Abaci structures of $(s,ms\pm1)$-core partitions

We develop a geometric approach to the study of $(s,ms-1)$-core and $(s,ms+1)$-core partitions through the associated $ms$-abaci. This perspective yields new proofs for results of H. Xiong and A. Straub (originally proposed by T. Amdeberhan) on the enumeration of $(s, s+1)$ and $(s,ms-1)$-core partitions with distinct parts. It also enumerates the $(s, ms+1)$-cores with distinct parts. Furthermore, we calculate the weight of the $(s, ms-1,ms+1)$-core partition with the largest number of parts. Finally we use 2-core partitions to enumerate self-conjugate core partitions with distinct parts. The central idea is that the $ms$-abaci of maximal $(s,ms\pm1)$-cores can be built up from $s$-abaci of $(s,s\pm 1)$-cores in an elegant way.


Partitions and abacus diagrams.
A partition λ of the positive integer n is a weakly decreasing sequence of positive integers which sum to n. We will call n the weight of λ. Each of the integers which make up the partition is known as a part of the partition. For example, (8, 6, 5, 5, 3, 2, 2, 2, 1) is a partition with weight n = 34, and is alternatively written as (8, 6, 5 2 , 3, 2 3 , 1).
A Young diagram is a pictorial representation of a partition. Simply put, it is a finite collection of boxes which are arranged in left-justified rows with the row lengths weakly decreasing (since each row of the Young diagram corresponds to a part in the partition). To each box in the Young diagram of λ we assign a hook, which is the set of boxes in the same row and to the right, and in the same column and below, as well as the box itself, which is called the corner of the hook. We use matrix notation to label the hooks: h ij is the hook whose corner is in the i-th row and the j-th column. The number of boxes |h ij | is the hook length of h ij . The first-column hook lengths are those that appear in the left-most column of the Young diagram.
The first-column hook lengths uniquely determine a partition λ. We can generalize the set of first column hooks using the notion of a bead set X corresponding to λ, where X = {0, · · · , k − 1, |h 11 | + k, |h 21 | + k, |h 31 | + k, · · · } for some non-negative integer k. It can also be seen as a finite set of non-negative integers, represented by beads at integral points of the x-axis, i.e., a bead at position x for each x in X and spacers at positions not in X. Then |X| is the number of beads that occur after the zero position, wherever that may fall. The minimal bead-set X of λ is one where 0 labels the first spacer, and is exactly the set of first-column hook lengths. {0, 3, 5, 7} and X ′′ = {0, 1, 2, 3, 2 + 4, 4 + 4, 6 + 4} = {0, 1, 2, 3, 6, 8, 10} are two bead sets that also correspond to λ.
The set of hooks {h ιγ } of λ correspond bijectively to pairs (x, y) where x ∈ X, y ∈ X and x > y; that is, a bead in a bead-set X of λ and a spacer to the left of it. Hooks of length s are those such that x − y = s.
The following result (Lemma 2.4, [14]) allows us to recover the size of the part from its corresponding bead. Lemma 1.2. Let X be a bead-set of a partition λ. The size of the part λ α of λ corresponding to the bead x ′ ∈ X is the number of spacers to the left of the bead, that is, λ α = |y ∈ X : y < x ′ |.
Given a fixed integer s, we can arrange the nonnegative integers into an s-grid, an array of s columns labeled from 0 ≤ i ≤ s − 1, and consider the columns as runners, on which beads are placed in their respective positions. This organizes a given bead-set by their values modulo s.

Definition 1.3 (s-abacus)
. Consider a bead-set X. Placing a bead in each position on the s-grid where there is a value x ∈ X gives the s-abacus diagram S of X. Positions not occupied by beads are spacers. A minimal s-abacus S corresponds to a minimal bead-set X (where the first spacer labels the zero position). Definition 1.4 (s-abacus position). Let S be the s-abacus associated to a bead-set X. We say that a bead x ∈ X has s-abacus position (i, j) ∈ S, where 0 ≤ i ≤ s − 1 and j ≥ 0 if and only if i + js = x ∈ X. Definition 1.5. A sub-abacus S ′ of an s-abacus S is a set of s-abacus positions (i, j) that obey the property that if (i, j) ∈ S ′ , then (i, j) ∈ S.
1.2. s-core and simultaneous (s, t)-core partitions. A s-core partition (or simply s-core) of n is a partition in which no hook of length s appears in the Young diagram. Note that a bead x in runner i with a spacer y one row below, but also in runner i, corresponds to an s-hook of λ. A partition λ is a s-core if and only if its s-abacus has the property that no spacer occurs below a bead in a given runner. This is expressed in the following lemma. Lemma 1.6. An s-abacus S corresponds to an s-core partition if and only (i, j) ∈ S and j > 0 implies that (i, j − 1) ∈ S.
