) = S(µ), and hence λ = µ.
Proposition 3.14. Suppose N ∈ N. The equivalence relation ≡ N on C s is preserved by the level t action ofS s .
So we can define a bijection ψ : S(w i λ) → S(w i µ) by s . For the rest of this section we specialise to the case where N is divisible by t, and we write nt instead of N. Our aim is to connect the level t action on C (nt) s to Conjecture 3.1 by showing that each orbit contains a unique (s, t)-core, and that the size of the orbit containing an s-core λ is inversely proportional to |Stab s,t (λ)|.
The first step is to compute the kernel of the action. For the next proposition we must exclude some cases where nt is very small. Proof. Take w ∈S s , and suppose first that w(m) ≡ m (mod s) for all m. We claim that for any λ ∈ C (nt) s we haveŵλ = λ if and only if w ∈ K (n) . By Lemma 3.4 we haveẘ(m) ≡ m (mod s) for all m, so the unique bijection φ : S(λ) → S(wλ) satisfying φ(x) ≡ x (mod s) for all x ∈ X is just the restriction ofẘ to S(λ). If w ∈ K (n) , then by Lemma 3.4ẘ(m) ≡ m (mod nst) for all m, so φ(x) ≡ x (mod nst) for all x ∈ S(λ). So λ ≡ ntw λ, and henceŵλ = λ. On the other hand, if w K (n) , choose an integer m such that w(m) m (mod ns), and let x be the element of S(λ) congruent to mt − s•t modulo s; then we haveẘ(x) x (mod nst), so φ(x) x (mod nst), and hencewλ nt λ, i.e.ŵλ λ. So our claim holds, and in particular w lies in the kernel of the level t action on C (nt) s if and only if w ∈ K (n) . Now suppose instead that there is an integer m such that w(m) m (mod s); letting x = mt − s•t, we haveẘ(x) x (mod s). Let y =ẘ(x) + s; then obviously we have y ≡ẘx (mod s) but (by the assumption that nt > 1) y ẘx (mod nst). If s 3, then since y x (mod s) there is an s-set X containing both x and y; the unique bijection φ : X →ẘ(X) satisfying φ(z) ≡ z (mod s) for all z ∈ X must map y toẘ(x), and in particular φ(y) y (mod nst). So there is no bijection from X toẘ(X) fixing every element modulo nst. So if µ is the s-core with s-set X, then µ ntw µ. Hence if λ is the unique element of C (t) s with λ ≡ nt µ, then λ ntw λ, and hence λ ŵλ So w is not in the kernel of the level t action ofS s on C (nt) s . It remains to consider the case s = 2. Taking x as above, consider y = 1 − x; then {x, y} is a 2-set, so ifẘ(x) 1 − x (mod 2nt), then we can repeat the argument from the paragraph above. So supposeẘ(x) ≡ 1 − x (mod 2nt); repeating the argument with x + 2 in place of x, we can also assume thatẘ(x + 2) ≡ −1 − x (mod 2nt). But now which gives 2nt 4, contradicting the assumptions on n.
Remark. It is easy to fill in the exceptional cases ruled out by the assumptions in Proposition 3.15. If n = t = 1, then C (nt) s = {∅}, so the action ofS s is trivial and the kernel of this action is the whole ofS s . The remaining case is where (s, t, n) = (2, 1, 2), so that C (nt) s = {∅, (1)}. These two partitions are interchanged byŵ 0 and fixed byŵ 1 . So the kernel of the level 1 action is the normal subgroup ofS 2 generated by w 1 , and one can check that this equals w ∈S 2 w(0) ≡ 0 or 1 (mod 4) . Proof. Suppose ξ is an (s, t)-core, and write the elements of S(ξ) in increasing order as x 1 , . . . , x s . Then [F1,Propositions 4.2,4.3] implies that x i+1 − x i t for each i, and this implies that ξ ∈ C (t) s , and hence ξ ∈ C (nt) s . For the second part of the lemma, let O be the orbit containing λ. Then by [F1,Propositions 4.2,4.3] there is w ∈S s such that µ :=wλ is the t-core of λ, and in particular µ is an (s, t)-core. Since by the first part of the lemma µ lies in C (nt) s , we have µ =ŵλ ∈ O, so O contains an (s, t)-core. For uniqueness, suppose O contains another (s, t)-core ν. Then ν =vµ for some v ∈S s , so ν ≡ ntv µ. The definitions ofv and the relation ≡ nt now imply that there is a bijection φ : S(µ) → S(ν) such that φ(x) ≡ x (mod t) for each x; so by [F1,Proposition 4.1] µ and ν have the same t-core. Since µ and ν are t-cores, this means that µ = ν. Now we look at orbit sizes. Recall that Stab s,t (λ) denotes the stabiliser of an s-core λ under the level t action ofS s on C s . If in addition w ∈ Stab s,t (λ), then we haveẘ(S(λ)) = S(λ). But the elements of S(λ) are pairwise incongruent modulo s, so in fact we must haveẘ(x) = x for every x ∈ S(λ). Since S(λ) contains one element in each congruence class modulo s, this gives w(m) = m for every integer m, so w = 1.

