Self-conjugate (s,s+d,…,s+pd) -core partitions and free rational Motzkin paths
aa r X i v : . [ m a t h . C O ] A p r Self-conjugate ( s , s + d , . . . , s + pd )-core partitions andfree rational Motzkin paths Hyunsoo Cho a , JiSun Huh b, ∗ a Institute of Mathematical Sciences, Ewha Womans University, Seoul 03760, Republic of Korea b Department of Mathematics, Ajou University, Suwon 16499, Republic of Korea
Abstract
A partition is called an ( s , s , . . . , s p )-core partition if it is simultaneously an s i -core for all i = , , . . . , p . Simul-taneous core partitions have been actively studied in various directions. In particular, researchers concerned withproperties of such partitions when the sequence of s i is an arithmetic progression.In this paper, for p ≥ s and d , we propose the ( s + d , d ; a )-abacus of a self-conjugate partition and establish a bijection between the set of self-conjugate ( s , s + d , . . . , s + pd )-core partitions andthe set of free rational Motzkin paths with appropriate conditions. For p = ,
3, we give formulae for the number ofself-conjugate ( s , s + d , . . . , s + pd )-core partitions and the number of self-conjugate ( s , s + , . . . , s + p )-core partitionswith m corners. Keywords: simultaneous core partitions, self-conjugate, free rational Motzkin paths
1. Introduction A partition λ of n is a finite non-increasing positive integer sequence ( λ , λ , . . . , λ ℓ ) such that the sum of all λ i is equal to n . For a partition λ of n , the Young diagram of λ is a collection of n boxes arranged in left-justified rows,with the i th row having λ i boxes. The hook length of a box in the Young diagram of λ is defined to be the number ofboxes weakly below and strictly to the right of the box. Figure 1 shows the Young diagram of λ and the hook lengthsof each box. For a positive integer s , a partition λ is an s-core (partition) if it has no box of hook length s . Referringto Figure 1, we can easily verify that λ = (5 , , ,
1) is an s -core for s = , , or s ≥
9. For a sequence ( s , s , . . . , s p )of distinct positive integers, we say that a partition λ is an ( s , s , . . . , s p ) -core if it is simultaneously an s -core, an s -core, . . . , and an s p -core. 8 6 4 3 16 4 2 13 11 Figure 1: The Young diagram and the hook lengths of λ = (5 , , , There has been considerable interest in recent years in core partitions, which have applications in representationtheory and number theory. Since the following result of Anderson [2], many researchers found results on ( s , t )-corepartitions (see [3, 5, 8, 9, 10, 11]). ∗ Corresponding author.
Email addresses: [email protected] (Hyunsoo Cho), [email protected] (JiSun Huh)
Preprint submitted to Elsevier April 14, 2020 heorem 1.1. [2, Theorem 1] For relatively prime positive integers s and t, the number of ( s , t ) -core partitions isgiven by s + t s + ts ! . In particular, the number of ( s , s + -core partitions is the sth Catalan number C s = s + (cid:16) ss (cid:17) = s + (cid:16) s + s (cid:17) . Going further, researchers considered simultaneous core partitions when the sequence of s i forms an arithmeticprogression (see [1, 4, 6, 16, 17, 18]). Recently, the authors [7] enumerated the ( s , s + d , . . . , s + pd )-core partitionsby giving a lattice path interpretation.A free rational Motzkin path of type ( s , t ) is a path from (0 ,
0) to ( s , t ) consisting of up steps U = (1 , D = (1 , − F = (1 , s , t ) by F ( s , t ). Inaddition, if a free rational Motzkin path of type ( s , t ) is allowed to stay only weakly above the line y = x , then it iscalled a rational Motzkin path of type ( s , t ). Theorem 1.2. [7, Theorem 1.5] Let s and d be relatively prime positive integers. For an integer p ≥ , the number of ( s , s + d , . . . , s + pd ) -core partitions is equal to the number of rational Motzkin paths of type ( s + d , − d ) without UF i Usteps for i = , , . . . , p − if p ≥ , that is given by s + d s + dd ! + ⌊ s / ⌋ X k = r X ℓ = k + d k + dk − ℓ ! k − ℓ ! s + d − ℓ ( p − − k + d − ! , where r = min( k − , ⌊ ( s − k ) / ( p − ⌋ ) . For a partition λ , the conjugate of λ is the partition whose Young diagram is the reflection along the main diagonalof the Young diagram of λ . A partition whose conjugate is equal to itself is called self-conjugate . Some researchershave considered with self-conjugate simultaneous core partitions whose cores line up with an arithmetic progression.We denote the set of all self-conjugate ( s , s + d , . . . , s + pd )-core partitions by SC ( s , s + d ,..., s + pd ) . Ford-Mai-Sze [11] firstinvestigated self-conjugate ( s , t )-core partitions and obtained the following result. Theorem 1.3. [11, Theorem 1] For relatively prime positive integers s and t, the number of self-conjugate ( s , t ) -corepartitions is given by ⌊ s ⌋ + ⌊ t ⌋⌊ s ⌋ ! . In particular, the number of self-conjugate ( s , s + -core partitions is equal to the number of symmetric Dyck pathsof order s, that is given by (cid:16) s ⌊ s / ⌋ (cid:17) . Motivated by Theorem 1.3, the authors [6] gave a formula for the number of self-conjugate ( s , s + , s + s . Theorem 1.4. [6, Theorem 4] For a positive integer s, the number of self-conjugate ( s , s + , s + -cores is X i ≥ ⌊ s ⌋ i ! i ⌊ i ⌋ ! , which counts the number of symmetric Motzkin paths of length s. The authors also suggested a conjecture about a relation between the set of self-conjugate ( s , s + , . . . , s + p )-core partitions and the set of symmetric ( s , p )-generalized Dyck paths (see [6, 17] for the definition of symmetric( s , p )-generalized Dyck paths). This conjecture was recently proved by Yan-Yu-Zhou [17]. Theorem 1.5. [17, Theorems 2.14, 2.19, and 2.22] For positive integers s and p, the number of self-conjugate ( s , s + , . . . , s + p ) -core partitions is equal to the number of symmetric ( s , p ) -generalized Dyck paths. By using the above theorem together with an another path interpretation, the authors [7] found a formula for thenumber of self-conjugate ( s , s + , . . . , s + p )-core partitions.2 heorem 1.6. [7, Theorem 3.9] For positive integers s and p ≥ , the number of self-conjugate ( s , s + , . . . , s + p ) -corepartitions is given by + ⌊ s / ⌋ X k = r X ℓ = ⌊ k − ⌋⌊ ℓ ⌋ ! ⌊ k ⌋⌊ ℓ + ⌋ ! ⌊ s − ℓ ( p − ⌋ k ! , where r = min( k − , ⌊ ( s − k ) / ( p − ⌋ ) . As there was no known result on self-conjugate ( s , s + d , . . . , s + pd )-core partitions with d ≥ s + d , d ; a )-abacus diagram” and define the “( s + d , d ; a )-abacus function” of a self-conjugate partition λ and then investigateproperties of the ( s + d , d ; a )-abacus function of a self-conjugate ( s , s + d , . . . , s + pd )-core partition. After that, inSection 3.1, we give a lattice path interpretation for a self-conjugate ( s , s + d , . . . , s + pd )-core partition by constructinginjections from the set of self-conjugate ( s , s + d , . . . , s + pd )-core partitions to the set of free rational Motzkin paths.The following theorem is the main result of this paper. Theorem 1.7.