We then have the following result. Corollary 1.7. An s-core partition is an ms-core partition for all m > 1.
Proof. An ms-hook on an s-abacus S is expressed as a bead in abacus position (i, j) and a spacer in position (i, j −m). Either there are no beads in positions (i, j −1), · · · , (i, j −m+1) or there is at least one. In the either case, we violate the condition of Lemma 1.6.
A result of Sylvester from 1884 gives us the size of the largest possible first-column hook length of a simultaneous (s, t)-core. Proposition 1.8. If gcd(s, t)=1, the largest possible hook of an (s, t)-core has length st−s−t.
In recent years, the study of core partitions has expanded to include partitions which are simultaneously cores for various integers. Anderson [5] first enumerated (s, t)-cores in the case when s and t are relatively prime. Subsequently, the work of Olsson and Stanton (and others) showed that, when gcd(s, t) = 1, there is a unique (s, t)-core with largest weight, denoted by κ s,t . We call such a simultaneous core maximal. Theorem 1.9 (J. Olsson and D. Stanton, Theorem 4.1, [15]). Let gcd(s, t) = 1. Then there is a unique maximal (s, t)-core κ s,t such that Using the notation κ s,t we restate a canonical result of J. Anderson (Proposition 1, [5]).
Proposition 1.10. Suppose gcd(s, t) = 1. The minimal s-abacus S of an (s, t)-core will be a sub-abacus of the minimal s-abacus K of κ s,t the maximal (s, t)-core partition. Furthermore, As a consequence of Proposition 1.8, Theorem 1.9, and Proposition 1.10, we have the following useful result.
Self-conjugate simultaneous core partitions are also of interest. B. Ford, H. Mai and L. Sze [8] have, in a manner analogous to Olsson-Stanton, enumerated the self-conjugate (s, t)-core partitions.
In Section 4, we study (s, ms ± 1)-cores with distinct parts; in Section 5 we apply our methods to analogize the results of Xiong and Straub and enumerate the self-conjugate simultaneous (s, s + 1)-core and (s, ms ± 1)-core partitions with distinct parts. Before we do, we give an overview of existing results on simultaneous core partitions with distinct parts.
1.3. Simultaneous (s, t)-cores with distinct parts. Simultaneous core partitions with distinct parts were first introduced as an object of study by T. Amdeberhan. One of the conjectures proposed by T. Amdeberhan (Conjecture 11.9, [3]) has lead to new results by H. Xiong and A. Straub in this area. More recently A. Zaleski ([21]) has published some on moments of their generating functions, building on work by S. Ekhad and D. Zeilberger [7]. Theorem 1.12 (H. Xiong, Theorem 1.1(1), [19]). Let s ≥ 1 and F s+1 be the (s + 1)st Fibonacci number. Then F s+1 is the number of (s, s + 1)-core partitions with distinct parts. Our paper develops a framework from which results of H. Xiong and A. Straub in this direction follow naturally. That is, we use the geometry of the s-abacus of the maximal (s, s+1)-core, and that of the ms-abacus of the maximal (s, ms−1)-core and (s, ms+1)-core partitions, to prove Theorems 1.12 and 1.13 in a uniform manner. Before proving Theorem 1.13, however, we enumerate (s, ms + 1)-core partitions with distinct parts (Theorem 1.14). In doing so, we provide a partition-theoretic meaning to a numerical relation first observed by Straub (see Lemma 4.3,[17]). This lays the groundwork for the proof of Theorem 1.13. Theorem 1.14. Let m, s ≥ 1. The number E + m (s) of (s, ms + 1)-core partitions into distinct parts is characterized by . These proofs appear in Section 4.
1.4. Simultaneous (s, ms−1, ms+1)-core partitions. Suppose s, t, u are positive integers such that gcd(s, t, u) = 1. Enumerating and calculating the weight of simultaneous (s, t, u)cores is more complicated than simultaneous (s, t)-cores, in part because no analogous result to Sylvester's characterization of the maximum possible hook length exists. However, for special cases, progress has been made. T. Amdeberhan and E. Leven [4], R. Nath and J. Sellers, [13], Xiong [19], and Yang-Zhang-Zhou [20] investigated (s − 1, s, s + 1)-cores, and both the weight of the maximal core and the number of such cores is known. V. Wang [18] has enumerated (s, s + d, s + 2d)-cores. A. Aggarwal [1] has also studied containment properties of (s, t, u)-cores.
