Simultaneous core partitions: parameterizations and sums

Fix coprime $s,t\ge1$. We re-prove, without Ehrhart reciprocity, a conjecture of Armstrong (recently verified by Johnson) that the finitely many simultaneous $(s,t)$-cores have average size $\frac{1}{24}(s-1)(t-1)(s+t+1)$, and that the subset of self-conjugate cores has the same average (first shown by Chen--Huang--Wang). We similarly prove a recent conjecture of Fayers that the average weighted by an inverse stabilizer---giving the"expected size of the $t$-core of a random $s$-core"---is $\frac{1}{24}(s-1)(t^2-1)$. We also prove Fayers' conjecture that the analogous self-conjugate average is the same if $t$ is odd, but instead $\frac{1}{24}(s-1)(t^2+2)$ if $t$ is even. In principle, our explicit methods---or implicit variants thereof---extend to averages of arbitrary powers. The main new observation is that the stabilizers appearing in Fayers' conjectures have simple formulas in Johnson's $z$-coordinates parameterization of $(s,t)$-cores. We also observe that the $z$-coordinates extend to parameterize general $t$-cores. As an example application with $t := s+d$, we count the number of $(s,s+d,s+2d)$-cores for coprime $s,d\ge1$, verifying a recent conjecture of Amdeberhan and Leven.

A partition λ has, associated to each square (r, c) ∈ [λ], a rim hook {(i, j) ∈ [λ] : i ≥ r, j ≥ c, and (i + 1, j + 1) / ∈ [λ]} of (positive) rim hook length 1 + (λ r − r) + (λ c − c)-the same as the hook length of the usual hook {(i, c) ∈ When s is a positive integer, we say a partition is an s-core if it has no hooks of length s, or equivalently no rim s-hooks (rim hooks of length s); following Fayers [15], we denote by C s the set of s-cores, and by D s ⊆ C s the set of self-conjugate s-cores. More generally, any partition λ has a unique s-core λ s ∈ C s , given by repeatedly removing rim s-hooks. To prove that this s-core operation λ → λ s is well-defined, one can use the beta-sets reviewed in Section 2.1, which also show that λ is an s-core if and only if it has no hook lengths divisible by s, unifying two common definitions of C s . These notions are connected to representation theory, symmetric function theory, and number theory (see e.g. [13,15,19]).
Going further, many authors (see e.g. [1,2,3,4,5,6,7,11,13,14,15,16,17,21,24,25,27,28,29]) have recently considered the interaction of s-cores and t-cores (both the partitions and operations), for two positive integers s, t. For example, Anderson [5] showed that for coprime s, t ≥ 1, the set C s ∩ C t of (simultaneous) (s, t)-cores has size equal to the number of (s, t)-Dyck paths, which Bizley [8] had earlier enumerated-via 'cyclic shifts'as the 'rational Catalan number' 1 s+t s+t t . 1 Ford, Mai, and Sze [17] later showed that for coprime s, t ≥ 1, the set D s ∩ D t of self-conjugate (s, t)-cores has size equal to the lattice path count s/2 + t/2 t/2 . In a different direction, Olsson [24] showed that the t-core of an s-core is an s-core, hence a simultaneous (s, t)-core (as it is a t-core by definition).
In this paper, we mainly focus on related conjectures of Armstrong from [6], and Fayers from [15], on certain weighted average sizes of (s, t)-cores when s, t are coprime. Chen, Huang, and Wang [11] established Armstrong's self-conjugate conjecture (Theorem 1.2 below) using the Ford-Mai-Sze bijection [17]. Using a poset formulation of Anderson's bijection [5], Stanley and Zanello [27] recursively established Armstrong's general conjecture (Theorem 1.1 below) for the 'Catalan case t = s + 1'; Aggarwal [1] generalized their method to the case t ≡ 1 (mod s). However, it is unclear whether a similar 'Catalan-like' recursive structure holds for other choices of s, t. Recently, by different means described below, Johnson [21] fully proved Theorem 1.1, and re-proved Theorem 1.2.  where the sums run over all self-conjugate (s, t)-core partitions.
Developing Olsson's [24] and his own [13] ideas, Fayers soon after conjectured weighted analogs (Theorems 1.3 and 1.4 below) of Armstrong's conjectures-in some sense giving the "expected size of the t-core of a random s-core" [15]. In this paper, we carry over explicit versions of Johnson's methods to establish both of Fayers' conjectures, despite the absence of an obvious 'exponential' analog of Ehrhart reciprocity. We also briefly explain, in Remark 8.3, how one could give more implicit or "conceptual" proofs if necessary. Theorem 1.3 (Fayers [15,Conjecture 3.1]). Fix coprime s, t ≥ 1. Then λ∈Cs∩Ct |Stab Gs,t (λ)| −1 · |λ| λ∈Cs∩Ct |Stab Gs,t (λ)| −1 · 1 where the sums run over all (s, t)-core partitions, and the stabilizers are defined in terms of Fayers' 'level t' group action on C s [13,15] reviewed in Definition 3.7.
Theorem 1.4 (Fayers [15,Conjecture 4.5]). Fix coprime s, t ≥ 1. Then λ∈Ds∩Dt |Stab Hs,t (λ)| −1 · |λ| λ∈Ds∩Dt |Stab Hs,t (λ)| −1 · 1 , where the sums run over all self-conjugate (s, t)-core partitions, and the stabilizers are defined in terms of Fayers' 'level t' group action on D s [15] reviewed in Definition 6.6. Remark 1.5. As Fayers notes in [15], the orbits of G s,t and H s,t are infinite, so one weights by the inverses of the finite stabilizers instead. However, by considering finite quotients acting on certain finite subsets of C s and D s , he also gives the weighted averages finite probabilistic interpretations that agree exactly, not just asymptotically, with the original averages.
Johnson's z-coordinates parameterization of (s, t)-cores, and our modest extension to general t-cores (see Proposition 4.2 for general cores and in Proposition 6.11 for the self-conjugate specialization)-which depends on a choice of s ≥ 1 coprime to t-plays a key role in our paper, which rests upon his cyclic shifts argument for general cores. The key argument (reviewed in Proposition 4.7) works because the size function for t-cores is cyclic (invariant under rotation of coordinates), and the z-coordinates are (not invariant, but still) well-behaved under rotation. However, whereas Johnson finishes by 'weighted Ehrhart reciprocity' (see [9] for an introduction to 'un-weighted' Ehrhart theory), we will stick to flexible direct toolsand implicit variants thereof-which can in principle evaluate sums of arbitrary powers of the partition sizes (see Remark 8.6).
The main new observation is that the stabilizer sizes appearing in Fayers' conjectures (Theorems 1.3 and 1.4) have symmetric (or almost symmetric) formulas in the z-coordinates, which simply re-index the restricted counts |S s (λ) ∩ (j + tZ)| of elements in Fayers' s-sets S s (λ) (from [13,15]). In fact, we first prove (in Propositions 3.4 and 3.6) that the sets S s (λ)∩(j +tZ) underlie the tools allowing us, in Fayers' words [13], to "[compare] the t-cores of different s-cores" in the first place-thus illustrating the significance of z-coordinates.
As an application of the cyclic shifts in the (extended) z-coordinates, we also parameterize and then enumerate the simultaneous (m, m+d, m+2d)-cores for coprime m, d ≥ 1, verifying the following recent conjecture of Amdeberhan and Leven [4] in two different ways.
Remark 1.8. A few days after the arXiv postings of v2 of the present paper and v1 of [2], Paul Johnson informed us (via private correspondence) that he had independently found our asymmetric proof (in Section 7) of Theorem 1.6. In fact, he proved the slightly stronger result that the term 1 m+d m+d i,i+d,m−2i counts the number of (m, m + d, m + 2d)-cores with exactly i hooks of length d; see the end of Remark 3.5 for a brief explanation.
1.2. Outline of paper. In Section 2, we review the relevant definitions, terminology, and basic results about s-core partitions and the s-core operation on partitions, mostly from Fayers [13,15] and Johnson [21]. Section 3 provides the fundamental results on t-cores of s-cores, giving the background needed to state Theorems 1.3 and 1.4 (Fayers' conjectures). We isolate our main new observation as Proposition 3.4, which first gives a cleaner proof of a key proposition from [13], and later features in our stabilizer computations. Section 4 describes the relevant computations for Section 5 (on general cores): we compute the sizes of the stabilizers appearing in Theorem 1.3 and express s-set data, stabilizer sizes, and partition sizes in Johnson's z-coordinates. We also extend the z-coordinates, and review the cyclic shifts used to compute z-coordinate sums of cyclic functions, such as the stabilizer and partition sizes.
Section 5 presents the main results of the paper, namely explicit proofs of Theorems 1.1 and 1.3 (general conjectures). Section 6 presents self-conjugate analogs of the general analysis, building up to explicit proofs of Theorems 1.2 and 1.4 (self-conjugate conjectures).
Section 7 uses cyclic shifts in the extended z-coordinates from Section 4.2 to quickly prove Theorem 1.6. Section 8 discusses possibilities for future work, including the question of calculating sums of higher powers (or moments) of the (s, t)-core partition sizes.