Let s and d be relatively prime positive integers. For a given integer p ≥ , the mapping φ ( s + d , d ) gives aone-to-one correspondence between SC ( s , s + d ,..., s + pd ) and the set of paths P ∈ F ( ⌊ s / ⌋ + ⌈ d / ⌉ , −⌈ d / ⌉ ) satisfying thati) P has no UF i U as a consecutive subpath for all i = , , . . . , p − if p ≥ ; ii) P starts with no F j U; iii) P endswith no UF k for all(a) j = , , . . . , ⌊ ( p − / ⌋ if p ≥ and k = , , . . . , ⌊ ( p − / ⌋ if p ≥ , when s is odd and d is even;(b) j = , , . . . , ⌊ ( p − / ⌋ if p ≥ and k = , , . . . , p − , when s is odd and d is odd;(c) j = , , . . . , ⌊ ( p − / ⌋ if p ≥ and k = , , . . . , p − , when s is even and d is odd. As a corollary, we give formulae for the number of self-conjugate ( s , s + d , s + d )-core partitions and the numberof self-conjugate ( s , s + d , s + d , s + d )-core partitions in Section 3.2. In Section 3.3, we focus on self-conjugate( s , s + , . . . s + p )-core partitions with m corners. We specify free rational Motzkin paths corresponding to self-conjugate ( s , s + , . . . , s + p )-core partitions with m corners. In particular, we count the number of self-conjugate( s , s + , s + m corners.
2. The ( s + d , d ; a )-abacus diagram Let λ be a partition. The s -abacus of λ , introduced by James-Kerber [13], has played important roles in the theoryof core partitions (see [3, 8, 9, 14, 15]). In [7], the authors introduced the ( s + d , d )-abacus of λ , which is useful fordetermining whether λ is an ( s , s + d , . . . , s + pd )-core. Now, we slightly modify the ( s + d , d )-abacus of λ to getan abacus, which is useful when we deal with self-conjugate ( s , s + d , . . . , s + pd )-core partitions. Note that if λ is aself-conjugate partition, then elements in MD ( λ ) are all distinct and odd, where MD ( λ ) denotes the set of the maindiagonal hook lengths of λ . Let Z denote the set of all integers. Definition 2.1.
Let s and d be relatively prime positive integers. We define the ( s + d , d ; a )-abacus diagram to be thebottom and left justified diagram with infinitely many rows labeled by i ∈ Z and ⌊ ( s + d + / ⌋ columns labeled byj ∈ { , , . . . , ⌊ ( s + d − / ⌋} whose position ( i , j ) is labeled by a + s + d ) i + d j, where a = a ( s , d ) denotes theinteger such that − a is the smallest odd number among s and s + d.For a self-conjugate partition λ , the ( s + d , d ; a )-abacus of λ is obtained from the ( s + d , d ; a ) -abacus diagram byplacing a bead on each position labeled by ℓ , where | ℓ | ∈ MD ( λ ) . A position without bead is called a spacer . The following proposition shows that, for self-conjugate partitions λ , the ( s + d , d ; a )-abacus of λ are well-definedand all distinct. In addition, if λ is an ( s , s + d , . . . , s + pd )-core, then there are exactly | MD ( λ ) | beads on the ( s + d , d ; a )-abacus of λ . Proposition 2.2.
Let s and d be relatively prime positive integers. For every positive odd integer h such that h . s + d (mod 2( s + d )) , there exist a unique position labeled by h or − h in the ( s + d , d ; a ) -abacus diagram. roof. For the ( s + d , d ; a )-abacus diagram, positions in column j are labeled by a + s + d ) i + d j for i ∈ Z . Note thatthe absolute value of these labels are congruent to a + d j or 2( s + d ) − a − d j modulo 2( s + d ). We claim that a + d j and 2( s + d ) − a − d j for j ∈ { , , . . . , ⌊ ( s + d − / ⌋} are all incongruent modulo 2( s + d ) except a + d j ≡ s + d (mod 2( s + d )). First, for 0 ≤ j < j ≤ ⌊ ( s + d − / ⌋ , it is clear that a + d j and a + d j are incongruent modulo2( s + d ). Now, suppose that a + d j ≡ s + d ) − a − d j (mod 2( s + d )) for some 0 ≤ j , j ≤ ⌊ ( s + d − / ⌋ . Itfollows that 2 a + d ( j + j ) is a multiple of 2( s + d ). We consider two cases according to the parity of s :If s is odd, a is supposed to be − s so that − s + d ( j + j ) is a multiple of 2( s + d ). It follows that d ( j + j + s + d . Since s and d are relatively prime, the only possibility is that j = j = ( s + d − / d iseven. In this case, − s + d ( s + d − / ≡ s + d (mod 2( s + d )).If s is even, a is supposed to be − s − d so that − s − d + d ( j + j ) is a multiple of 2( s + d ). It follows that d ( j + j ) is a multiple of s + d . Since s and d are relatively prime, the only possibility is that j = j =
0. Note that − s − d ≡ s + d (mod 2( s + d )).This completes the proof of the claim. From the claim we have that, for every odd integer h , there exists j ∈{ , , . . . , ⌊ ( s + d − / ⌋} such that h is congruent to a + d j or 2( s + d ) − a − d j modulo 2( s + d ). In addition, if h . s + d (mod 2( s + d )), then there exists a unique position labeled by h or − h in the ( s + d , d ; a )-abacus diagram.By using the following proposition, we investigate the properties of the ( s + d , d ; a )-abacus of a self-conjugate( s , s + d , . . . , s + pd )-core partition according to the parity of s and d . Proposition 2.3. [11, Proposition 3] Let λ be a self-conjugate partition. Then λ is an s-core if and only if both of thefollowing hold:(a) If h ∈ MD ( λ ) with h > s, then h − s ∈ MD ( λ ) .(b) If h , h ∈ MD ( λ ) , then h + h . s ) . Corollary 2.4. If λ is a self-conjugate ( s , t ) -core partition, then s + t < MD ( λ ) .Proof. We may assume that s < t . Suppose that s + t ∈ MD ( λ ), it follows from Proposition 2.3 (a) that t − s ∈ MD ( λ ).But this gives a contradiction to Proposition 2.3 (b) since ( s + t ) + ( t − s ) = t . Thus, s + t < MD ( λ ).For the ( s + d , d ; a )-abacus of a self-conjugate ( s , s + d , . . . , s + pd )-core partition λ with p ≥
2, let r ( j ) denotethe row number such that position ( r ( j ) , j ) is labeled by a positive integer while position ( r ( j ) − , j ) is labeled by anegative integer. Lemma 2.5.
