aa r X i v : . [ m a t h . C O ] J u l Parking functions on toppling matrices
Jun Ma a, ∗ Yeong-Nan Yeh b, † a Department of Mathematics, Shanghai Jiaotong University, Shanghai b Institute of Mathematics, Academia Sinica, Taipei
Abstract
Let ∆ be an integer n × n -matrix which satisfies the conditions: det ∆ = 0, ∆ ij ≤ i = j, and there exists a vector r = ( r , . . . , r n ) > r ∆ ≥
0. Herethe notation r > r i > i , and r ≥ r ′ means that r i ≥ r ′ i forevery i . Let R (∆) be the set of vectors r such that r > r ∆ ≥
0. In this paper,(∆ , r )-parking functions are defined for any r ∈ R (∆). It is proved that the set of(∆ , r )-parking functions is independent of r for any r ∈ R (∆). For this reason, (∆ , r )-parking functions are simply called ∆-parking functions. It is shown that the numberof ∆-parking functions is less than or equal to the determinant of ∆. Moreover, thedefinition of (∆ , r )-recurrent configurations are given for any r ∈ R (∆). It is provedthat the set of (∆ , r )-recurrent configurations is independent of r for any r ∈ R (∆).Hence, (∆ , r )-recurrent configurations are simply called ∆-recurrent configurations. Itis obtained that the number of ∆-recurrent configurations is larger than or equal tothe determinant of ∆. A simple bijection from ∆-parking functions to ∆-recurrentconfigurations is established. It follows from this bijection that the number of ∆-parking functions and the number of ∆-recurrent configurations are both equal to thedeterminant of ∆. MSC:
Keywords: chip-firing game; parking function; sandpile model;
The classical parking functions are defined as follows. There are n parking spaces which arearranged in a line, numbered 0 to n − n drivers labeled 1 , . . . , n . Eachdriver i has an initial parking preference a i . Drivers enter the parking area in the order inwhich they are labeled. Each driver proceeds to his preferred space and parks here if it isfree, or parks at the next unoccupied space to the right. If all the drivers park successfullyby this rule, then the sequence ( a , . . . , a n ) is called a parking function.Konheim and Weiss [8] introduced the conception of parking functions in the study ofthe linear probes of random hashing function. Riordan [13] studied a relation of parking ∗ Email address of the corresponding author: [email protected] supported by SRFDP 20110073120068 † Partially supported by NSC 98-2115-M-001-010-MY3 roblems to ballot problems. The most notable result about parking functions is a bijectionfrom the set of classical parking functions of length n to the set of labeled trees on n + 1vertices.There are many generalizations of parking functions. Please refer to [2, 9, 11, 14, 15, 16].Postnikov and Shapiro [12] introduced a new generalization, the G -parking functions, in thestudy of certain quotients of the polynomial ring. Let G be a connected digraph with vertexset V ( G ) = { , , , . . . , n } and edge set E ( G ). We allow G to have multiple edges and loops.For any I ⊆ V ( G ) \ { } and v ∈ I , define outdeg I,G ( v ) to be the number of edges from thevertex v to a vertex outside of the subset I in G . G -parking functions are defined as follows. • A G -parking function is a function f : V ( G ) \ { } → { , , , . . . } , such that for every I ⊆ V ( G ) \ { } there exists a vertex v ∈ I such that 0 ≤ f ( v ) < outdeg I,G ( v ).For the complete graph G = K n +1 on n + 1 vertices, K n +1 -parking functions are exactly theclassical parking functions.Chebikin and Pylyavskyy [1] gave a family of bijections from the set of G -parking func-tions to the set of the (oriented) spanning trees of G . Let L G be the Laplace matrix thatcorresponds to the connected digraph G and L the truncated Laplace matrix obtained fromthe matrix L G by deleting the rows and columns indexed by 0. It follows from the matrix-treetheorem that the number of G -parking functions is equal to det L .One of the objective of the present paper is to generalize the G -parking functions asso-ciated to an integer n × n -matrix ∆ which satisfy the following conditions:det ∆ = 0;∆ ij ≤ i = j ;there exists a vector r = ( r , . . . , r n ) > r ∆ ≥ . (1)Here the notation r > r i > i and r ≥ r ′ means that r i ≥ r ′ i for every i .Let R (∆) = { r ∈ Z n | r ∆ ≥ r > } where Z is the set of integers. For any r ∈ R (∆), let c = c ( r ) = ( c , . . . , c n ) = r ∆ , m = m ( r ) = n X i =1 r i . Denote by Ω( r ) the set of integer vectors χ = ( χ (1) , · · · , χ ( n ))such that 0 ≤ χ ( i ) ≤ r i for every i and χ ( i ) = 0 for some i. Let ∆ j = (∆ j , . . . , ∆ nj ) T be the j -th column of ∆. There is a standard inner product h X, Y i = n P i =1 X i Y i on integer vectors of length n . We define (∆ , r )-parking functions asfollows: 2 efinition 1.1. Let r ∈ R (∆) . A (∆ , r ) -parking function is a function f : { , , . . . , n } →{ , , , . . . } , such that for any χ ∈ Ω( r ) , there exists a vertex j with χ ( j ) ≥ such that ≤ f ( j ) < h χ, ∆ j i . Denote by P (∆ , r ) the set of (∆ , r ) -parking functions. Example 1.2.