Background: s-core partitions and operation
Experts can quickly skim this section for the notation used in our paper (as the literature seems to have many different conventions), particularly the framework of beta-sets (Section 2.1) for studying hooks, and two parameterizations of s-cores: Johnson's c-coordinates (Section 2.2) from [21], and Fayers' s-sets and a-coordinates (Section 2.4) from [13,15].
Recall the following definition from the introduction.
Side Definition 2.1. Fix s ≥ 1. As in [15], let C s denote the set of s-cores λ, i.e. partitions with no rim s-hooks (or equivalently, no traditional hooks of length s). For any s, t ≥ 1, we often call C s ∩ C t the set of (simultaneous) (s, t)-cores.
Side Definition 2.2. For an arbitrary partition λ, say an s-core partition µ is an s-core of λ if it can be obtained (starting from λ itself) by repeatedly removing rim s-hooks. (Clearly λ is an s-core if and only if λ is an s-core of λ itself.) · · · Figure 2. Russian notation for λ = (3, 2, 2, 0, . . .), from Johnson's arXiv source [21]. (As pointed out by an anonymous expert, the idea itself has been used earlier by Okounkov and Reshetikhin.) The infinite rim boundary is thickened. A rim edge has a filled circle below if and only if the edge slopes upwards. By convention, our labeling increases from right to left (opposite the usual Cartesian x-axis).
Side Remark 2.3. As remarked in the introduction, there is in fact a unique such s-core µ (of λ) by Proposition 2.10, which we denote by λ s . So λ is an s-core if and only if λ = λ s .

Beta-sets.
To each partition one naturally associates a beta-set illuminating the hook length structure. It is often easier to work with beta-sets than with partitions themselves.
Proof. In Figure 2, a rim s-hook is parameterized by an upwards-sloping rim edge at position x + 1 2 (i.e. x ∈ B λ ) followed by a downwards-sloping rim edge at position Side Remark 2.4. The s-core criterion B λ − s ⊆ B λ is due to Robinson [26,2.8] (according to [15]). As mentioned in the introduction, it shows that λ ∈ C s if and only if λ has no hook of length divisible by s, establishing an alternative common definition of C s .
Side Remark 2.5. Fix coprime s, t ≥ 1. Then via beta-sets (more precisely, after negation and suitable translation), the numerical semigroups (subsets of Z ≥0 that contain 0, are closed under addition, and contain all sufficiently large integers) containing s, t inject into the set of (s, t)-cores-as mentioned in the introduction and the future work sections.  [19]; [12]). Call a set S ⊆ Z good if S ∩ Z >0 and Z <0 \ S are both finite. For good S, define the signed s-charge measure ] for any s ≥ 1 and i ∈ Z/sZ. Then i∈Z/sZ c s,i (S) = c 1,0 (S) is the total charge of S.
For any partition λ, we may define c s,i (λ) := c s,i (B λ ), since B λ is good.
Remark 2.5. We will not use the notions of electron, positron, and Maya diagram from Johnson's exposition.
The basic importance of charge is given by the following charge condition.   Proof. In Figure 2, a simple geometric argument shows that removing the rim s-hook parameterized by x ∈ B λ \ (B λ + s) simply swaps the slopes (up or down) of the rim edges at positions x + 1 2 and x − s + 1 2 , while preserving the slopes at all other positions. This corresponds to replacing x ∈ B λ by x − s, and leaving the other elements alone.
Furthermore, replacing x ∈ B λ \ (B λ + s) by x − s preserves the charge s-tuple (c s,i ) i∈Z/sZ ; by Definition 2.4, there is nothing to check when −1 − i ≡ x (mod s). One can then check that c s,−1−x stays constant.
Side Remark 2.6. In the infinite s-abacus terminology in the literature-see e.g. [13,21]the s-push is described as "pushing the bead at x by s to the unfilled position x − s"; or in the electron diagram (see Figure 2) of [21] as "inverting the filled energy state x + 1 2 with the unfilled energy state x − s + 1 2 " (the authors of [12] would likely also describe the operation this way). Abaci.]). The s-push of a good set S is the well-defined result of repeatedly applying the s-pushes defined in Proposition 2.8. Explicitly, the s-push of S can be described as follows: • Fix a residue class i + sZ; then S ∩ {i + sZ} takes the form {. . . , x − 2s, x − s, x, x + α 1 s, . . . , x + α k s} for any sufficiently small x ∈ S ∩ {i + sZ}.
Repeatedly applying Proposition 2.8 shows that the s-core operation is well-defined.
In particular, i∈Z/sZ c s,i = 0 (Proposition 2.6) is equivalent to i∈Z/sZ a s,i = s 2 . Proposition 2.11 (Fayers [13]; [15,Section 3.3]). Fix s ≥ 1. The s-cores are parameterized (uniquely) by s-sets S s = {a s,i } i∈Z/sZ of a-coordinates summing to s 2 with a s,i ≡ i (mod s) for all i ∈ Z. Explicitly, S s (λ) := (B λ + s) \ B λ for λ ∈ C s . Remark 2.12. In view of c s,i (λ) = c s,i (λ s ) from Proposition 2.10, it would be meaningful to define a s,i (λ) := a s,i (λ s ) for any λ. However, we will only speak of s-sets and a-coordinates of s-cores. Charge will suffice for our greater needs in Proposition 6.2 and Corollary 6.3.