Let λ be a self-conjugate partition. For relatively prime positive integers s and d, if λ is an ( s , s + d , . . . , s + pd ) -core, then the ( s + d , d ; a ) -abacus of λ satisfies the following.(a) If a bead is placed on position ( i , j ) such that i > r ( j ) , then a bead is also placed on each of positions ( i − , j ) , ( i − , j ) , . . . , ( r ( j ) , j ) .(b) If a bead is placed on position ( i , j ) such that i < r ( j ) − , then a bead is also placed on each of positions ( i + , j ) , ( i + , j ) , . . . , ( r ( j ) − , j ) .(c) A bead can be placed on at most one of the two positions ( r ( j ) , j ) and ( r ( j ) − , j ) .Proof. Fix a column number j .(a) A bead is placed on position ( i , j ) with i > r ( j ) means that a + s + d ) i + d j ∈ MD ( λ ). Since λ is an( s + d )-core, a + s + d )( i − + d j belongs to MD ( λ ) by Proposition 2.3 (a). In a similar way, we also have a + s + d )( i − + d j , . . . , a + s + d ) r ( j ) + d j ∈ MD ( λ ). Hence, a bead is placed on each of positions( i − , j ) , ( i − , j ) , . . . , ( r ( j ) , j ).(b) A bead is placed on position ( i , j ) such that i < r ( j ) − − a − s + d ) i − d j ∈ MD ( λ ). Again, it followsfrom Proposition 2.3 (a) that − a − s + d )( i + − d j belongs to MD ( λ ). In a similar way, we also have − a − s + d )( i + − d j , . . . , − a − s + d )( r ( j ) − − d j ∈ MD ( λ ). Hence, a bead is placed on each ofpositions ( i + , j ) , ( i + , j ) , . . . , ( r ( j ) − , j ).(c) By Proposition 2.3 (b), at most one of a + s + d ) r ( j ) + d j and − a − s + d )( r ( j ) − − d j belongs to MD ( λ )since the sum of those two numbers is 2( s + d ). Therefore, a bead can be placed on at most one of the positions( r ( j ) , j ) and ( r ( j ) − , j ). 4or p ≥
2, let λ be a self-conjugate ( s , s + d , . . . , s + pd )-core partition. In order to explain the properties of the( s + d , d ; a )-abacus of λ more simply, we define the ( s + d , d ; a ) -abacus function of λ f : { , , . . . , ⌊ ( s + d − / ⌋} → Z as follows: If a bead is placed on a position in column j being labeled by a positive integer, then f ( j ) is defined to bethe largest number i such that a bead is placed on position ( i , j ); Otherwise, f ( j ) is defined to be the largest number i such that position ( i , j ) is a spacer being labeled by a negative integer.The following proposition gives some basic properties of the ( s + d , d ; a )-abacus function of a self-conjugate( s , s + d , . . . , s + pd )-core partition. Proposition 2.6.
Let s and d be relatively prime positive integers. For p ≥ , if λ is a self-conjugate ( s , s + d , . . . , s + pd ) -core partition, then the ( s + d , d ; a ) -abacus function f of λ satisfies the following.(a) f (0) = .(b) f ( j − is equal to one of the three values f ( j ) − , f ( j ) , and f ( j ) + , for j = , . . . , ( s + d − / .(c) If p ≥ and f ( j − = f ( j ) − , then f ( j − p + , f ( j − p + , . . ., f ( j − ≥ f ( j − , for j = p − , . . . , ( s + d − / .Proof. We consider the ( s + d , d ; a )-abacus of λ .(a) Since position (0 ,
0) is labeled by a = − s (resp. a = − s − d ) when s is odd (resp. even) and position (1 ,
0) islabeled by s + d (resp. s + d ), both of positions are spacers and r (0) =
1. It follows from Lemma 2.5 that thereis no bead in column 0. Hence, f (0) = j , let f ( j ) = i .Suppose that a bead is placed on position ( i , j ) so that it is labeled by a positive integer. We first show that f ( j − ≥ f ( j ) − i − , j − i − , j −
1) is labeled by a positive integer,then a bead is placed on position ( i − , j −
1) by Proposition 2.3 (a) as λ is an ( s + d )-core. Otherwise, ifposition ( i − , j −
1) is labeled by a negative integer, then position ( i − , j −
1) is a spacer by Proposition 2.3 (b).In any case, it follows from the definition of f that f ( j − ≥ f ( j ) −
1. Now, we show that f ( j − ≤ f ( j ) + i + , j − f ( j ) = i and a bead is placed on position ( i , j ), position ( i + , j )is a spacer. Therefore, position ( i + , j −
1) is a spacer by Proposition 2.3 (a) as λ is an s -core. Hence, f ( j − ≤ f ( j ) + i , j ) is a spacer so that it is labeled by a negative integer. Since position ( i − , j − λ is an ( s + d )-core. Therefore, f ( j − ≥ f ( j ) −
1. To completethe proof, we assume that f ( j − ≥ i +
2. If position ( i + , j −
1) is labeled by a positive integer, then a bead isplaced on this position by Lemma 2.5 (a). In this case, either a bead is placed on position ( i + , j ) being labeledby a positive integer or position ( i + , j ) is a spacer being labeled by a negative integer by Proposition 2.3 (a)and (b) as λ is an s -core. It contradicts to f ( j ) = i . Otherwise, if position ( i + , j −
1) is labeled by a negativeinteger, then it is a spacer. Therefore, position ( i + , j ) is a spacer by Proposition 2.3 (a). Also, it contradicts to f ( j ) = i . Hence, f ( j − ≤ f ( j ) + j , let f ( j ) = i . It su ffi ces to show that f ( j − k ) ≥ f ( j ) −
1, for k = , . . . , p −
1. Note that λ isan ( s + ( k + d )-core partition. If position ( i − , j − k ) is labeled by a negative integer, then it is a spacerregardless of the presence of a bead on position ( i , j ) by Proposition 2.3 (a) and (b). If position ( i − , j − k )is labeled by a positive integer, then a bead is placed on position ( i , j ) being labeled by a positive integer. ByProposition 2.3 (a), a bead is placed on position ( i − , j − k ). In any case, we conclude that f ( j − k ) ≥ f ( j ) − s + d , d ; a )-abacus function f .In the following three subsections, we cover other properties which is depending on the parity of s and d andgive several examples of the ( s + d , d ; a )-abacus diagram and the ( s + d , d ; a )-abacus function of a self-conjugate( s , s + d , . . . , s + pd )-core partition. Note that we do not need to consider the case when both of s and d are evenbecause s and d are supposed to be relatively prime positive integers.5 .1. Self-conjugate (s,s + d,. . . ,s + pd)-cores with odd s and even d First, we consider the case where s is odd and d is even. In the following proposition, we give additional propertiesof the ( s + d , d ; a )-abacus function f of a self-conjugate ( s , s + d , . . . , s + pd )-core partition λ , where s is odd and d iseven. Recall that a = a ( s , d ) = − s in this case. Proposition 2.7.