Let us consider a connected digraph G with vertex set { , , . . . , n } . Thetransposed matrix of the truncated Laplace matrix L of G satisfies the conditions in (1) andthe vector ∈ R ( L T ) , where the notation denotes a row vector of length n in which allcoordinate have value . ( L T , ) -parking functions are exactly G -parking functions. Example 1.3.
The matrix ∆ and the vector r are given as follows: ∆ = − − ! , r = (2 , . Then c = r ∆ = (1 , , m = 3 , and P (∆ , r ) = { (0 , , (0 , , (0 , , (1 , , (1 , } . A very interesting result is obtained: the set of (∆ , r )-parking functions is independentof r for any r ∈ R (∆). For this reason, (∆ , r )-parking functions are simply called ∆-parkingfunctions and denote by P (∆) the set of ∆-parking functions.Let h ∆ i = Z ∆ ⊕ Z ∆ ⊕ · · · ⊕ Z ∆ n be the sublattice in Z n spanned by the vectors ∆ i ,where ∆ i = (∆ i , . . . , ∆ in ) be the i -th row of ∆. We define an equivalence relation ∼ on Z n by declaring that f ∼ f ′ if and only if f − f ′ ∈ h ∆ i . It is proved that distinct ∆-parkingfunctions cannot be equivalent. Thus, every equivalent class of Z n contains at most one∆-parking function. Since the order of the quotient of the integer lattice Z n / h ∆ i is det ∆,it follows that the number of ∆-parking functions is less than or equal to det ∆.Now we turn to the abelian sandpile model, also known as the chip-firing game. It wasintroduced by Dhar [4] and was studied by many authors. Gabrielov [6] introduced thesandpile model for a class of toppling matrices, which is more general than in [4]. We statethis model as follows.An integer n × n -matrix ∆ is a toppling matrix if it satisfies the following conditions:∆ ij ≤ i = j ;there exists a vector h > h > . These matrices is called avalanche-finite redistribution matrices in [6].We list some properties of toppling matrices as follows.
Proposition 1.4. (Gabrielov, [6]) (1) A matrix ∆ is a toppling matrix if and only if its transposed matrix ∆ T is a topplingmatrix.
2) If ∆ is a toppling matrix, then all principal minors of ∆ are strictly positive.(3) Every integer matrix ∆ such that ∆ ij ≤ for i = j ; n X j =1 ∆ ij ≥ for all i ; and det ∆ = 0 . is a toppling matrix. For a toppling matrix ∆, let ∆ i = (∆ i , . . . , ∆ in ) be the i -th row of ∆. A row vector u = ( u , . . . , u n ) is called a configuration if u i ≥ i . For any vertex i , if u i ≥ ∆ ii ,we say that the vertex i is critical. A configuration u is called stable if no vertex is critical,i.e., 0 ≤ u i < ∆ ii for all vertices i . A critical vertex i is toppled, that is a subtraction thevector ∆ i from the vector u . Furthermore, a sequence of topplings is a sequence of vertices i , i , . . . , i k such that i j is a critical vertex of u − ∆ i − · · · − ∆ i j − for any 1 ≤ j ≤ k . Arepresentation vector for the sequence of topplings is a vector r = ( r , . . . , r n ) with r s = |{ j | i j = s, ≤ j ≤ k }| . Clearly, u − k P j =1 ∆ i j = u − r ∆. Proposition 1.5. (Dhar, [4])
Every configuration can be transformed into a stable configu-ration by a sequence of topplings. This stable configuration does not depend on the order inwhich topplings are performed.