Background: t-cores of s-cores
In this section, we go through the fundamental results on t-cores of s-cores. In particular, we isolate the crucial Proposition 3.4, meanwhile giving a cleaner proof of a key result of Fayers (see Proposition 3.6). This motivates Fayers' 'level t action on s-cores' (Definition 3.7), defining the stabilizers in Theorem 1.3 (Fayers' general conjecture).

3.1.
When is a t-core an s-core? The following simple criterion parameterizes C s ∩ C t in a t -coordinates. It implicitly appears throughout this paper and elsewhere. . In Fayers' a-coordinates (see Proposition 2.11), the set C s ∩ C t of s-cores within the affine lattice C t of t-cores is defined by the system of inequalities a t,i ≥ a t,i+s − s for i ∈ Z/tZ.
Proof. The proof is the same as the first half of the proof of Proposition 3.4. Suppose λ ∈ C t , i.e. the t-core criterion B λ − t ⊆ B λ holds (from Proposition 2.3). Then by definition of a t,i ≡ i (mod t) and a t,i+s ≡ i + s (mod t), we have for any i ∈ Z. Thus the s-core criterion B λ − s ⊆ B λ holds if and only if the inequalities a t,i+s − s ≤ a t,i hold for all i ∈ Z/tZ.

3.2.
Comparing t-cores of different s-cores. Propositions 3.2, 3.4, and 3.6 below are the key conceptual inputs for comparing t-cores of various s-cores. We start by extending Olsson's theorem [24], following Fayers [13]. . Fix any s, t ≥ 1 and λ ∈ C s . Then λ t ∈ C s ∩ C t , and furthermore S s (λ t ) ≡ S s (λ) (mod t) (viewed as multisets of residues).
Remark 3.3. One can avoid induction by first comparing (B λ −s)∩(j +tZ) with B λ ∩(j +tZ) as j ∈ Z varies, and then (in view of Proposition 2.10) the t-pushes (B λ t − s) ∩ (j + tZ) and Alternative proof avoiding induction. Suppose λ ∈ C s , i.e. the s-core criterion B λ − s ⊆ B λ holds (from Proposition 2.3), and fix an integer j. But B λ t is the t-push of B λ by Proposition 2.10, so the previous computations translate to Varying over all j shows that B λ t − s ⊆ B λ t (as k ≤ always), so the t-core λ t is indeed still an s-core, and furthermore, We isolate our main new observation as the following proposition, which first gives a cleaner proof of a key result from [13] (see Proposition 3.6), and later features in our stabilizer computations, namely Propositions 4.1 (general case) and 6.12 (self-conjugate analog).
Proposition 3.4. Fix any s, t ≥ 1 and λ ∈ C s . Then Proof. Proposition 3.2 says λ t ∈ C s and S s (λ) ≡ S s (λ t ) (mod t). Thus it suffices to prove the result with λ replaced by its t-core λ t .
In other words, we may without loss of generality assume λ ∈ C s ∩ C t . Recall the t-core criterion B λ − t ⊆ B λ from Proposition 2.3. Then by definition of a t,j ≡ j (mod t) and a t,j+s ≡ j + s (mod t), we have . Remark 3.5. Compare with both the statement and proof of Lemma 3.1, which can be rephrased in terms of the quantities ω : . When s, t are coprime, these re-index in Corollary 4.4 to form Johnson's z-coordinates for (s, t)-cores λ (as vaguely mentioned in the introduction).
Furthermore, for arbitrary t-cores λ, the quantity ω is meaningful not only when . The latter observation is essentially due to Paul Johnson (via private correspondence), and gives combinatorial significance to our modest extension of his z-coordinates (see the second halves of Propositions 4.2 and 6.11). Proposition 3.4 cleanly proves a result of Fayers "crucial" for "comparing the t-cores of different s-cores" [13]. . Fix coprime s, t ≥ 1. Then the t-core λ t of an s-core λ is uniquely determined by the multiset of modulo t residues S s (λ) (mod t). Combined with Proposition 3.2, we conclude that λ, µ ∈ C s have the same t-core if and only if the multisets of modulo t residues S s (λ), S s (µ) (mod t) are congruent.
Since s, t are coprime, it follows that S s (λ) (mod t) determines S t (λ t ) = {a t,j (λ t )} up to translation. The sum condition j∈Z/tZ a t,j (λ t ) = t 2 (from Proposition 2.11) then singles out a unique translate equal to S t (λ t ), which corresponds under Proposition 2.11 to a unique t-core λ t .