Let s and d be relatively prime positive integers such that s is odd and d is even. For p ≥ , if λ is aself-conjugate ( s , s + d , . . . , s + pd ) -core partition, then the ( s + d , d ; − s ) -abacus function f of λ satisfies the following.(a) f (( s + d − / = − d / .(b) If p ≥ , then f (( s + d − / − k − ≥ − d / , for k = , , . . . , ⌊ ( p − / ⌋ .(c) If p ≥ , then f ( ℓ + ≤ , for ℓ = , , . . . , ⌊ ( p − / ⌋ .Proof. (a) Since position ( − d / , ( s + d − /
2) is labeled by − s − d and position ( − d / + , ( s + d − /
2) is labeledby s + d , both of them are spacers. By Lemma 2.5 (a) and (b), there is no bead in column ( s + d − /
2. Hence, f (( s + d − / = − d / s + d , d ; − s )-abacus function f of λ .(b) Note that position ( − d / , ( s + d − / − k −
1) is a spacer because it is labeled by − s − (2 k + d and λ is an( s + (2 k + d )-core, where k = , , . . . , ⌊ ( p − / ⌋ . Hence, f (( s + d − / − k − ≥ − d / , ℓ +
1) is a spacer because it is labeled by s + (2 ℓ + d and λ is an ( s + (2 ℓ + d )-core,where ℓ = , , . . . , ⌊ ( p − / ⌋ . Hence, f ( ℓ + ≤ Example 2.8.
Let λ be the self-conjugate partition with MD ( λ ) = { , , , , , , , } . It follows from Propo-sition 2.3 that λ can be considered as a (21 , , , , -core partition. Figure 2 shows the (25 , − -abacus of λ and the path obtained by connecting each pair of the two points ( j − , f ( j − and ( j , f ( j )) with a straight linesegment for j = , . . . , . Note that the (25 , − -abacus function f of λ is given byf (0) = f (1) = , f (2) = − , f (3) = f (4) = f (5) = , f (6) = , f (7) = , f (8) = − , f (9) = − , f (10) = − , f (11) = f (12) = − . Indeed, one can see that the (25 , , − -abacus function f of λ agrees with all the properties given in Lemma 2.5 andPropositions 2.6 and 2.7. − − − / j 0 1 2 3 4 5 6 7 8 9 10 11 12 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − ...... Figure 2: The (25 , − λ with MD ( λ ) = { , , , , , , , } Remark 2.9.
For relatively prime positive integers s and d such that s is odd and d is even, let f be a functionsatisfying all the conditions in Propositions 2.6 and 2.7. We remark that there exist a unique self-conjugate ( s , s + d , . . . , s + pd ) -core partition λ such that f is the ( s + d , d ; − s ) -abacus function of λ by Lemma 2.5 and the constructionof f . From this fact together with Proposition 2.2, we can conclude that there is a one-to-one correspondence betweenthe set of self-conjugate ( s , s + d , . . . , s + pd ) -core partitions and the set of functions satisfying all the conditions inPropositions 2.6 and 2.7. .2. Self-conjugate ( s , s + d , . . . , s + pd ) -cores with odd s and odd d Now, we consider the case where both of s and d are odd. Let λ be a self-conjugate partition. The followingproposition gives several additional properties of the ( s + d , d ; a )-abacus function f of λ , where both of s and d areodd. In this case, a = a ( s , d ) = − s as well. Proposition 2.10.
Let s and d be relatively prime positive integers such that both of s and d are odd. For p ≥ , if λ is a self-conjugate ( s , s + d , . . . , s + pd ) -core partition, then the ( s + d , d ; − s ) -abacus function f of λ satisfies thefollowing.(a) f (( s + d − / = − ( d − / or − ( d + / .(b) If p ≥ , then f (( s + d − / − k − ≥ − ( d + / , for k = , , . . . , p − .(c) If p ≥ , then f ( ℓ + ≤ , for ℓ = , , . . . , ⌊ ( p − / ⌋ .Proof. (a) Since position ( − ( d − / , ( s + d − /
2) is labeled by − d , position ( − ( d − / + , ( s + d − / s + d , and position ( − ( d − / − , ( s + d − /
2) is labeled by − s − d . By Corollary 2.4,2 s + d = s + ( s + d ) , s + d = ( s + d ) + ( s + d ) < MD ( λ ). It follows from Lemma 2.5 that there is at most onebead which is labeled by − d in column ( s + d − /
2. Hence, f (( s + d − / = − ( d − / − ( d + / − ( d + / , ( s + d − / − k −
1) is a spacer because it is labeled by − s − (2 k + d = −{ s + ( k + d } + { s + ( k + d } and λ is an ( s + ( k + d , s + ( k + d )-core, where k = , , . . . , p −
3. Hence, f (( s + d − / − k − ≥ − ( d + / , ℓ +
1) is a spacer because it is labeled by s + (2 ℓ + d and λ is an ( s + (2 ℓ + d )-core, where ℓ = , , . . . , ⌊ ( p − / ⌋ . Hence, f ( ℓ + ≤ Example 2.11.
Let µ be the self-conjugate partition with MD ( µ ) = { , , , , , , , , , , } . It followsfrom Proposition 2.3 that µ can be considered as a (23 , , , -core partition. Figure 3 shows the (26 , − -abacus of µ and the path obtained by connecting each pair of the two points ( j − , f ( j − and ( j , f ( j )) with astraight line segment for j = , . . . , . Note that the (26 , − -abacus function f of µ is given byf (0) = f (1) = , f (2) = − , f (3) = f (4) = f (5) = , f (6) = , f (7) = , f (8) = − , f (9) = − , f (10) = − , f (11) = f (12) = − , and the (26 , , − -abacus function f of µ agrees with all the properties given in Lemma 2.5 and Propositions 2.6and 2.10. − − − / j 0 1 2 3 4 5 6 7 8 9 10 11 12 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − ...... Figure 3: The (26 , − µ with MD ( µ ) = { , , , , , , , , , , } emark 2.12. For relatively prime positive integers s and d such that both s and d are odd, let f be a functionsatisfying all the conditions in Propositions 2.6 and 2.10. We remark that there exist a unique self-conjugate ( s , s + d , . . . , s + pd ) -core partition λ such that f is the ( s + d , d ; − s ) -abacus function of λ by Lemma 2.5 and the constructionof f . From this fact together with Proposition 2.2, we can conclude that there is a one-to-one correspondence betweenthe set of self-conjugate ( s , s + d , . . . , s + pd ) -core partitions and the set of functions satisfying all the conditions inPropositions 2.6 and 2.10.2.3. Self-conjugate ( s , s + d , . . . , s + pd ) -cores with even s and odd d Finally, we consider the case where s is even and d is odd. We give additional properties of the ( s + d , d ; a )-abacusfunction f of a self-conjugate ( s , s + d , . . . , s + pd )-core partition λ , where s is even and d is odd so that a = − ( s + d ). Proposition 2.13.