For any 1 ≤ i ≤ n , the operator A i is given by increasing u i by 1, and then performinga sequence of topplings that lead to a new stable configuration. So the avalanche operators A , · · · , A n map the set of stable configurations to itself. Dhar [4] proved that the avalancheoperators A , . . . , A n commute pairwise.A stable configuration u is called recurrent if there are positive integers c i such that A c i i u = u for all i . Dhar [4] also showed that the number of recurrent configurations equalsdet ∆ . A configuration u is allowed if there exists j ∈ I such that u j ≥ X i ∈ I \{ j } ( − ∆ i,j )for any nonempty subset I of vertices. Dhar obtained a more explicit characterization ofrecurrent configurations: • Every recurrent configuration is allowed.Dhar suggested that a configuration is recurrent if and only if it is stable and allowed.Gabrielov [5] found that this statement is not true in general, and proved the conjecture fora toppling matrix ∆ which has nonnegative column sums. For symmetric ∆ = L , Dhar’sconjecture was proved in [3, 7, 10], where G is an undirected graph and L is the truncated4aplace matrix of G . Postnikov and Shapiro [12] gave a bijection from G -parking functionsto recurrent configurations for the toppling matrix ∆ = L .Let ∆ be an integer n × n -matrix and satisfy the condition in (1). Another objective of thepresent paper is to show how ∆-parking functions are related to the sandpile model. First,we show that an integer matrix ∆ is a toppling matrix if and only if it satisfies the conditionsin (1). Then for any r ∈ R (∆) we define (∆ , r )-recurrent configurations as follows. Definition 1.6.
Let u be a configuration and r ∈ R (∆) . We say that u is a (∆ , r ) -recurrentconfiguration if u is stable and the configuration u + r ∆ can be transformed into u by asequence of topplings. Denote by R (∆ , r ) the set of (∆ , r ) -recurrent configurations. Example 1.7.
The matrix ∆ and the vector r are given as those in Example 1.3. Then R (∆ , r ) = { (1 , , (1 , , (1 , , (0 , , (0 , } . Let d = d (∆) = (∆ − , ∆ − , . . . , ∆ nn − . For any r ∈ R (∆), we prove that a configuration u is a (∆ , r )-recurrent configuration if andonly if d − u is a (∆ , r )-parking function. This gives a bijection from (∆ , r )-recurrent configu-rations to (∆ , r )-parking functions and implies that the set of (∆ , r )-recurrent configurationsis independent of r for any r ∈ R (∆). Hence, (∆ , r )-recurrent configurations are simplycalled ∆-recurrent configurations and denote by R (∆) the set of ∆-parking functions. Wealso show that every equivalent class of Z n contains at least one ∆-recurrent configuration.So the number of ∆-recurrent configurations is larger than or equal to det ∆. Combiningthe results about ∆-parking functions, we obtain the number of ∆-parking functions and thenumber of ∆-recurrent configurations are both equal to det ∆.Note that recurrent configurations for a toppling matrix ∆ are exactly ∆-recurrent con-figurations. Thus, with the benefit of the bijection from (∆ , r )-recurrent configurations to(∆ , r )-parking functions, we give explicit characterization of recurrent configurations in thesandpile model.The rest of this paper is organized as follows. In Section 2, we study ∆-parking functions.In Section 3, we study ∆-recurrent configurations. ∆ -parking functions In this section, we always let ∆ = (∆ ij ) ≤ i,j ≤ n be an integer n × n -matrix and satisfy theconditions in (1). For any r ∈ R (∆), denote by ˜∆ = ˜∆( r ) = ( ˜∆ i,j ) ≤ i,j ≤ n the followingmatrix ˜∆ = ˜∆( r ) = r · · · r . . . . . . . . . . . . . . . . . . r n ∆ = r ∆ r ∆ . . . r ∆ n r ∆ r ∆ . . . r ∆ n . . . . . . . . . . . .r n ∆ n r n ∆ n . . . r n ∆ nn . r ) are nonnegative since r ∆ ≥
0, i.e., n P i =1 ˜∆ i,j ≥ j. We construct a digraph D = D (∆ , r ) with vertex set [0 , n ] as follows:(a) For any 1 ≤ i, j ≤ n and i = j , we connect i to j by − ˜∆ i,j edges directed from j to i ;(b) For every j ∈ { , , . . . , n } , we connect j to 0 by n P i =1 ˜∆ i,j ( ≥
0) edges directed from j to0.Let u = ( u , . . . , u n ) is a vector of length n and A = ( a ij ) an n × n matrix over a set { , , . . . , n } . For each nonempty subset I ⊆ { , , . . . , n } , we denote by A [ I ] the submatrixof A obtained by deleting the rows and columns whose indices are in { , , . . . , n } \ I and by u [ I ] the vector obtained from u by deleting the entries whose indices are in { , , . . . , n } \ I . Lemma 2.1.