Proof. Clearly both K (n) and Stab s,t (λ) lie inside Stab (n)
s,t (λ), so K (n) Stab s,t (λ) Stab (n) s,t (λ). Conversely, suppose w ∈ Stab (n) s,t (λ). Then by definitionwλ ≡ nt λ; let φ denote the bijection S(λ) → S(wλ) such that φ(x) ≡ x (mod nst) for all x. Then there is y ∈S s such that φ is just the restriction to S(λ) of y. Moreover, we have y(m) ≡ m (mod t) for all m ∈ Z, so by Proposition 3.5 there is v ∈S s such that y =v. Following the construction of v given in the proof of Proposition 3.5, we see that since y(m) ≡ m (mod nst) for every m, we have v(m) ≡ m (mod ns) for every m; that is, v ∈ K (n) .
The last two results show that if λ ∈ C (nt) s then the size of the orbit containing λ is inversely proportional to |Stab s,t (λ)|. In fact, we can be more precise, given the following lemma. In the case n = 1, this lemma is very well known in Lie theory; it arises from the fact that the affine symmetric group is the semidirect product of the finite symmetric group with its root lattice.
To go from the case n = 1 to the general case, we just need to show that |K (1) : K (n) | = n s−1 . But K (1) is a free abelian group of rank s − 1, and K (n) consists of the nth powers of the elements in this group, which gives the result.
This yields the following result giving the sizes of level t orbits in C Proof. The cases where n = t = 1 or (s, t, n) = (2, 1, 2) are easy to deal with, so we assume that nt > 1 and nst > 4, which enables us to use Proposition 3.15. LetS , by Lemma 3.18. Hence the order of this stabiliser is so by the Orbit-Stabiliser Theorem the size of the orbit containing λ is This result enables us to make precise our informal motivation from Section 3.2 concerning random s-cores. We now haveS s acting on a finite set C (nt) s , and we can select an s-core uniformly randomly from this set. By Lemma 3.16 each orbit contains a unique (s, t)-core, which is the common t-core of all the partitions in this orbit. Hence if λ ∈ C s ∩ C t then the probability of choosing an s-core whose t-core is λ is proportional to the size of the orbit containing λ, which in turn is inversely proportional to |Stab s,t (λ)|. So the left-hand side of Conjecture 3.1 gives the expected size of the t-core of λ.

The denominator
Another consequence of the results in Section 3.6 is a formula for the denominator appearing in Conjecture 3.1.

Proposition 3.21.
Proof. We specialise the results of Section 3.6 to the case n = 1. By Lemma 3.12, |C (t) s | = t s−1 , and this is the sum of the sizes of the orbits ofS s on C (t) s . Each of these orbits contains a unique (s, t)-core, so we just sum the result of Corollary 3.20 over all (s, t)-cores λ. We obtain which gives the result.