Let s and d be relatively prime positive integers such that s is even and d is odd. For p ≥ , if λ is a self-conjugate ( s , s + d , . . . , s + pd ) -core partition, then the ( s + d , d ; − s − d ) -abacus function f of λ satisfies thefollowing.(a) f (( s + d − / = − ( d − / or − ( d + / .(b) If p ≥ , then f (( s + d − / − k − ≥ − ( d + / , for k = , , . . . , p − .(c) If p ≥ , then f ( ℓ + ≤ , for ℓ = , , . . . , ⌊ ( p − / ⌋ .Proof. (a) Since position ( − ( d − / , ( s + d − /
2) is labeled by − d , position ( − ( d − / + , ( s + d − / s + d , and position ( − ( d − / − , ( s + d − /
2) is labeled by − s − d . By Corollary 2.4,2 s + d = s + ( s + d ) , s + d = ( s + d ) + ( s + d ) < MD ( λ ). It follows from Lemma 2.5 that there is at most onebead which is labeled by − d in column ( s + d − /
2. Hence, f (( s + d − / = − ( d − / − ( d + / − ( d + / , ( s + d − / − k −
1) is a spacer because it is labeled by − s − (2 k + d = −{ s + ( k + d } + { s + ( k + d } and λ is an ( s + ( k + d , s + ( k + d )-core, where k = , , . . . , p −
2. Hence, f (( s + d − / − k − ≥ − ( d + / , ℓ +
1) is a spacer because it is labeled by s + (2 ℓ + d and λ is an ( s + (2 ℓ + d )-core, where ℓ = , , . . . , ⌊ ( p − / ⌋ . Hence, f ( ℓ + ≤ Example 2.14.
Let ν be the self-conjugate partition with MD ( ν ) = { , , , , , , , , , } . It follows fromProposition 2.3 that ν can be considered as a (22 , , , -core partition. Figure 4 shows the (25 , − -abacusof ν and the path obtained by connecting each pair of the two points ( j − , f ( j − and ( j , f ( j )) with a straight linesegment for j = , . . . , . Note that the (25 , − -abacus function f of ν is given byf (0) = f (1) = , f (2) = − , f (3) = f (4) = f (5) = , f (6) = , f (7) = , f (8) = − , f (9) = − , f (10) = − , f (11) = f (12) = − , and the (25 , , − -abacus function f of ν agrees with all the properties given in Lemma 2.5 and Propositions 2.6and 2.13. Remark 2.15.
For relatively prime positive integers s and d such that s is even and d is odd, let f be a functionsatisfying all the conditions in Propositions 2.6 and 2.13. We remark that there exist a unique self-conjugate ( s , s + d , . . . , s + pd ) -core partition λ such that f is the ( s + d , d ; − s − d ) -abacus function of λ by Lemma 2.5 and the constructionof f . From this fact together with Proposition 2.2, we can conclude that there is a one-to-one correspondence betweenthe set of self-conjugate ( s , s + d , . . . , s + pd ) -core partitions and the set of functions satisfying all the conditions inPropositions 2.6 and 2.13.
3. Free rational Motzkin paths of type ( s , t ) with restrictions In this section, we give a lattice path interpretation of self-conjugate ( s , s + d , . . . , s + pd )-core partitions.8 − − − / j 0 1 2 3 4 5 6 7 8 9 10 11 12 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − ...... Figure 4: The (25 , − ν with MD ( ν ) = { , , , , , , , , , } For relatively prime positive integers s and d , we construct a mapping φ ( s + d , d ) : SC ( s , s + d , s + d ) → F ( ⌊ s / ⌋ + ⌈ d / ⌉ , −⌈ d / ⌉ )associated with the ( s + d , d ; a )-abacus function f of λ ∈ SC ( s , s + d , s + d ) as follows: First, for convenience, we set f ( ⌊ s / ⌋ + ( d + / = − ( d + / d is odd. The path φ ( s + d , d ) ( λ ) starts from (0 ,
0) and its j th step is (1 , f ( j ) − f ( j − j = , . . . , ⌊ s / ⌋ + ⌈ d / ⌉ .From our first setting and Propositions 2.7 (a) and 2.10 (a), the last step of the path is either D or F when d is odd.From this fact together with Proposition 2.6 (b), we can say that the path φ ( s + d , d ) ( λ ) consist of up steps U = (1 , D = (1 , − F = (1 , f (0) = f ( ⌊ s / ⌋ + ⌈ d / ⌉ ) = −⌈ d / ⌉ by Proposition 2.7 (a) and by our first setting, we conclude that φ ( s + d , d ) ( λ ) is a free rational Motzkin path of type( ⌊ s / ⌋ + ⌈ d / ⌉ ) , −⌈ d / ⌉ ). Hence, the mapping φ ( s + d , d ) is well-defined. We note that it is possible that the two paths φ ( s + d , d ) ( µ ) and φ ( s + d , d ) ( ν ) are the same, while φ ( s + d , d ) is injective for fixed s and d .In Remarks 2.9, 2.12, and 2.15, for given appropriate integers s , d , and p , we showed that there is a one-to-onecorrespondence between the set SC ( s , s + d ,..., s + pd ) and the set of functions f with necessary conditions. Now, we areready to prove our main result. Proof of Theorem 1.7.
It follows almost directly from the construction of φ ( s + d , d ) and Propositions 2.6, 2.7, 2.10, and2.13. It is clear that φ ( s + d , d ) ( λ ) ∈ F ( ⌊ s / ⌋ + ⌈ d / ⌉ , −⌈ d / ⌉ ) for λ ∈ SC ( s , s + d ,..., s + pd ) . By Proposition 2.6 (c), p ≥ f ( j − = f ( j ) − f ( j − p + , f ( j − p + , . . . , f ( j − ≥ f ( j − p − ≤ j ≤ ( s + d − /
2. Itfollows from the construction of φ ( s + d , d ) that φ ( s + d , d ) ( λ ) does not contain UF i U steps for all i = , , . . . , p − p ≥ f (( s + d − / − k − ≥ − d /
2, for k = , , . . . , ⌊ ( p − / ⌋ if p ≥
3. It follows from the construction of φ ( s + d , d ) that φ ( s + d , d ) ( λ ) never ends with UF k steps for k = , , . . . , ⌊ ( p − / ⌋ if p ≥
3. By Proposition 2.7 (c), f ( ℓ + ≤
0, for all ℓ = , , . . . , ⌊ ( p − / ⌋ if p ≥
4. This implies that φ ( s + d , d ) ( λ )cannot start with F j U steps for all j = , , . . . , ⌊ ( p − / ⌋ if p ≥
4. Note that (b) and (c) can be proved in a similarmanner by using Propositions 2.10 (b) and (c), and 2.13 (b) and (c), respectively.
Example 3.1.
Figure 5 shows the corresponding free rational Motzkin paths associated with the ( s + d , d ; a ) -abacusfunctions of the self-conjugate partitions λ , µ , and ν in the previous examples. Let P = φ (25 , ( λ ) . Then P = FDUFFUDDDDUF ∈ F (12 , − is a path satisfying that i) P starts with no U, ii) P ends with no U or UF, iii) P hasno UU or UFU as a consecutive subpath. Similarly, one can check that φ (26 , ( µ ) = φ (25 , ( ν ) = FDUFFUDDDDUFF ∈ F (13 , − satisfies conditions given in Theorem 1.7. From Theorem 1.7, we easily get the following result.
Corollary 3.2.
Let s and d be relatively prime positive integers. If d is odd and s , p are even, then the number ofself-conjugate ( s , s + d , . . . , s + pd ) -core partitions is equal to that of ( s + , s + d + , . . . , s + pd + -core partitions. φ (25 , ( λ ) xy φ (26 , ( µ ) and φ (25 , ( ν ) xy Figure 5: The corresponding free rational Motzkin paths of λ , µ , and ν ( s , s + d , s + d ) -core partitions and self-conjugate ( s , s + d , s + d , s + d ) -core partitions In particular, we give a closed formula for the number of self-conjugate ( s , s + d , s + d )-core partitions. Theorem 3.3.