Let ∆ = (∆ ij ) ≤ i,j ≤ n be an integer n × n -matrix and satisfy the conditions: ∆ ij ≤ for i = j and there exists a row vector r > such that r ∆ ≥ . Then the submatrix ∆[ I ] of ∆ also satisfies the conditions above and det ∆[ I ] ≥ for eachnonempty subset I ⊆ { , , . . . , n } .Proof. Let r = ( r , . . . , r n ) > r ∆ ≥
0. Fix a nonempty subset I ⊆ { , , . . . , n } and let J = { , , . . . , n } \ I . For any i ∈ I , we have r [ J ]∆ i [ J ] ≤ i is the i -th column of ∆. Hence, r [ I ]∆ i [ I ] = r ∆ i − r [ J ]∆ i [ J ] ≥ r ∆ i ≥ r [ I ]∆[ I ] ≥ r )[ I ] and the graph D = D (∆[ I ] , r [ I ]). By thematrix-tree theorem, the number of sink spanning trees rooted at 0 in D isdet ˜∆ = Y i ∈ I r i det ∆[ I ] . Hence, we must have det ∆[ I ] ≥ . Proposition 2.2.
Let ∆ = (∆ ij ) ≤ i,j ≤ n be an integer n × n -matrix. Then ∆ satisfies the con-ditions in (1) if and only if the matrix ∆[ I ] satisfies the conditions in (1) for each nonemptysubset I ⊆ { , , · · · , n } . roof. Suppose that ∆ satisfies the conditions in (1). Let us consider the matrix ˜∆ = ˜∆( r )and the graph D = D (∆ , r ). By the matrix-tree theorem, the number of sink spanning treesrooted at 0 in D is det ˜∆ = r · · · r n det ∆ . Hence, we must have det ∆ > = 0. This also implies that the number of sink spanning forests rooted at J in D is larger than 0, where J = { , , . . . , n } \ I . By the matrix-forest theorem, we havedet ˜∆[ I ] = Y i ∈ I r i det ∆[ I ] > I ] > . Hence, the matrix ∆[ I ] satisfy the conditions in (1) by Lemma 2.1. Corollary 2.3.
Let ∆ = (∆ ij ) ≤ i,j ≤ n be an integer n × n -matrix and satisfy the conditionsin (1). Then det ∆[ I ] > for each nonempty subset I ⊆ { , , . . . , n } . Denote by V ( r ) a multiset with r i copies of i for every i ∈ { , , · · · , n } . For any χ ∈ Ω( r )we can obtain a submultiset of V ( r ) by giving χ ( i ) copies of i for every i ∈ { , , · · · , n } .Thus, we call χ the characteristic function of W . Conversely, for any submultiset W of V ( r ),let χ ( i ) be the occurrence number of sites i in W for every i ∈ { , , · · · , n } . Then χ ∈ Ω( r ). Lemma 2.4.