A weighted version of Armstrong's conjecture for self-conjugate (s, t)-cores
Now we consider analogues of the results and conjectures in the previous section for selfconjugate cores. The structure of this section is largely the same as in Section 3, though we will be able to be briefer by using some results from that section.
Throughout this section let D s denote the set of all self-conjugate s-cores. Recall that we define u = ⌊s/2⌋.

The affine hyperoctahedral group
We begin by defining a subgroup ofS s that fixes D s , and which will take the place ofS s in this section.
. ThenH s is isomorphic to the affine hyperoctahedral group of degree u, i.e. the affine Coxeter group of typeC u .

Example. If s = 4, thenH s is generated by
In either case, we have and in fact these are defining relations forH s .
It will be helpful to describeH s explicitly in terms of permutations. We will prove by induction on M(w) that w ∈H s . In the case M(w) = 0, we have w(i) = i for i = 0, . . . , u − 1; the fact that w is s-periodic and w(−1 − m) = −1 − w(m) for all m ∈ Z then means that w(m) = m for all m ∈ Z, so w is the identity permutation, which lies inH s . For the inductive step, assume M(w) > 0 and suppose first that there is a ∈ {1, . . . , u − 1} for which w(a) < w(a − 1). Let w ′ = wv a ; then for i ∈ {0, . . . , u − 1} we have . w ′ lies in H, so by induction w ′ lies inH s , and hence so does w. So we may assume that w(0) < w(1) < · · · < w(u − 1). Since M(w) > 0, this means in particular that either w(0) < 0 or w(u − 1) > u − 1. In the first case, let w ′ = wv 0 ; then we have , and again we can use the induction hypothesis.
Finally suppose that we are in the case where w(u − 1) > u − 1. Note that if s is odd then in fact w(u − 1) u + 1; this is because when s is odd the conditions on w give w(u) = u, so w(u − 1) cannot equal u. Whether s is even or odd, we let w ′ = wv u . Now we find that w ′ which in either case is negative, so again we can apply the inductive hypothesis.
By restricting the level t action ofS s on Z, we obtain a level t action ofH s on Z. As withS s , we can describe the image of this action explicitly.

Proposition 4.2. The image of the level t action ofH s on
Proof. LetH s,t denote the image ofH s under the level t action ofH s on Z, and let H denote the given subgroup ofH s . Recall thatv denotes the image of v ∈H s under the level t action on Z. From Proposition 3.5 we know thatv i ∈S s andv i (m) ≡ m (mod t) for all i ∈ Z/sZ and m ∈ Z.

It is easy to check that in additionv
For the converse, we follow the proof of Proposition 3.5. Suppose we are given x ∈ H; for . Then (from the proof of Proposition 3.5) w ∈S s and x =ẘ. Moreover, one can easily check that since x ∈H s we have w ∈H s too. So x lies inH s,t . Now we consider the action ofH s on s-cores. We begin with the following lemma; note that this is where we really require the term −s•t in the definition of the level t action ofS s on Z.