Let s and d be relatively prime positive integers. The number of self-conjugate ( s , s + d , s + d ) -corepartitions is given by ⌊ s ⌋ X i = s + d − i , d + i , s − − i ! , if d is even; ⌊ s ⌋ X i = ⌊ s + d − ⌋⌊ i ⌋ , ⌊ d + i ⌋ , ⌊ s ⌋ − i ! , if d is odd.Proof. If s = r + d = c , then the number of self-conjugate ( s , s + d , s + d )-core partitions is equal to thenumber of free rational Motzkin paths of type ( r + c , − c ) by Theorem 1.7. For 0 ≤ i ≤ ⌊ r / ⌋ , the number of freerational Motzkin paths of type ( r + c , − c ) having i up steps (so that it has c + i down steps and r − i flat steps) is givenby (cid:16) r + ci , c + i , r − i (cid:17) . Hence, the number of free rational Motzkin paths of type ( r + c , − c ) is given by ⌊ r ⌋ X i = r + ci , c + i , r − i ! = ⌊ s ⌋ X i = s + d − i , d + i , s − − i ! . On the other hand, Theorem 1.7 says that if s = r + d = c −
1, then the number of self-conjugate( s , s + d , s + d )-core partitions is equal to the number of free rational Motzkin paths of type ( r + c , − c ) for whichends with either a down step or a flat step. Since the number of free rational Motzkin paths of type ( r + c , − c ) with k up steps for which ends with a down (resp. flat) step is given by (cid:16) r + c − k , c + k − , r − k (cid:17) (resp. (cid:16) r + c − k , c + k , r − k − (cid:17) ), the total number ofsuch paths is ⌊ r ⌋ X k = r + c − k , c + ( k − , r − k ! + ⌊ r − ⌋ X k = r + c − k , c + k , r − (2 k + ! = r X i = r + c − ⌊ i ⌋ , c + ⌊ i − ⌋ , r − i ! . It follows from Corollary 3.2 that if s = r and d = c −
1, then the number of self-conjugate ( s , s + d , s + d )-corepartitions is also given by r X i = r + c − ⌊ i ⌋ , c + ⌊ i − ⌋ , r − i ! . Hence, we conclude that for odd d , the number of self-conjugate ( s , s + d , s + d )-core partitions is given by ⌊ s ⌋ X i = ⌊ s + d − ⌋⌊ i ⌋ , ⌊ d + i ⌋ , ⌊ s ⌋ − i ! . This completes the proof 10ndeed, the result by putting d = s , s + d , s + d , s + d )-core partitions. Theorem 3.4.
Let s and d be relatively prime positive integers. The number of self-conjugate ( s , s + d , s + d , s + d ) -core partitions is given by ⌊ s ⌋ X i = s + d − − i s − − i ! s + d − − ii ! , if d is even; ⌊ s ⌋ X i = ⌊ s + d − ⌋ − ⌊ i ⌋⌊ s ⌋ − i ! ⌊ s + d ⌋ − ⌊ i + ⌋⌊ i ⌋ ! , if d is odd.Proof. First, we consider the case where s = r + d = c . It follows from Theorem 1.7 that the number ofself-conjugate ( s , s + d , s + d , s + d )-core partitions is equal to the number of free rational Motzkin paths of type( r + c , − c ) for which ends with no U and has no UU as a consecutive subpath. Among these corresponding paths,we focus on the paths P with i up steps. A path P can be obtained as follows: For a given path Q = Q · · · Q r + c − i consisting of c + i down steps and r − i flat steps, insert i up steps in Q satisfying that i) there is at most one up stepbefore Q ; ii) there is at most one up step between Q j and Q j + for j = , . . . , r + c − i −
1; iii) there is no up stepafter Q r + c − i . Then we have a free rational Motzkin paths of type ( r + c , − c ) for which ends with no U and has no UU as we desired. Note that there are (cid:16) r + c − ir − i (cid:17) ways to choose Q and there are (cid:16) r + c − ii (cid:17) ways to insert i up steps satisfying theconditions. Hence, the total number of such P ’s is given by ⌊ r ⌋ X i = r + c − ir − i ! r + c − ii ! = ⌊ s ⌋ X i = s + d − − i s − − i ! s + d − − ii ! . Now, we consider the case where s = r + d = c −
1. Theorem 1.7 gives that the number of self-conjugate( s , s + d , s + d , s + d )-core partitions is equal to the number of free rational Motzkin paths of type ( r + c , − c ) for whichends with no U or UF and has no UU as a consecutive subpath. Among these corresponding paths, we focus on thepaths P with k up steps as well. In this case, a path P can be obtained as follows: For a given path Q = Q · · · Q r + c − k consisting of c + k down steps and r − k flat steps, insert k up steps in Q satisfying that i) there is at most one up stepbefore Q ; ii) there is at most one up step between Q j and Q j + for j = , . . . , r + c − k −
1; iii) there is no up stepafter Q r + c − k ; in addition, iv) there is no up step between Q r + c − k − and Q r + c − k if Q ends with a flat step. Note that thereare (cid:16) r + c − k − r − k (cid:17) ways to choose Q ending with a down step and there are (cid:16) r + c − kk (cid:17) ways to insert k up steps satisfying theconditions. On the other hand, there are (cid:16) r + c − k − r − k − (cid:17) ways to choose Q ending with a flat step and there are (cid:16) r + c − k − k (cid:17) waysto insert k up steps satisfying the conditions. Hence, the total number of such P ’s is given by ⌊ r ⌋ X k = r + c − k − r − k ! r + c − kk ! + ⌊ r − ⌋ X k = r + c − k − r − (2 k + ! r + c − k − k ! = r X i = r + c − ⌊ i ⌋ − r − i ! r + c − ⌊ i + ⌋⌊ i ⌋ ! . When s = r and d = c −
1, the paths P can be obtained in a similar way. In this case, the paths are not supposedto start with U . Except for this, all conditions are the same as for the case where s = r − d = c −
1. Hence,the number of ( s , s + d , s + d , s + d )-core partitions is given by ⌊ r ⌋ X k = r + c − k − r − k ! r + c − k − k ! + ⌊ r − ⌋ X k = r + c − k − r − (2 k + ! r + c − k − k ! = r X i = r + c − ⌊ i ⌋ − r − i ! r + c − ⌊ i + ⌋ − ⌊ i ⌋ ! . We note that for odd d , the number of self-conjugate ( s , s + d , s + d , s + d )-cores can be written as ⌊ s ⌋ X i = ⌊ s + d − ⌋ − ⌊ i ⌋⌊ s ⌋ − i ! ⌊ s + d ⌋ − ⌊ i + ⌋⌊ i ⌋ ! . .3. Self-conjugate ( s , s + , . . . , s + p ) -core partitions with m corners For a partition λ , the number of corners in the Young diagram of λ is equal to the number of distinct parts in λ .Huang-Wang [12] proved that the number of ( s , s + m corners is equal to the Narayana number N ( s , m + = s (cid:16) sm + (cid:17)(cid:16) sm (cid:17) , and the number of ( s , s + , s + m corners is equal to (cid:16) s m (cid:17) C m , where C m is the m th Catalan number. In [7], the authors extended this result. Corollary 3.5. [7, Corollary 3.1] For positive integers s, p ≥ , and ≤ m ≤ ⌊ s / ⌋ , the number of ( s , s + , . . . , s + p ) -core partitions with m corners is r X ℓ = N ( m , ℓ + s − ℓ ( p − m ! , where r = min( m − , ⌊ ( s − m ) / ( p − ⌋ ) . In this subsection, we focus on self-conjugate ( s , s + , . . . , s + p )-core partitions with m corners. For simplicity,let F ( s , p ) : = φ ( s + , ( SC ( s , s + ,..., s + p ) ). Recall that, for positive integers s and p ≥
2, the set F ( s , p ) consists of paths P ∈ F ( ⌊ s / ⌋ + , −
1) satisfying that i) P has no UF i U as a consecutive subpath for all i = , , . . . , p − p ≥
3, ii) P starts with no F j U for all j = , , . . . , ⌊ ( p − / ⌋ if p ≥ j = , , . . . , ⌊ ( p − / ⌋ if p ≥ P ends withno UF k for all k = , , . . . , p − s is even (resp. odd). As a corollary of Theorem 1.7, we have a one-to-onecorrespondence between the set SC ( s , s + ,..., s + p ) and the set F ( s , p ). Now we refine this one-to-one correspondenceaccording to the number of corners in a self-conjugate ( s , s + , . . . , s + p )-core partition. For A ∈ { F , D } , let F A ( s , p )denote the set of paths in the set F ( s , p ) for which ends with a step A . Lemma 3.6.