For any r ∈ R (∆) , let m = m ( r ) = n P i =1 r i . Then f is a (∆ , r ) -parking functionif and only if there is a sequence of vertices in the multiset V ( r ) π (1) , . . . , π ( m ) such that for every i ∈ { , , · · · , m } , ≤ f ( π ( i )) < h χ i , ∆ π ( i ) i where χ i is the characteristic function of the multiset { π ( i ) , π ( i + 1) , . . . , π ( m ) } . Proof.
Suppose that f is a (∆ , r )-parking function. We construct a sequence π (1) , π (2) , . . . , π ( m )of vertices in V ( r ) by the following algorithm. Algorithm A. • Step 1. Let W = V ( r ), χ the characteristic function of W and U = { j ∈ W | ≤ f ( j ) < h χ , ∆ j i} . Set π (1) ∈ U . 7 Step 2. At time i ≥
2, suppose π (1) , . . . , π ( i −
1) are determined. Let W i = V ( r ) \ { π ( j ) | j = 1 , . . . , i − } ,χ i the characteristic function of W i and U i = { j ∈ W i | ≤ f ( j ) < h χ i , ∆ j i} . Set π ( i ) ∈ U i .By Algorithm A, iterating Step 2 until i = m , we obtain the sequence of vertices as desired.Conversely, suppose that there is a sequence of vertices in V ( r ) π (1) , . . . , π ( m )such that for every i ∈ { , , . . . , m } ,0 ≤ f ( π ( i )) < h χ i , ∆ π ( i ) i = χ i ( π ( i ))∆ π ( i ) ,π ( i ) − X j = π ( i ) χ i ( j )( − ∆ j,π ( i ) ) , where χ i is the characteristic function of { π ( i ) , π ( i + 1) , . . . , π ( m ) } .Let χ ∈ Ω( r ) and U = { j | χ ( j ) = 0 } . For every j ∈ U , use θ ( j ) to denote the uniqueindex k ∈ { , , . . . , m } such that π ( k ) = j and χ k ( j ) = χ ( j ). Let i = min { θ ( j ) | j ∈ U } .Then χ i ( π ( i )) = χ ( π ( i )) = 0 and χ i ( k ) ≥ χ ( k ) for every k = π ( i ). Thus f ( π ( i )) < χ i ( π ( i ))∆ π ( i ) ,π ( i ) − X j = π ( i ) χ i ( j )( − ∆ j,π ( i ) )= χ ( π ( i ))∆ π ( i ) ,π ( i ) − X j = π ( i ) χ i ( j )( − ∆ j,π ( i ) ) ≤ χ ( π ( i ))∆ π ( i ) ,π ( i ) − X j = π ( i ) χ ( j )( − ∆ j,π ( i ) )= h χ, ∆ π ( i ) i . By Definition 1.1, f is a (∆ , r )-parking function. Lemma 2.5.
Suppose that r , r ′ ∈ R (∆) and r ≤ r ′ . Then P (∆ , r ′ ) ⊆ P (∆ , r ) . Proof.
Note that Ω( r ) ⊆ Ω( r ′ ) since r ≤ r ′ . So f is a (∆ , r )-parking function if it is a(∆ , r ′ )-parking function. Hence we have P (∆ , r ′ ) ⊆ P (∆ , r ) . Lemma 2.6.
Suppose that r , r ′ ∈ R (∆) . Then P (∆ , r + r ′ ) = P (∆ , r ) ∩ P (∆ , r ′ ) . Proof.