Lemma 4.3. Suppose λ is an s-core and i ∈ Z/sZ. Under the level t action ofS s on C s we have
Proof. It is well-known and easy to prove that for any partition λ. The result follows from the definition of the level t action ofS s via beta-sets.
Corollary 4.4. If λ ∈ D s and w ∈H s , then under the level t action ofS s on C s we havewλ ∈ D s .
In fact, it is not hard to show that (for any value of t)H s is the setwise stabiliser of D s under the level t action ofS s on C s . So we have a level t action ofH s on D s , which we may also denote w →w. Given λ ∈ D s , let StabSC s,t (λ) denote the stabiliser of λ under this action. Now we can state our main conjecture for self-conjugate cores.
The rest of this section follows the structure of Section 3: we begin by giving a formula for |StabSC s,t (λ)|, and examining the connection to Johnson's lattice of s-cores. We then consider actions ofH s on finite sets of self-conjugate s-cores, which will enable us to phrase Conjecture 4.5 in terms of random self-conjugate s-cores, and to give an explicit formula for the denominator in the weighted average in Conjecture 4.5.
We begin by showing that, as in the non-self-conjugate case, the level t orbit containing a self-conjugate s-core λ is determined by the t-core of λ. First we make a definition: say that an s-set X is symmetric if s − 1 − x ∈ X for every x ∈ X.
Proof. The relationship between B λ and B λ ′ given in the proof of Lemma 4.3 yields S(λ ′ ) = { s − 1 − x | x ∈ S(λ)} for any λ ∈ C s . The result follows.
Proposition 4.7. If λ ∈ D s , then the t-core of λ lies in the same orbit as λ under the level t action ofH s on D s .

Proof.
We follow the last part of the proof of [F1,Proposition 4.3]. Let O be the orbit containing λ, and let ν be a partition in this orbit for which the sum k∈S(ν) k 2 is minimised. If we can show that ν is a t-core, then ν must be the t-core of λ (since the level t action ofS s on C s preserves the t-core of an s-core).
Suppose for a contradiction that ν is not a t-core. For each i ∈ Z/sZ let k i be the unique element of S(ν) ∩ i. By Lemma 3.8, there must be some j ∈ Z/sZ such that k j > k j−t + t. By Lemma 4.6 S(ν) is symmetric, so we also have k t−j−1 > k −j−1 + t. Let i ∈ Z/sZ be such that j = it − s•t, and consider several cases.
• Suppose i = 0 or s is even and i = u. Then w i = v i ∈H s , sow i ν lies in O. Applyingẘ i to S(ν) amounts to replacing k j and k j−t with k j − t and k j−t + t. But then contradicting the choice of ν.
• Suppose i = l or −l, where 1 l u − 1, and considerv i ν =w iw−i ν. The conditions on l mean that j, j − t, t − j − 1, − j − 1 are distinct, so applyingv i to S(ν) amounts to replacing As in the previous case we get k∈S(v i ν) k 2 < k∈S(ν) k 2 , a contradiction. • Suppose s is odd and i = u. Now considerv i ν =w uw−uwu ν. We now have j = u, so that j = − j − 1. Hence applyingv i to S(ν) amounts to replacing k j+t and k j−t with k j+t − 2t and k j−t + 2t. As in the previous cases we reach a contradiction.
• Finally suppose s is odd and i = −u. As in the previous case, we can applyv i and reach a contradiction.
Hence we get the following analogue of Proposition 3.3. Proof. If λ and µ lie in the same level t orbit ofH s , then they lie in the same level t orbit ofS s and so have the same t-core by Proposition 3.3. The converse follows from Proposition 4.7.