3.3.
Level t action and statement of Fayers' general conjecture. Proposition 3.6 motivates the following group action on C s , for which Corollary 3.11 will hold almost by definition. It is easy to check that G s,t has a group structure, and that it acts in the obvious way on the set of s-sets S s (λ) (of s-cores λ), or equivalently on beta-sets B λ (of s-cores λ). This induces an action on C s , via Proposition 2.11. Remark 3.8 (Different but equivalent definitions). We have not given Fayers' actual definition of the 'level t action of the s-affine symmetric group' (based on the t = 1 case from [23]), but rather one equivalent by [15,Proposition 3.5], and easier to work with for our purposes. In practice one often encounters group elements by their restrictions to s-sets. Proof. The uniqueness is clear: the s-periodicity condition uniquely determines f on the modulo s residue classes a s,i (λ) + sZ = i + sZ, hence on the whole set of integers. Explicitly, It is then easy to check that this (uniquely determined) f is actually a permutation of Z (it permutes the residue classes modulo s, because φ permutes S s (λ)) preserving residues modulo t (because φ preserves residues modulo t, we have f (m) = m + [φ(a s,m (λ)) − a s,m (λ)] ≡ m (mod t)) and satisfying the sum condition hence an element of G s,t (again, as defined in Definition 3.7). . Fix coprime s, t ≥ 1. Then λ, µ ∈ C s lie in the same G s,t -orbit if and only if λ t = µ t . In other words, each G s,t -orbit of C s contains a unique t-core (hence an (s, t)-core), and any λ ∈ C s lies in G s,t λ t .
Sketch of more direct proof. Fix s-cores λ, µ. Proposition 3.6 followed by Proposition 3.10 gives equivalence of λ t = µ t and µ ∈ G s,t λ. The re-phrasing follows by specializing to µ := λ t (where λ t ∈ C s follows from Proposition 3.2).
Full direct proof. Let λ, µ be two s-cores. Proposition 3.6 states that λ t = µ t if and only if we have a congruence of s-sets S s (λ) ≡ S s (µ) (mod t), i.e. there exists a bijection φ : S s (λ) → S s (µ) (of s-sets) preserving residues modulo t. By Proposition 3.10, such bijections φ correspond to group elements f ∈ G s,t with restriction f | Ss(λ) = φ. So λ t = µ t if and only if µ = f λ for some f ∈ G s,t , i.e. λ, µ lie in the same G s,t -orbit.
Definition 3.7 and Corollary 3.11 provide the background and context for Theorem 1.3 (stated in the introduction).

Key inputs for computation
In this section, we describe all the computational methods and results used to compute the sums in Theorems 1.1 and 1.3. First we compute the sizes of the stabilizers appearing in Theorem 1.3. Section 4.2 compares Fayers' a-coordinates with a modest extension of Johnson's z-coordinates. In Section 4.4 we give an explicit formula for the size of a t-core, and in Section 4.3 we explain the standard cyclic shifts used to compute z-coordinate sums of cyclic functions, such as the stabilizer and partition sizes.

4.1.
Size of the stabilizer of an s-core. Most of the proof ideas for Theorem 1.3 come from Johnson [21] and Fayers [13,15]. The key new observation is the following computational simplification of Fayers' formula for the size of the stabilizer of an s-core, based on Proposition 3.4, which will simplify even further once we translate to Johnson's z-coordinates (see Corollary 4.4).
To prove this, take f ∈ G s,t (as defined in Definition 3.7); then in particular, f preserves residue classes modulo t. By definition, f lies in the stabilizer Stab(λ) if and only if f fixes the s-set S s (λ) (of the s-core λ), i.e. f restricts to a permutation π on the elements of S s (λ) also preserving residues modulo t. Observe that • Any such permutation π uniquely extends to an element f ∈ G s,t . Indeed, this is just Proposition 3.10 applied to the bijection π : S s (λ) → S s (λ) (of s-sets). • Such permutations π of S s (λ) correspond to (disjoint) products of permutations of S s (λ) ∩ (j + tZ) (on the individual residue classes j + tZ). Substituting in Proposition 3.4 gives the result.
Side Remark 4.1. We will only explicitly use this result for (s, t)-cores λ ∈ C s ∩ C t , when s, t are coprime, in the proof of Fayers' general conjecture (Theorem 1.3). More precisely, we will use the z-coordinate translation given in Corollary 4.4.