By Lemma 2.5, we have P (∆ , r + r ′ ) ⊆ P (∆ , r ) and P (∆ , r + r ′ ) ⊆ P (∆ , r ′ ). So, P (∆ , r + r ′ ) ⊆ P (∆ , r ) ∩ P (∆ , r ′ ) . Conversely, let m = n P i =1 r i and m ′ = n P i =1 r ′ i . For any f ∈ P (∆ , r ) ∩ P (∆ , r ′ ), by Lemma2.4, there are a sequence π (1) , . . . , π ( m )8f vertices in V ( r ) such that 0 ≤ f ( π ( i )) < h χ i , ∆ π ( i ) i for every i ∈ { , , · · · , m } and a sequence π ′ (1) , . . . , π ′ ( m ′ )of vertices in V ( r ′ ) such that 0 ≤ f ( π ′ ( i )) < h χ ′ i , ∆ π ′ ( i ) i for every i ∈ { , , · · · , m ′ } where χ i and χ ′ i are the characteristic functions of { π ( i ) , π ( i +1) , . . . , π ( m ) } and { π ′ ( i ) , π ′ ( i + 1) , . . . , π ′ ( m ′ ) respectively.Let us consider the following sequence σ (1) , . . . , σ ( m ) , σ ( m + 1) , . . . , σ ( m + m ′ )where σ ( i ) = ( π ( i ) if 1 ≤ i ≤ mπ ′ ( i − m ) if 1 + m ≤ i ≤ m + m ′ For every i = 1 , , · · · , m + m ′ , let ˆ χ i is the characteristic functions of { σ ( i ) , · · · , σ ( m + m ′ ) } .Then we have f ( σ ( i )) = ( f ( π ( i )) if 1 ≤ i ≤ mf ( π ′ ( i − m )) if 1 + m ≤ i ≤ m + m ′ and h ˆ χ i , ∆ σ ( i ) i = ( h χ i + r ′ , ∆ σ ( i ) i = h χ i , ∆ σ ( i ) i + h r ′ , ∆ σ ( i ) i if 1 ≤ i ≤ m h χ ′ i , ∆ σ ( i ) i if 1 + m ≤ i ≤ m + m ′ Since r ′ ∆ ≥
0, we have f ( σ ( i )) < h ˆ χ i , ∆ σ ( i ) i for every i = 1 , , . . . , m + m ′ . By Lemma 2.4, f is a (∆ , r + r ′ )-parking function. Hence, P (∆ , r + r ′ ) = P (∆ , r ) ∩ P (∆ , r ′ ) . Corollary 2.7. (1) Suppose that r ∈ R (∆) and b is a positive integer. Then P (∆ , b r ) = P (∆ , r ) . (2) Suppose that r , r , · · · , r k ∈ R (∆) and b , b , · · · , b k are k positive integers. Then P (∆ , b r + b r + · · · + b k r k ) = k \ i =1 P (∆ , r i ) . Theorem 2.8.
For any r , r ′ ∈ R (∆) , P (∆ , r ) = P (∆ , r ′ ) . Proof.
Note that there is a positive b such that b r ≥ r ′ since r >
0. By Lemma 2.5 andCorollary 2.7(1), we have P (∆ , r ) = P (∆ , b r ) ⊆ P (∆ , r ′ ) . Similarly, we have P (∆ , r ′ ) ⊆ P (∆ , r ) . Hence, P (∆ , r ) = P (∆ , r ′ ).9heorem 2.8 tells us that the set of (∆ , r )-parking functions is independent of r for any r ∈ R (∆). So, (∆ , r )-parking functions are simply called ∆-parking functions. Lemma 2.9.
Let r ∈ R (∆) . Suppose f and f ′ are two (∆ , r ) -parking functions. If f ′ − f ∈h ∆ i , then f ′ = f .Proof. Assume that f ′ = f . Then f ′ − f = x ∆ and x = 0. By symmetry, we may supposethat x j > j ∈ { , , . . . , n } .Let b be a positive integer such thatmin { br i | i = 1 , , · · · , n } ≥ max { x j | x j > ≤ j ≤ n } . Let χ ( j ) = ( x j if x j >
00 if x j ≤ j = 1 , , · · · , n . Then χ ∈ Ω(∆ , b r ) and for any j with χ ( j ) > ≤ f ( j ) = f ′ ( j ) − n X k =1 x k ∆ k,j ≤ f ′ ( j ) − n X k =1 xk> x k ∆ k,j = f ′ ( j ) − h χ, ∆ j i . So, for any j with χ ( j ) >
0, we have f ′ ( j ) ≥ h χ, ∆ j i . Hence f ′ is not a (∆ , b r )-parkingfunction since χ ∈ Ω( b r ). But Corollary 2.7(1) implies that f ′ is a (∆ , b r )-parking functionssince f ′ is a (∆ , r )-parking functions, a contradiction.Lemma 2.9 implies distinct ∆-parking functions cannot be equivalent and every equivalentclass of Z n contains at most one ∆-parking function. So we obtain the following corollary. Corollary 2.10.