s-sets and stabilisers
Next we show how to compute |StabSC s,t (λ)| from the s-set for λ, when λ is a self-conjugate s-core. The method here is the same as in Proposition 3.7, but the statement is more complicated. Given j ∈ Z/tZ, let S(λ) j = S(λ) ∩ j. Then any good permutation must permute S(λ) j . If j s − 1 − j, then any permutation of S(λ) j can occur as the restriction of a good permutation v, and the condition that v(s − 1 − i) = s − 1 − v(i) for all i then uniquely determines the restriction of v to S(λ) s−1−j . On the other hand, if j = s − 1 − j, then the restriction of a good permutation v to S(λ) j must itself satisfy v(s − 1 − i) = s − 1 − v(i) for all i; the number of permutations of S(λ) j achieving this is 2 y y!, where y = ⌊ 1 2 |S(λ) j |⌋. Combining these observations with an analysis of when j equals s − 1 − j (which depends on the parities of s and t) yields the formulae in the proposition. Now as in Section 3.5 we connect this result to the lattice of s-cores; in the interests of brevity, we omit some of the details here. Recall the affine space and the lattice of s-cores By Lemma 4.6, an s-core λ is self-conjugate if and only if x λ ∈ Q s . So the set { x λ | λ ∈ D s } is the lattice Λ s ∩ Q s , which we call the lattice of self-conjugate s-cores.
As with Λ s , we can identity the set { x λ | λ ∈ D s ∩ D t } geometrically. Recall that H j denotes the hyperplane in P s defined by the equation x j − x j−t = t, and that SC s (t) is the simplex bounded by these hyperplanes. Define J j := H j ∩ Q s and SD s (t) := SC s (t) ∩ Q s . Then SD s (t) is bounded by the hyperplanes J j in Q s , and it is immediate from Lemma 3.8 that if λ ∈ D s , then λ is a t-core if and only if x λ lies in SD s (t).
Note that J j = J t−1−j for each j, so there are only u + 1 distinct hyperplanes J j . Since Q s is a u-dimensional space, this means that SD s (t) is a simplex in Q s .

Example.
Consider the case (s, t) = (4, 5). In Figure 4 we illustrate part of the lattice of selfconjugate 4-cores, labelling each point with its coordinates x 0 , x 1 , x 2 , x 3 and with the corresponding 4-core. The lines drawn are the hyperplanes J 0 , J 1 = J 3 and J 2 . The triangle bounded by these lines is SD 4 (5), and the points of Λ s it contains are precisely the points x λ for λ a self-conjugate (4, 5)-core. We illustrate the case (s, t) = (5, 4) similarly in Figure 5.

Actions on finite sets of self-conjugate cores
Now we consider actions on finite sets of self-conjugate s-cores, following the approach in Section 3.6. Given N ∈ N, let D (N) s = C (N) s ∩ D s ; that is, the set of self-conjugate s-cores λ such that k − l < Ns for all k, l ∈ S(λ).
As with C amounts to choosing a symmetric s-set whose elements differ by less than Ns. We define a shifted doubled symmetric s-set to be a set of s integers all of the same parity and pairwise incongruent modulo 2s, which is fixed by the map x → −x.
There is an obvious bijection from symmetric s-sets to shifted doubled symmetric s-sets, which sends an s-set X to { 2x − s + 1 | x ∈ X}. Moreover, the elements of X differ by less than Ns if and only if the elements of the corresponding shifted doubled symmetric s-set differ by less than 2Ns. So it suffices to count the shifted doubled symmetric s-sets contained in [1 − Ns, Ns − 1]. So suppose Y is such a set.
Suppose first that s is odd. Then |Y| is odd, so Y must contain 0. Furthermore, for each i ∈ {2, 4, . . . , s − 1} Y must contain exactly one integer in [1 − Ns, Ns − 1] congruent to i modulo 2s, and must also contain the negative of this integer. So there are N (s−1)/2 possibilities for Y. Now suppose s is even. Then the elements of Y must be odd: if the elements of Y are even, then one of them, say y, is divisible by s; but then y and −y are congruent modulo 2s and both lie in Y, which is a contradiction unless y = 0, but this would imply that |Y| is odd, also a contradiction. Now we can see that there are N s/2 possibilities for Y: for each i ∈ {1, 3, . . . , s − 1}, Y must contain exactly one of the N integers in [1 − Ns, Ns − 1] congruent to i modulo 2s, and must also contain the negative of this integer.
Next we want to show that the level t action ofS s on C (N) s restricts to an action ofH s on D (N) s . To do this, recall the equivalence relation ≡ N from Section 3.6. We have the following analogue of Proposition 3.13.
Conversely, suppose w lies in the kernel of the level 1 action on D Note that the reductions modulo 2s of the two s-sets S x − and S x + have only i and s − 1 − i in common; since w fixes both S x − and S x + modulo 2s, we must therefore have w(i) ≡ i