The methods described herein also allow us to study another family of triply simultaneous cores: in particular, we calculate the weight of the longest (s, ms − 1, ms + 1)-core partition (that is, the core partition with the largest number of parts).
Theorem 1.15. The weight of the longest (s, ms − 1, ms + 1)-core is We also conjecture that this is the weight of any maximal (s, ms − 1, ms + 1)-core.
The key observation we utilize in proving all of our results is the way in which the ms-abaci of maximal (s, ms ± 1)-cores are built up from the s-abaci of (s, s ± 1)-cores and other objects. Hence, we now transition to a detailed description of the relevant s-abaci and ms-abaci.
Proof. Suppose S is the s-abacus of an (s + 1)-core. Then (i, j) ∈ S if and only if there if a bead in a position s + 1 steps to the left, wrapping down-and-around-to-the-right the abacus when necessary. This is exactly the statement of the lemma.

Lemma 2.2. An s-abacus S represents an
Proof. The argument is identical to that of Lemma 2.1 replacing s + 1 by s − 1, with the caveat that a bead in position (s − 1, 0) is not permitted.
A crucial part of our argument below will involve the following two abaci constructions.
The proofs of Lemmas 2.7 and 2.8 follow from Definition 2.6. Details are left to the reader.
Proof. Part (1) is immediate. Part (2) ensures that when moving s positions to the left of (i, j) wraps around-and-down the ms-abacus, a bead occupies the relevant abacus position.
The next corollary follows from Corollary 1.7 and the definition of an ms-abacus.     Definition 3.7. Suppose A and B are s-abaci and t-abaci respectively. We denote by A ∧ B the (s + t)-abacus whose 0 ≤ i ≤ s − 1 runners correspond to the s − 1 runners of A, and whose s ≤ i ≤ s + t − 1 runners of correspond to the t − 1 runners of B. This will be called appending B to A on the right. When we append A to itself m times, we will use the notation ∧ m A = A ∧ · · · ∧ A m . Lemma 3.8. The following relations hold. ( . Consider the projection map π ℓ that takes (i + ℓs, j) to (i, j), where 1 ≤ i ≤ s − 1. We prove each case separately.
We are now in a position to prove Theorem 3.6, which is critical for the rest of our results.
Proof of Theorem 3.6. The abaci above are minimal, by construction. It remains to show they satisfy the relevant core properties and that they are of maximal weight. We consider each case separately.
is an s-core, we have to satisfy the conditions of Lemma 3.1. Suppose j > 0. If (i, j) is in the copy of B 1 (s), then (i − s, j) ∈ E − m (s), since B 1 (s) is a sub-abacus of B 0 (s) by Lemma 2.12. If (i, j) is in one of the rightmost m − 2 copies of B 1 (s), then, (i − s, j) ∈ E − m (s). If (i, j) is in the leftmost copy B 1 (s), notice that π 0 (i) = i. It is enough to see that there exists an (i + (m − 1)s, j − 1) in the rightmost copy of B 1 (s), using π m−1 and Lemma 2.8.
(2) The proof is analogous to (1); the abacus position (ms − 1, s − 2) corresponds to the maximal bead value in the underlying bead-set.
4. ms-abaci of (s, ms ± 1)-cores with distinct parts We now wish to turn our attention to simultaneous cores with distinct parts. This will allow us to provide unified proofs of Theorems 1.12, 1.13 and 1.14. (1) (i − 1, j) ∈ S and (i + 1, j) ∈ S if 1 < i < s − 1, and Proof. A partition has distinct parts if and only if its minimal bead-set X satisfies the following property: if x, y ∈ X and x > y, then x − y = 1. This is exactly the statement of the lemma when translated into s-abaci.
The combination of Lemma 4.1 and Theorem 3.6 allow us to study the abaci of certain simultaneous core partitions with distinct parts. (1) Suppose (i, j) ∈ S such that j > 0. Then (i, j − 1) ∈ S and (i − 1, j − 1) ∈ S by Proposition 1.10, Lemma 2.1, Lemma 2.2, Lemma 2.5. This is a contradiction.
We now possess all of the necessary tools to prove Theorems 1.12-1.14 in a unified, combinatorial fashion. As noted earlier, we switch our convention and prove Theorem 1.14 first; Theorem 1.13 then follows from Theorem 4.3 and some manipulation.
Proof of Theorem 1.12. There is only one simultaneous (1, 2)-core partition with distinct parts, the empty partition. There are two simultaneous (2, 3)-core partitions with distinct parts: the empty partition, and λ = (1). This gives us the initial conditions, F 2 = 1 and F 3 = 2.