4.2.
Johnson's z-coordinates versus Fayers' t-sets. As reflected by the simple translation Corollary 4.4, Proposition 4.1 and Remark 3.5 provide one source of motivation for the following choice of coordinates. We not only review Johnson's parameterization of (s, t)cores, but also extend it to arbitrary t-cores, given a parameter s ≥ 1 coprime to t. Fix coprime s, t ≥ 1. The set C s ∩ C t of (s, t)-cores (viewed as s-cores within the set of t-cores) is parameterized by either of the sets A t (s), TD t (s), described as follows.
• By Lemma 3.1, the set of t-sets S t (λ) of (s, t)-cores λ, i.e. the set of points A t (s) = {(a t,i ) i∈Z/tZ } defined by the inequalities a t,i ≥ a t,i+s −s, the sum condition i∈Z/tZ a t,i = t 2 , and congruence conditions a t,i ≡ i (mod t).
• Johnson's trivial determinant representations set TD t (s) = {(z t,i ) i∈Z/tZ } defined by the inequalities z t,i ≥ 0, the sum condition i∈Z/tZ z t,i = s, and congruence conditions z t,i ≡ 0 (mod 1) and i∈Z/tZ iz t,i ≡ 0 (mod t).
An isomorphism (also preserving the ambient linear and simplex structures) is given by the invertible affine change of variables z t,j : The inverse map can be described by a t,k+ s − t−1 Under the same affine change of variables, the larger set C t of t-cores is parameterized by either of the following sets.
• By Proposition 2.11, the set of t-sets S t (λ) of t-cores λ, i.e. the set of points C t = {(a t,i ) i∈Z/tZ } defined by the sum condition i∈Z/tZ a t,i = t 2 , and congruence conditions a t,i ≡ i (mod t).
• In z-coordinates, the set of points C t = {(z t,i ) i∈Z/tZ } defined by the sum condition i∈Z/tZ z t,i = s, and congruence conditions z t,i ≡ 0 (mod 1) and i∈Z/tZ iz t,i ≡ 0 (mod t).
Side Remark 4.2. In the z-coordinate parameterization of the larger set C t (as opposed to C s ∩ C t ), one may think of s as a "purely algebraic parameter" coprime to t, with applications to (for instance) the "symmetric proof" of Theorem 1.6 given in Section 7.
Remark 4.3. Although it will only matter for the asymmetric Theorem 1.3, not the symmetric Theorem 1.1, we use, in Johnson's notation, the parameterization TD t (s), instead of TD s (t) as Johnson might for Armstrong's conjectures [21].
Before proving the result, we first translate s-set and stabilizer data to z-coordinates.
where k is the constant 1 2 (s + 1)(t − 1). Furthermore, Proposition 4.1 translates to Side Remark 4.3. Propositions 4.1 and 3.4 are not the only way to motivate the z-coordinates-Johnson considers them in [21] for aesthetic and practical reasons (i.e. repeatedly "simplifying" the parameterization of (s, t)-cores)-but the propositions do give the coordinates additional significance.
Proof of a-versus-z isomorphism in Proposition 4.2. Let φ be the affine map sending a point (x i ) i∈Z/tZ on the (t−1)-dimensional plane i∈Z/tZ X i = t are coprime, it is easy to check that φ : Perhaps the most natural description of the inverse φ −1 is given (for ∈ Z/tZ, noting that → s + k is surjective modulo t) by evaluating t−1 j=0 ( t−1 2 − j)y j+ · t, i.e. the sum This yields the equality t−1 Having analyzed the ambient affine space, we now wish to show that φ restricts to a set bijection A t (s) → TD t (s) for C s ∩ C t , as well as the analogous bijection for C t . Suppose (x i ) ∈ { i∈Z/tZ X i = t 2 } corresponds under φ to (y j ) ∈ { j∈Z/tZ Y j = s}; then we make the following observations.
(1) Since s, t are coprime, the inequalities x i ≥ x i+s − s hold for all i ∈ Z/tZ if and only if y j ≥ 0 for all j ∈ Z/tZ; (2) The identity t−1 j=0 jy j+ = k − x k+s follows (for any ∈ Z/tZ) from the telescoping sum above (substituting t−1 j=0 y j+ = s and k = t−1 , and further y j ≡ 0 (mod 1) for all j ∈ Z/tZ. Then for any ∈ Z, we have so x i ≡ i (mod t) for all i ∈ Z/tZ (as s, t are coprime). (Alternatively, we could look at the partial sums z 0 + · · · + z −1 = 1 t (x k − x k+ s + s).) The first and third items show that φ maps A t (s) into TD t (s). The first and fourth items show that φ −1 maps TD t (s) into A t (s). (We are using the fact that φ bijects a superset of A t (s) to a superset of TD t (s).) So φ bijects A t (s) to TD t (s), establishing the change of coordinates for C s ∩ C t . Similarly, considering only the third and fourth items (ignoring the first item) establishes the change of coordinates for C t .
Finally, the explicit formula for the inverse φ −1 establishes the desired description of the inverse (a t,i ) i∈Z/tZ = φ −1 (z t,j ) j∈Z/tZ of the isomorphism.
Remark 4.5. Already in this proof we have seen the 'cyclic shifts identity' t−1 j=0 jy j+ ≡ −s + j∈Z/tZ jy j (mod t) for t integers (y j ) j∈Z/tZ summing to s, which will feature more prominently in Proposition 4.7.
Corollary 4.6 (Weak version of Anderson's theorem [5]). A t (s) and TD t (s) are discrete bounded sets, hence finite. In particular, there are finitely many (s, t)-cores, so the sums in Theorems 1.1, 1.2, 1.3, 1.4 are finite and well-defined.
Side Remark 4.4. As mentioned in the introduction, Anderson [5] actually computes the exact number of (s, t)-cores as the 'rational Catalan number' 1 s+t s+t s , by bijecting the set of (s, t)-cores to down-right lattice paths from (0, s) to (t, 0) staying below the connecting line (or tuples (w 1 , . . . , w s+t ) ∈ {s, −t} s+t summing to 0, with all nonnegative partial sums, where w i = s corresponds to a rightward step and w i = −t corresponds to a downward step); these are essentially (s, t)-Dyck paths, which Bizley [8] counts via 'cyclic shifts of order s+t'.
One route to the bijection (mentioned in the introduction) goes from (s, t)-cores to betasets closed under subtraction by s, t, which correspond under negation and suitable translation to subsets of Z ≥0 that contain 0 and are closed under addition by s, t, which correspond to the desired lattice paths (see e.g. [18]).
At the beginning of our proof of Theorem 1.1 we implicitly give Johnson's variant using 'cyclic shifts of order t' in the asymmetric z-coordinates. The criterion for z-coordinates to correspond (s, t)-cores can be geometrically framed as a divisibility condition on "area under lattice paths from (0, s) to (t − 1, 0)" modulo t, in contrast to the Catalan-like condition for w-coordinates.

4.3.
Cyclic shifts in the z-coordinates. Johnson [21] relies on cyclic symmetry in his proofs of Anderson's theorem [5] (that there are exactly 1 s+t s+t s distinct simultaneous (s, t)cores) and Armstrong's general conjecture (Theorem 1.1). For Theorems 1.1 and 1.3, we also rely on the following 'cyclic shifts' argument. Let f (X 0 , . . . , X t−1 ) be a cyclic complex-valued function (i.e. for all i ∈ Z/tZ we have f (X 0 , . . . , X t−1 ) = f (X i , . . . , X i+t−1 ), with indices taken modulo t). Then Proof. For any t nonnegative integers x 0 , . . . , x t−1 ≥ 0 (indexed modulo t) summing to s, the cyclic permutations (x r , . . . , x r+t−1 ) leave distinct residues via the 'cyclic shifts identity' since s is coprime to t. Thus each orbit of the cyclic Z/tZ-action contains exactly one point of TD t (s), and since f is cyclic (and the sums are over the finite sets TD t (s) and {(x j ) j∈Z/tZ ∈ Z t ≥0 : j∈Z/tZ x j = s}, by Corollary 4.6), the result follows. Corollary 4.8 (Relevant cyclic sums). Fix coprime s, t ≥ 1. Then we evaluate the sum (z t,j )∈TDt(s) f (z t,0 , . . . , z t,t−1 ) in the cases listed below, where the most important terms have been boxed. For a vector or weak composition x = (x i ) i∈Z/tZ with t components, define the sum |x| := i∈Z/tZ x i and (if appropriate) the multinomial coefficient |x| x := |x| x 1 ,...,xt . First we look at "exponential" cases.
t i∈Z/tZ x i x i+r for some r ≡ 0 (mod t) (quadratic). Next we look at "ordinary" cases.
when f = 1 t i∈Z/tZ x i x i+r for some r ≡ 0 (mod t) (mixed quadratic). Remark 4.9. We will use these explicit (generating function) calculations below in the proofs of Theorems 1.1 and 1.3, instead of the coarser Ehrhart and Euler-Maclaurin theory language of Johnson [21]. (See Remarks 8.3 and 8.6 for further conceptual discussion.) (However, the difference is probably not as large as it might sound: Ehrhart theory simply captures many of the most important qualitative aspects of the generating functions related to lattice point geometry; see the textbook [9] for more details.) Proof of "exponential" cases. First we use Proposition 4.7 to reduce the cyclic sums in question over TD t (s) to sums over the easier domain of {x i ≥ 0 : |x| = s}. On this easier domain, the standard tool of exponential generating functions ( i∈Z/tZ exp(Z i T )) suffices. Equivalently, one may directly differentiate the multinomial |x|=s |x| Size of a t-core. The c-and a-coordinate versions of the following lemma could have been placed earlier, but we wish to use the z-coordinates, most relevant right before Section 5. The c-coordinate version has been used in [19] and [12] to deduce certain properties of the generating function for t-cores. • In Johnson's c t -coordinates,  [12], up to the linear relation i∈Z/tZ c t,i = 0.
Side Remark 4.5. We will only explicitly use this result for (s, t)-cores λ ∈ C s ∩ C t , when s, t are coprime, in the proof of Theorems 1.1 and 1.3. But the general t-core version may help for other problems.
Side Remark 4.6. In principle we could compute the coefficients of M 2 more explicitly, but by the computations in Corollary 4.8, we will not need to explicitly distinguish the different terms of M 2 .  [21].) Due to the cyclic nature of the change of variables relating the z-and a-coordinates (see Proposition 4.2), it suffices to understand why |λ| is symmetric in the a-coordinates.
The following proof directly explains the (complete) symmetry in the a-coordinates, but one could also convert from the c-coordinates using Section 2.4.
Discrete calculus proof for a-coordinates. Split the partition into two halves by cutting along the y-axis (see Figure 3). We calculate the size of the partition (or area of the partition diagram) as x + 1 2 .
As written this holds for any partition, but we want to simplify it further for t-cores λ. As we are working with the a t,i -coordinates, we will break up the contributions by residue class modulo t. Fix 0 ≤ i ≤ t − 1, and define the i + tZ area contribution F (a t,i ) (i.e. contribution in the sums above from x congruent to i (mod t)); we will study how it changes as a t,i varies along the residue class i + tZ. We have 2 (note that this holds both in the 'positive/left case' a t,i ≥ i + t and the 'negative/right case' a t,i ≤ i).