The number of ∆ -parking functions is less than or equal to det ∆ .Proof. Since the order of the quotient of the integer lattice Z n / h ∆ i is det ∆, it follows fromLemma 2.9 and Theorem 2.8 that |P (∆) | ≤ det ∆. ∆ -recurrent configurations In this section, we shall give the definition of ∆-recurrent configurations and study theirproperties, where the matrix ∆ satisfies the conditions in (1).
Proposition 3.1.
A matrix ∆ is a toppling matrix if and only if it satisfies the conditionsin (1). roof. Suppose that ∆ satisfies the conditions in (1). Then there exists a vector r = ( r , · · · , r n ) > r ∆ ≥ . Let˜∆ = r · · · r · · · · · · · · · · · · · · · · · · r n ∆ = r ∆ r ∆ · · · r ∆ n r ∆ r ∆ · · · r ∆ n · · · · · · · · · · · · r n ∆ n r n ∆ n · · · r n ∆ nn By Proposition 1.4(1) and (3), ˜∆ is a toppling matrix. There exists a column vector h > h >
0. Suppose v = ( v , v , . . . , v n ) T = ∆ h . We have˜∆ h = ( r v , r v , . . . , r n v n ) T > . This implies ∆ h > =0. By Proposition 1.4(1), its transposed matrix ∆ T is a toppling matrix. There exists acolumn vector h > T h >
0. So h T ∆ > Proposition 3.2.
A matrix ∆ is a toppling matrix if and only if all principal minors of ∆ are toppling matrices.Proof. Let ∆ be a toppling matrix and h = ( h , · · · , h n ) > h T >
0. For each nonempty subset I ⊆ { , , · · · , n } , let J = { , , · · · , n } \ I and supposethat ∆ i is the i -th row of ∆. Then for any i ∈ I , ∆ i [ J ] h [ J ] T ≤ I ] i h [ I ] T = ∆ i [ I ] h [ I ] T = ∆ i h T − ∆ i [ J ] h [ J ] T ≥ ∆ i h T > . This implies that ∆[ I ] h [ I ] T > I ] is a toppling matrix. Proposition 3.3.
Let ∆ = (∆ ij ) ≤ i,j ≤ n be an integer n × n -matrix with ∆ ij ≤ for i = j and adj (∆) = ( A ij ) ≤ i,j ≤ n the adjugate of ∆ . Then ∆ is a toppling matrix if and only if det ∆ > , A ii > and A ij ≥ for any i = j .Proof. Suppose that det ∆ > A ii > A ij ≥ i = j . Let h = adj (∆) T . Then h > h = ∆ adj (∆) T = (det ∆) T > . Hence, ∆ is a toppling matrix.Conversely, suppose ∆ is a toppling matrix. Proposition 1.4(2) implies det ∆ > u . It follows from the definition of recurrent configurationsthat for every i there is a positive integer c i such that A c i i u = u . This means that theconfiguration u + c i e i can be transformed into u by a sequence of topplings, where e i is a11ow vector of length n in which the i -th coordinate has value 1 and the other coordinate hasvalue 0. Suppose that representation vector for the sequence of topplings is r i = ( r i , . . . , r in ) . Then r ii ≥ , r ij ≥ i = j and r i ∆ = c i e i . So r i = c i det ∆ e i adj (∆) = c i det ∆ ( A i , · · · , A in ) . Hence, A ii > A ij ≥ i = j .Now, we always let ∆ be an n × n integer matrix satisfying the condition in (1). Lemma 3.4.