For positive integers s and p ≥ , the mapping φ ( s + , gives a one-to-one correspondence between theset of self-conjugate ( s , s + , . . . , s + p ) -core partitions with an even (resp. odd) number of corners and the set F D ( s , p ) (resp. F F ( s , p ) ).Proof. Let λ be a self-conjugate ( s , s + , . . . , s + p )-core partition and let P = φ ( s + , ( λ ). We first note that λ has anodd number of corners if and only if 1 ∈ MD ( λ ).Let r = ⌊ s / ⌋ . Note that, for the ( s + , − r − , r ) is labeled by − r where beads are allowed to be placed. Hence, for the ( s + , − r − f of λ , the value f ( r ) is either − ∈ MD ( λ ) or 0 if 1 < MD ( λ ). It follows from the construction of P , P ends witheither a down step if 1 < MD ( λ ) or a flat step if 1 ∈ MD ( λ ). This completes the proof. Lemma 3.7.
Let s, p, and D be positive integers with D ≥ . For a self-conjugate ( s , s + , . . . , s + p ) -core partition λ with MD ( λ ) = { d , d , . . . , d D } , where d > d > · · · > d D ≥ , let ˜ λ denote the self-conjugate partition withMD ( ˜ λ ) = { d , . . . , d D } . If P = φ ( s + , ( λ ) and ˜ P = φ ( s + , ( ˜ λ ) , then we have the following.(a) If d = d + , then P and ˜ P have the same number of flat steps.(b) If d > d + , then P has two less flat steps than ˜ P.Proof. If s is even, it can be proven in a similar way to odd, so we only prove the odd case here. Let s = r + s + , − s − i , j ) be labeled by either d or − d and let f and ˜ f be the( s + , − s − λ and ˜ λ , respectively. We note that 1 ≤ j ≤ r − f ( x ) = ˜ f ( x ) for all0 ≤ x ≤ r + x = j in any case.We first suppose that position ( i , j ) is labeled by d so that position ( i , j −
1) is labeled by d − • When d = d + f ( j − = f ( j ) = i , f ( j + = i − f ( j − = i , ˜ f ( j ) = ˜ f ( j + = i −
1. Hence, thepaths can be written as P = P · · · P j − FDP j + · · · P r + and ˜ P = P · · · P j − DFP j + · · · P r + . • When d > d + f ( j − = i − , f ( j ) = i , f ( j + = i − f ( j − = ˜ f ( j ) = ˜ f ( j + = i −
1. In thiscase, the paths can be written as P = P · · · P j − UDP j + · · · P r + and ˜ P = P · · · P j − FFP j + · · · P r + .We now suppose that position ( i , j ) is labeled by − d so that position ( i , j +
1) is labeled by − ( d − • When d = d + f ( j − = i , f ( j ) = f ( j + = i − f ( j − = ˜ f ( j ) = i , ˜ f ( j + = i −
1. Hence, thepaths can be written as P = P · · · P j − DFP j + · · · P r + and ˜ P = P · · · P j − FDP j + · · · P r + .12 When d > d + f ( j − = i , f ( j ) = i − , f ( j + = i and ˜ f ( j − = ˜ f ( j ) = ˜ f ( j + = i . In this case, thepaths can be written as P = P · · · P j − DUP j + · · · P r + and ˜ P = P · · · P j − FFP j + · · · P r + .This completes the proof. Theorem 3.8.
For positive integers s and p ≥ , the mapping φ ( s + , gives a one-to-one correspondence between theset of self-conjugate ( s , s + , . . . , s + p ) -core partitions with an even (resp. odd) number m of corners and the set ofpaths in F D ( s , p ) (resp. F F ( s , p ) ) having ⌊ s / ⌋ − m (resp. ⌊ s / ⌋ − m + ) flat steps.Proof. Let r = ⌊ s / ⌋ . For a self-conjugate partition λ with MD ( λ ) = { d , . . . , d D } , where d > · · · > d D , we denote theself-conjugate partition with MD ( ˜ λ ) = { d , . . . , d D } by ˜ λ .From Lemma 3.6, we have learned that there is a one-to-one correspondence between the set of self-conjugate( s , s + , . . . , s + p )-core partitions with an even (resp. odd) number of corners and the set F D ( s , p ) (resp. F F ( s , p )).To prove the remainder of the theorem, we claim that if λ is a self-conjugate ( s , s + , . . . , s + p )-core partition withan even (resp. odd) number of corners, say m , then its corresponding path P = φ ( s + , ( λ ) has r − m (resp. r − m + | MD ( λ ) | . Let f be the ( s + , − r − λ .If | MD ( λ ) | =
0, then λ is an empty partition so that it has 0 corner. Since there is no bead on the ( s + , − r − λ , f ( j ) = j except f ( r + = −
1. Hence, the path P = F · · · FD has r − | MD ( λ ) | =
1. Let λ be a self-conjugate ( s , s + , . . . , s + p )-core partition with MD ( λ ) = { d } . We note that d is less than 2 s , because that d − s ∈ MD ( λ ) otherwise. Hence, if position ( i , j ) is labeledby either d or − d , then i must be either 0 or 1. First, we suppose that d = j = r and f (0) = · · · = f ( r − = f ( r ) = f ( r + = −
1. In this case, λ = (1) has one corner and the corresponding path P = F · · · FDF has r − + d ,
1, then λ has two corners and its corresponding path can be written as either P = F · · · FUDF · · · FD or P = F · · · FDUF · · · FD so that it has r − λ with | MD ( λ ) | = D − D ≥
2. Let λ be a self-conjugate( s , s + , . . . , s + p )-core partition with m corners and MD ( λ ) = { d , . . . , d D } , where d > · · · > d D . When d = d + λ has m corners and its corresponding path ˜ P has r − m (resp. r − m +
1) flat steps if m is even (resp. odd) by theinduction hypothesis. It follows from Lemma 3.7 (a) that P also has r − m (resp. r − m +
1) flat steps if m is even (resp.odd). When d > d +
2, ˜ λ has m − P has r − m + r − m +
3) flat steps if m is even (resp. odd) by the induction hypothesis. It follows from Lemma 3.7 (b) that P has r − m (resp. r − m + m is even (resp. odd). This completes the proof of the claim.In particular, we obtain formulae for the number of self-conjugate ( s , s + , . . . , s + p )-core partitions with m cornersfor p = Proposition 3.9.