Standard discrete calculus yields
To finish, we sum over all 0 ≤ i ≤ t−1, using the identity 1 2t . Discrete calculus proof for c-coordinates. Mimicking the proof in the a-coordinates, the area contribution from c t,i is ( 2 in the front is just a correction term, as when −c t,i = 0, the contribution should be 0. To finish, we sum the contributions over all 0 ≤ i ≤ t − 1. Proof of a-to-z translation. Write |λ| = − 1 24 (t 2 − 1) + 1 2t · P for convenience. Proposition 4.2 (which holds for t-cores, not just (s, t)-cores) gives a t,k+ s − t−1 2 = t−1 j=0 ( t−1 2 − j)z t,j+ for all ∈ Z/tZ, since s, t are coprime. Then is a cyclic quadratic polynomial in the z t, , so P = C(t) ∈Z/tZ z 2 t, + H(z t,0 , . . . , z t,t−1 ) for some constant C depending only on t, and a cyclic homogeneous quadratic H ∈ Z[X 0 , . . . , X t−1 ] with no 'square terms' (i.e. X 2 0 , . . . , X 2 t−1 ). We make the following calculations. • P vanishes when a t, = t−1 2 for all ∈ Z/tZ, or equivalently when the z-coordinates are all equal. So the sum of z-coefficients of |λ| + 1 . Substituting P into |λ| = − 1 24 (t 2 − 1) + 1 2t · P finishes the job.

Proofs of general conjectures
In this section, we first give a proof of Armstrong's general conjecture by direct computation. We then use the same methods to prove Fayers' general conjecture.
Proof of Theorem 1.1 by direct computation. We use z-coordinates. By Corollary 4.8, the denominator is just TDt(s) 1 = 1 t s+t−1 t−1 (the number of (s, t)-cores). Using Lemma 4.10 and Corollary 4.8 with the identity Proof of Theorem 1.3. In z-coordinates, the stabilizer Stab Gs,t (λ) has size i∈Z/tZ z t,i ! (by Corollary 4.4). By Corollary 4.8, the denominator times s! is just TDt(s) t ·t s = t s−1 . Using Lemma 4.10 and Corollary 4.8 with the identity x 2 = x(x − 1) + x, the numerator times s!, i.e. TDt(s) . Finally, dividing (s! times the) numerator by (s! times the) denominator yields the desired ratio of 1 24 (s − 1)(t 2 − 1). Remark 5.1. Conceptually, the s(s − 1)t s−2 coefficients cancel because they collect to form the sum of the coefficients of |λ| + 1 24 (t 2 − 1), which is 0 as noted in Lemma 4.10.