For any r ∈ R (∆) , a configuration u is a (∆ , r ) -recurrent configuration if andonly if d − u is a (∆ , r ) -parking function.Proof. Let m = m ( r ) = n P j =1 r j . Suppose that u is a (∆ , r )-recurrent configuration. ByDefinition 1.6, the configuration u + r ∆ can be transformed into u by a sequence i , i , . . . , i m of topplings. Note that r is the representation vector for the sequence i , i , . . . , i m . For every j ∈ { , , . . . , m } , let χ j be the characteristic function of the multiset { i j , i j +1 , . . . , i m } . Thenwe have u i j + m X k = j ∆ i k ,i j ≥ ∆ i j ,i j and ( d − u ) i j = ∆ i j ,i j − − u i j ≤ m X k = j ∆ i k ,i j − h χ j , ∆ i j i − < h χ j , ∆ i j i . It follows from Lemma 2.4 that d − u is a (∆ , r )-parking function.Conversely, suppose f = d − u is a (∆ , r )-parking function. By Proposition 2.4, there isa sequence of vertices in V ( r ) π (1) , . . . , π ( m )such that for every i ∈ { , , . . . , m } ≤ f ( π ( i )) < h χ i , ∆ π ( i ) i where χ i is the characteristic function of { π ( i ) , π ( i + 1) , . . . , π ( m ) } . So, u π ( i ) = ∆ π ( i ) ,π ( i ) − − f ( π ( i )) > ∆ π ( i ) ,π ( i ) − − h χ i , ∆ π ( i ) i = ∆ π ( i ) ,π ( i ) − − m X k = i ∆ π ( k ) ,π ( i ) and u π ( i ) + m X k = i ∆ π ( k ) ,π ( i ) ≥ ∆ π ( i ) ,π ( i ) . This implies that u + r ∆ can be transformed into u by the sequence π (1) , π (2) , . . . , π ( m ) oftopplings. 12 heorem 3.5. For any r ∈ R (∆) , R (∆ , r ) = R (∆ , r ′ ) .Proof. The required results follows from Lemma 3.4 and Theorem 2.8.Theorem 3.5 tells us that the set of (∆ , r )-recurrent configurations is independent of r forany r ∈ R (∆). So, (∆ , r )-parking functions are simply called ∆-recurrent configurations. Lemma 3.6.
Let r ∈ R (∆) . For any integer vector v = ( v , . . . , v n ) , there exists a (∆ , r ) -recurrent configuration u such that v − u ∈ h ∆ i .Proof. Note that det ∆ > = ( adj (∆))∆ ∈ h ∆ i . For any integer vector v = ( v , . . . , v n ), there exists a positive integer k such that v + k (det ∆) >
0. It is sufficientto prove for any configuration v = ( v , . . . , v n ), there exists a (∆ , r )-recurrent configuration u such that v − u ∈ h ∆ i .We now suppose v is a configuration. By Proposition 1.5, we start from v , increase v i by ( r ∆) i for all i ∈ { , , · · · , n } and then transform v + r ∆ into a stable configuration by asequence of topplings. If we repeat the process, we shall reach another stable configuration.This procedure can be repeated as often as we please, whereas the number of stable config-urations is finite. So at least one of them must recur. This means that there exists a stableconfiguration u for which u + b · r ∆ can be transformed into u by a sequence of topplings.Hence, u is a (∆ , b r )-recurrent configuration. By Corollary 2.7 and Lemma 3.4, we have u is a (∆ , r )-recurrent configuration and u − v ∈ h ∆ i .Lemma 3.6 implies that every equivalent class of Z n contains at least one ∆-recurrentconfiguration. So we have the following corollary. Corollary 3.7.
The number of ∆ -recurrent configuration is larger than or equal to det ∆ .Proof. Since the order of the quotient of the integer lattice Z n / h ∆ i is det ∆, it follows fromLemma 3.6 and Theorem 3.5 that |R (∆) | ≥ det ∆. Theorem 3.8. |P (∆) | = |R (∆) | = det ∆ .Proof. Combining Corollaries 2.10, 3.7, Lemma 3.4 and Theorem 3.5, we have |P (∆) | = |R (∆) | = det ∆.Note that recurrent configurations for a toppling matrix ∆ are exactly ∆-recurrent con-figurations. Let r ∈ R (∆). We say that a configuration u = ( u , u , · · · , u n ) is r -allowed iffor any χ ∈ Ω(∆ , r ), there exists a vertex j with χ ( j ) ≥ u j ≥ ∆ j,j − h χ, ∆ j i . Corollary 3.9.
Let r ∈ R (∆) . A configuration u is a recurrent configuration if and only ifit is stable and r -allowed. eferences [1] D. Chebikin, P. Pylyavskyy, A Family of Bijections Between G-Parking Functions andSpanning Trees, Journal of Combinatorial Theory A, 110 (2005), no. 1, 31-41.[2] R. Cori, D. Poulalhon, Enumeration of ( p, qp, q