The number of self-conjugate ( s , s + , s + -core partitions with m corners is given by ⌊ s ⌋⌊ m ⌋ , ⌊ m + ⌋ , ⌊ s ⌋ − m ! . Proof.
Let r = ⌊ s / ⌋ . By Theorem 3.8, if m is even (resp. odd), then the number of self-conjugate ( s , s + , s + m corners is equal to the number of free rational Motzkin paths of type ( r + , −
1) with r − m (resp. r − m +
1) flat steps for which ends with a down (resp. flat) step. Now, we enumerate these Motzkin paths.When m = k , each of the paths consists of k up steps, k + r − k flat steps and ends with a downstep. Therefore, the number of such paths is (cid:16) rk , k , r − k (cid:17) .When m = k +
1, each of the paths consists of k up steps, k + r − k flat steps and ends with aflat step. Therefore, the number of such paths is (cid:16) rk , k + , r − k − (cid:17) .Thus, the number of self-conjugate ( s , s + , s + m corners is given by r ⌊ m ⌋ , ⌊ m + ⌋ , r − m ! and this completes the proof. 13 roposition 3.10. The number of self-conjugate ( s , s + , s + , s + -core partitions with m corners is given by ⌊ s ⌋ − ⌊ m ⌋⌊ s ⌋ − m ! ⌊ s + ⌋ − ⌊ m + ⌋⌊ m ⌋ ! . Proof.
First, we consider the case where s = r + m = k . It follows from Theorems 1.7 and 3.8 that the numberof self-conjugate ( s , s + , s + , s + m corners is equal to the number of free rational Motzkinpaths of type ( r + , −
1) with r − k flat steps for which ends with a down step and has no UU as a consecutive subpath.It can be found in the proof of Theorem 3.4 that the number of such paths is (cid:16) r − kr − k (cid:17)(cid:16) r + − kk (cid:17) .We now consider the case where s = r + m = k +
1. By Theorems 1.7 and 3.8, the number of self-conjugate ( s , s + , s + , s + m corners is equal to the number of free rational Motzkin pathsof type ( r + , −
1) with r − k flat steps for which ends with a flat step, does not end with UF , and has no UU as aconsecutive subpath. Similarly, the number of such paths is (cid:16) r − kr − (2 k + (cid:17)(cid:16) r − kk (cid:17) by the proof of Theorem 3.4.Therefore, the number of self-conjugate ( s , s + , s + , s + m corners is given by r − ⌊ m ⌋ r − m ! r + − ⌊ m + ⌋⌊ m ⌋ ! , where s = r + s = r , it is similar to the case where s = r + s , s + , s + , s + r − ⌊ m ⌋ r − m ! r − ⌊ m + ⌋⌊ m ⌋ ! . Thus, the number of self-conjugate ( s , s + , s + , s + ⌊ s ⌋ − ⌊ m ⌋⌊ s ⌋ − m ! ⌊ s + ⌋ − ⌊ m + ⌋⌊ m ⌋ ! . Acknowledgments
Hyunsoo Cho was supported by Basic Science Research Program through the National Research Foundationof Korea(NRF) funded by the Ministry of Education (Grant No. 2019R1A6A1A11051177). JiSun Huh was sup-ported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No.2020R1C1C1A01008524).
References [1] Tewodros Amdeberhan and Emily Sergel Leven. Multi-cores, posets, and lattice paths.
Adv. in Appl. Math. , 71:1–13, 2015.[2] Jaclyn Anderson. Partitions which are simultaneously t - and t -core. Discrete Math. , 248(1-3):237–243, 2002.[3] Drew Armstrong, Christopher R. H. Hanusa, and Brant C. Jones. Results and conjectures on simultaneous core partitions.
European J.Combin. , 41:205–220, 2014.[4] Jineon Baek, Hayan Nam, and Myungjun Yu. Johnson’s bijections and their application to counting simultaneous core partitions.
EuropeanJ. Combin. , 75:43–54, 2019.[5] William Y. C. Chen, Harry H. Y. Huang, and Larry X. W. Wang. Average size of a self-conjugate ( s , t )-core partition. Proc. Amer. Math. Soc. ,144(4):1391–1399, 2016.[6] Hyunsoo Cho, JiSun Huh, and Jaebum Sohn. Counting self-conjugate ( s , s + , s + arXiv preprint arXiv:1904.02313 , 2019.[7] Hyunsoo Cho, JiSun Huh, and Jaebum Sohn. The ( s , s + d , s + d , . . . , s + pd )-core partitions and rational motzkin paths. arXiv preprintarXiv:2001.06651 , 2020.[8] Matthew Fayers. The t -core of an s -core. J. Combin. Theory Ser. A , 118(5):1525–1539, 2011.[9] Matthew Fayers. A generalisation of core partitions.
J. Combin. Theory Ser. A , 127:58–84, 2014.[10] Susanna Fishel and Monica Vazirani. A bijection between dominant Shi regions and core partitions.
European J. Combin. , 31(8):2087–2101,2010.
11] Ben Ford, Ho`ang Mai, and Lawrence Sze. Self-conjugate simultaneous p - and q -core partitions and blocks of A n . J. Number Theory ,129(4):858–865, 2009.[12] Harry H. Y. Huang and Larry X. W. Wang. The corners of core partitions.
SIAM J. Discrete Math. , 32(3):1887–1902, 2018.[13] Gordon James and Adalbert Kerber.
The representation theory of the symmetric group , volume 16 of
Encyclopedia of Mathematics and itsApplications . Addison-Wesley Publishing Co., Reading, Mass., 1981.[14] Paul Johnson. Simultaneous cores with restrictions and a question of zaleski and zeilberger. arXiv preprint arXiv:1802.09621 , 2018.[15] Rishi Nath and James A. Sellers. A combinatorial proof of a relationship between maximal (2 k − , k + k − , k , k + Electron. J. Combin. , 23(1):Paper 1.13, 11, 2016.[16] Victor Y. Wang. Simultaneous core partitions: parameterizations and sums.
Electron. J. Combin. , 23(1):Paper 1.4, 34, 2016.[17] Sherry H. F. Yan, Yao Yu, and Hao Zhou. On self-conjugate ( s , s + , . . . , s + k )-core partitions. Adv. in Appl. Math. , 113:101975, 20, 2020.[18] Jane Y. X. Yang, Michael X. X. Zhong, and Robin D. P. Zhou. On the enumeration of ( s , s + , s + European J. Combin. ,49:203–217, 2015.,49:203–217, 2015.