Self-conjugate analogs
Using almost the same methods as before, we build up to proofs of Theorems 1.2 and 1.4 in Section 6.4. However, we no longer need a cyclic shifts argument, because the parameterization of self-conjugate cores (see Proposition 6.11) is simpler.
6.1. Background: beta-sets, charges, s-sets, and conjugation.  The c-to-a translation of Section 2.4 gives another corollary of Proposition 6.2. If λ is an s-core, then (suppressing notation for clarity) so for the conjugate λ we have (again suppressing notation for clarity) Thus λ is also an s-core (which also follows from the original hook length definition), with a s,i (λ) = s − 1 − a s,−1−i (λ) for all i ∈ Z/sZ.
6.2.1. Comparing t-cores of different self-conjugate s-cores. The only necessary modification from the general case is the following self-conjugate version of Proposition 3.2.
Direct alternative proof. Proposition 3.2 already shows everything except that λ t is selfconjugate. Strictly speaking, [15,Proposition 4.7] does prove this indirectly, but it is cleaner to use λ = λ and Corollary 6.3 to get λ t = λ t = λ t .
6.2.2. Self-conjugate level t action. We now present a self-conjugate analog of the group G s,t from Definition 3.7. It is easy to check that H s,t has a group structure, and that it acts in the obvious way on the set of symmetric s-sets S s (λ), or equivalently suitable beta-sets. This induces an action on D s , via Corollary 6.4. Remark 6.7 (Different but equivalent definitions). We have not given Fayers' actual definition of the 'level t action of the s-affine hyperoctahedral group', but rather one equivalent by [15,Propositions 4.1 and 4.2], and easier to work with for our purposes.  We can now state a self-conjugate analog of Corollary 3.11. Corollary 6.10 ([15, Corollary 4.8]). Fix coprime s, t ≥ 1. Then λ, µ ∈ D s lie in the same H s,t -orbit if and only if λ t = µ t . In other words, each H s,t -orbit of D s contains a unique t-core (hence self-conjugate (s, t)-core), and any λ ∈ D s lies in H s,t λ t .
Sketch of more direct proof. Fix λ, µ ∈ D s . Proposition 3.6 followed by Proposition 6.9 gives equivalence of λ t = µ t and µ ∈ H s,t λ. The re-phrasing follows by specializing to µ := λ t (where λ t ∈ D s follows from Proposition 6.5).
Full direct proof. Let λ, µ be two self-conjugate s-cores. Proposition 3.6 states that λ t = µ t if and only if we have a congruence of (symmetric) s-sets S s (λ) ≡ S s (µ) (mod t), i.e. there exists a bijection φ : S s (λ) → S s (µ) (of symmetric s-sets) preserving residues modulo t. Since S s (λ) and S s (µ) are symmetric s-sets, we may further specify this bijection φ to satisfy φ(m) + φ(s − 1 − m) = s − 1 for all m ∈ S s (λ). By Proposition 6.9, such bijections φ correspond to group elements f ∈ H s,t with restriction f | Ss(λ) = φ. So λ t = µ t if and only if µ = f λ for some f ∈ H s,t , i.e. λ, µ lie in the same H s,t -orbit.
Finally, for any self-conjugate s-core λ ∈ D s , we have λ t ∈ G s,t λ by specializing the previous paragraph to µ := λ t ∈ D s . Definition 6.6 and Corollary 6.10 provide the background and context for Theorem 1.4 (stated in the introduction).
6.3. Key inputs for computation. In this section, we describe all the computational methods and results used to compute the sums in Theorems 1.2 and 1.4, roughly following the structure of Section 4. 6.3.1. Johnson's u-coordinates versus Fayers' t-sets. We will use the z-coordinates again (and the closely related u-coordinates) to get a simple expression (Proposition 6.12) for the H s,t -stabilizers. Again, we not only review Johnson's (s, t)-core parameterization, but also give a modest extension to general t-cores. Fix coprime s, t ≥ 1. For convenience, let s = s/2 and t = t/2 . The set D s ∩ D t of self-conjugate (s, t)-cores (viewed as self-conjugate s-cores within the set of t-cores) is parameterized by either of the sets B t (s), U t (s), described as follows.
• By Lemma 3.1, the set of t-sets S t (λ) of self-conjugate (s, t)-cores λ, i.e. the subset B t (s) of points (a t,i ) i∈Z/tZ ∈ A t (s) (specializing a-coordinates from Proposition 4.2) satisfying the additional symmetry a t,i + a t,−1−i = t − 1 from Corollary 6.4-this is [15,Lemma 4.6]. • The subset U t (s) of points (z t,i ) i∈Z/tZ ∈ TD t (s) (specializing z-coordinates from Proposition 4.2) satisfying the additional symmetry z t,i = z t,−i -this is [21, Lemma 3.9].
• In fact, as implicitly used in [21, proof of Lemma 3.10], the set U t (s) is canonically isomorphic to the set of lattice points {(u 0 , u 1 , . . . , u t ) ∈ Z 1+t ≥0 : u 0 +u 1 +· · ·+u t = s }, with the isomorphism defined by u 0 := z t,0 /2 = s − t i=1 u i and the following additional relations: u i := z t,i (for 1 ≤ i ≤ t ) if t is odd; and u t := z t,t /2 and u i := z t,i (for 1 ≤ i ≤ t − 1) if t is even. The isomorphism between these sets (also preserving the ambient linear and simplex structures) is the same as that described in Proposition 4.2.
Under the same affine change of coordinates, the set D t of self-conjugate t-cores can be parameterized as follows.
• By Proposition 2.11, the set of t-sets S t (λ) of self-conjugate t-cores λ, i.e. the subset of points (a t,i ) i∈Z/tZ ∈ C t satisfying the additional symmetry a t,i + a t,−1−i = t − 1 from Corollary 6.4. • In z-coordinates, the subset of points (z t,i ) i∈Z/tZ ∈ C t satisfying the additional symmetry z t,i = z t,−i . • In u-coordinates, the set of lattice points {(u 0 , u 1 , . . . , u t ) ∈ Z 1+t : u 0 +u 1 +· · ·+u t = s }, with the isomorphism defined by u 0 := z t,0 /2 = s − t i=1 u i and the following additional relations: u i := z t,i (for 1 ≤ i ≤ t ) if t is odd; and u t := z t,t /2 and Proof of a-to-z translation. As usual, let k = 1 2 (s + 1)(t − 1). Define the map φ as in Proposition 4.2, so for any i ∈ Z/tZ, the difference t · (y i − y −i ) evaluates to Observe that (si + k) + (−si + s + k) = (−si + k) + (si + s + k) = s + 2k ≡ −1 (mod t) for all i ∈ Z/tZ. It follows (since s is coprime to t) that y i = y −i for all i ∈ Z/tZ if and only if x i + x −1−i is constant over i ∈ Z/tZ; if and only if x i + x −1−i = t − 1 for all i ∈ Z/tZ (because we are working in the plane i∈Z/tZ (x i + x −1−i ) = t(t − 1)).
Proof of z-to-u translation. If y i = y −i for all i ∈ Z/tZ, then 0y 0 = 0 and iy i + (t − i)y t−i ≡ 0 (mod t) for all i ∈ Z/tZ.
If t is odd, then we automatically get i∈Z/tZ iy i ≡ 0 (mod t), clearly establishing U t (s) = {(u 0 , u 1 , . . . , u t ) ∈ Z 1+t ≥0 : u 0 + u 1 + · · · + u t = s } under the canonical map given by u i := z t,i for 1 ≤ i ≤ t and u 0 := z t,0 /2 = s − t i=1 u i (from i∈Z/tZ z t,i = s). If t is even, then the vanishing of i∈Z/tZ iy i ≡ (t/2)y t/2 (mod t) modulo t is equivalent to y t ≡ 0 (mod 2). This establishes the isomorphism U t (s) = {(u 0 , u 1 , . . . , u t ) ∈ Z 1+t ≥0 : u 0 + u 1 + · · · + u t = s } under the canonical map given by u t = z t,t /2; u i := z t,i for 1 ≤ i ≤ t − 1; and and u 0 := z t,0 /2 = s − t i=1 u i (from i∈Z/tZ z t,i = s). The previous two paragraphs prove the result for D s ∩ D t . The same z-to-u conversion works for D t as well, except one ignores the inequality conditions u i ≥ 0 (which come from the conditions z t,i ≥ 0 of TD t (s)).
Side Remark 6.2. The simplicity of U t (s) may seem somewhat mysterious, but note that regardless of the choice of the constant k when defining z-coordinates, it should still be intuitively clear that some condition like y i = y α−i should hold identically for some constant α (depending on k), in which case (at least for t odd) we have i∈Z/tZ iy i ≡ t 2 α i∈Z/tZ y i = αst 2 (mod t) independent of the lattice point (y i ) i∈Z/tZ (as long as i∈Z/tZ y i = s).
Side Corollary 6.3 (Weak version of Ford-Mai-Sze's theorem [17]). B t (s) and U t (s) are discrete bounded sets, hence finite. In particular, there are finitely many self-conjugate (s, t)cores, so the sums in Theorems 1.2 and 1.4 are finite and well-defined.
Side Remark 6.4. Ford-Mai-Sze [17] actually computes the exact number of self-conjugate (s, t)-cores as s/2 + t/2 s/2 , by bijecting to lattice paths. At the beginning of the proof of Theorem 1.2 we implicitly give Johnson's variant using the u-coordinates.