By Lemma 4.2(1), for any s-abacus S of an (s, s + 1)-core with distinct parts, if (i, j) ∈ S then j = 0 where 0 ≤ i ≤ s − 1. We divide the count into two cases, depending on whether or not (s − 1, 0) ∈ S.
If (s − 1, 0) ∈ S, then by Lemma 4.1, (s − 2, 0) ∈ S. By considering only the runners 0 ≤ i ≤ s − 3, we conclude there are F s−1 possible s-abacus arrangements for S with (s − 1, 0) ∈ S. If (s − 1, 0) ∈ S, then by considering only the runners 0 ≤ i ≤ s − 2, we can conclude that there are F s possible s-abacus arrangements for S with (s − 1, 0) ∈ S. Hence the total number of acceptable s-abacus arrangements for an (s, s + 1)-core with distinct parts is F s+1 = F s + F s−1 . This completes the proof.
Proof of Theorem 1.14. There is only one simultaneous (1, m + 1)-core; the empty partition. By Lemma 3.8 and   4)). Expanding, we have 5. the ms-abacus of the longest (s, ms − 1, ms + 1)-core We now move to discuss triply simultaneous core partitions. Lemmas 5.3, 5.5, 5.8 and Corollary 5.7 follow from the relevant definitions. In the interest of brevity the proofs are omitted.
Definition 5.1. Let A and B each be s-abaci. Then the intersection of A and B, denoted  A ∩ B, is the sub-abacus of all beads in both A and B. We let C k (s) = A(s) ∩ B k (s).
Proof. This follows from Definitions 3.7 and 5.1. Proof. This follows from Corollary 2.12 and the definitions of C 1 (s) and C 0 (s).
The simultaneous (a, b, c)-core partition with the most parts is called the longest one. Proof. It is enough to show that L m (s) is an (s, ms − 1, ms + 1)-core, and that the inclusion of beads in any other abacus positions in E − m (s) or E + m (s) will violate the (ms−1) or (ms+1)core condition. To see it is an s-core, we consider a bead in three abacus positions: in the rightmost C 1 (s), the leftmost C 0 (s), or one of the m − 2 wedge-copies of C 0 (s) in the middle. For a bead in the rightmost C 1 (s); by Lemma 5.12, there is a bead s-positions to the left and in the same row, since C 1 (s) is a sub-abacus of C 0 (s). The same argument applies to beads in the middle m − 2 copies of C 0 (s). Suppose a bead is in the leftmost copy of C 0 (s) with abacus position (i, j), where j > 0. Then it is enough that ((ms − 1) − i − 1, j − 1) ∈ L m (s). This follows by the construction of C 1 (s), and the projection map π m−1 .
In order to determine the total weight of this core, we simply sum (1) above over all relevant rows of the abacus. This yields using well-known results on sums of integer powers. Replacing s by 2t − 1 or 2t − 2 yields the results of this theorem after elementary simplification.
Conjecture 5.14. The size of a maximal (s, ms − 1, ms + 1)-core is There are two such maximal partitions; one corresponding to L(s), and one corresponding to its conjugate.
If Conjecture 5.14 is true, then we have the following elegant corollary.
Corollary 5.15. Let s be even. The weight of the maximal (s, ms − 1, ms + 1)-core partition is divisible by m 2 .
6. ms-abaci of self-conjugate (s, ms ± 1)-core partitions with distinct parts We close this paper by applying our tools to prove results on self-conjugate simultaneous core partitions with distinct parts. The following is a well-known lemma.
Lemma 6.1. The 2-core partitions are exactly those of the form (k, k − 1, k − 2, · · · , 1). The bead-sets of the 2-cores are of the form {∪ ℓ≤0 2ℓ + 1}. Lemma 6.2. Let X be a bead set of a self-conjugate partition. Then there exists a halfinteger θ such that if x ∈ X and x > θ then there exists a y ∈ X such that |y − θ| = |x − θ|.
Proof. See Corollary 3.4 in [12]. Proof. Every 2-core partition is clearly a self-conjugate partitions with distinct parts. Now suppose we have a self-conjugate partition λ with distinct parts. Then it must have a beadset X that consists of alternating spacer-and-beads. Suppose not. If two beads occur in a row, we know that it violates having distinct parts. Suppose two spacers occur in a row. If y, y + 1 ∈ X and both y, y + 1 < θ or both y, y + 1 > θ then, by Lemma 6.2, there will be two beads in succession on the other side of θ. If y < θ and θ < y + 1, then by Lemma 6.2 λ is not self-conjugate.
With the results of the previous sections and the lemmas above, we can consider selfconjugate simultaneous core partitions with distinct parts.