6.3.2.
Size of the stabilizer of a self-conjugate s-core. The main new observation for Theorem 1.4 is the following computational simplification of Fayers' formula for the size of the stabilizer of a self-conjugate s-core, based on Proposition 3.4. This is the self-conjugate analog of Corollary 4.4 to Proposition 4.1.
(1) If t is odd, then |Stab Hs,t (λ)| equals (2) If t is even (so s is odd), then |Stab Hs,t (λ)| equals Proof. Fix λ ∈ D s . Fayers uses Proposition 6.9 to compute |Stab Hs,t (λ)| as the number of permutations π of the symmetric s-set S s (λ) with π(s−1−m) = s−1−π(m) and π(m) ≡ m (mod t) for all m ∈ S s (λ), i.e. the product of the following individual contributions: Fayers evaluates the product in his own way, but it will be easier for us to directly translate to z-coordinates via , and then to u-coordinates via Proposition 6.11 (as we have λ t ∈ D s ∩ D t by Proposition 6.5). Observe that 2s + 2k ≡ s − 1 (mod t).
Side Remark 6.5. We will only explicitly use this result for self-conjugate (s, t)-cores λ ∈ D s ∩ D t , when s, t are coprime, in the proof Theorem 1.4.

6.3.3.
Evaluating quadratic sums in u-coordinates. We will evaluate several special u-coordinate sums in our proofs of Theorems 1.2 and 1.4 using the following self-conjugate analog of Corollary 4.8.
Next we look at 'modified exponential' cases for even t, freely using z t,0 = 2u 0 + [s] 2 and z t,t = 2u t .
(1) s +t t when f = 1 ; Side Remark 6.6. Again, we will use these explicit calculations below in the proofs of Theorems 1.2 and 1.4, instead of the indirect approaches taken by Johnson [21].
Proof of "exponential" cases. We can mimic the methods in Corollary 4.8, using the exponential generating function exp(2 −1 Z 0 T ) t i=1 exp(Z i T ) when t is odd, but instead the exponential generating function exp(2 −1 when t is even. Proof of "ordinary" cases. We can mimic the methods in Corollary 4.8, using the ordinary Size of a self-conjugate t-core. Lemma 4.10 (for C t ) simplifies in u-coordinates for D t . Lemma 6.14. Fix coprime s, t ≥ 1 and λ ∈ D t . For convenience, set s := s/2 and t := t/2 . We divide into cases based on the parity of t, but in both cases, the nonconstant (i.e. homogeneous quadratic) part of the quadratic, namely the polynomial given by |λ| + 1 24 (t 2 − 1), has coefficients summing to 0.
Side Remark 6.8. In principle we could compute the coefficients more explicitly, but by the computations in Proposition 6.13, we will not need to.
Recap of z-coordinate preliminaries. Write |λ| = − 1 24 (t 2 − 1) + 1 2t · P for convenience. Since λ ∈ D t ⊆ C t and s, t are coprime, Equation (1) (from the proof of Lemma 4.10) gives the ato-z translation P : We finish by breaking into cases based on parity of t and using Proposition 6.11 (which applies to D t , not just D s ∩ D t ). For convenience, let [x] t ∈ {0, 1, . . . , t − 1} denote the least nonnegative residue of x (mod t).
Proof of Lemma 6.14 for odd t. In the u-coordinates (using u i = z t,i = z t,−i for 1 ≤ i ≤ t−1 2 = t ), the polynomial P takes the form We make the following calculations, often suppressing z i := z t,i .
Proof of Lemma 6.14 for even t. In the u-coordinates (using u i = z t,i = z t,−i for 1 ≤ i ≤ t 2 − 1 = t − 1), the polynomial P takes the form We make the following calculations, often suppressing z i := z t,i .
6.4. Proofs of self-conjugate conjectures. This section is the self-conjugate analog of Section 5, addressing Armstrong's and Fayers' self-conjugate conjectures.
Side Remark 6.9. In principle we could give a more direct proof for even t along the same lines as the odd t proof, but we do not do so since our main focus is on Fayers' conjectures and not Armstrong's conjectures.
The same techniques prove Fayers' self-conjugate conjecture.
Our second proof uses the more familiar form of Proposition 4.2, for (s, t)-cores.
Asymmetric proof. Let (s, t) = (m + d, m), so s, t ≥ 1 are coprime. By Lemma 3.1, a t-core λ lies in C s = C m+d if and only if a t,i ≥ a t,i+s − s for all i ∈ Z/tZ, and λ ∈ C s+d = C m+2d = C 2s−t if and only if a t,i ≥ a t,i+2s −(2s−t) for all i. Thus λ ∈ C t is a (t, s, 2s−t) = (m, m+d, m+2d)core if and only if a t,i − [a t,i+s − s] ≥ 0 and a t,i − [a t,i+2s − 2s] ≥ t for all i. By Proposition 4.2 (applied to (s, t) = (m + d, m)) parameterizing C s ∩ C t = C m+d ∩ C m , these inequalities translate in z t -coordinates-after division by t-to z t,j + z t,j+1 ≥ 1, along with the usual z t,j ≥ 0; j∈Z/tZ z t,j = s = m + d; z t,j ≡ 0 (mod 1); and j∈Z/tZ jz t,j ≡ 0 (mod t).
A cyclic shifts argument analogous to Proposition 4.7 yields where in the last step we count valid sequences (x j ) j∈Z/tZ by first choosing the i pairwise non-neighboring 0 terms, and then the remaining t − i positive integers summing to s.