- Email: [email protected]
Parking functions on nonsingular M-matrices
Linear Algebra and its Applications 489 (2016) 1–14
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
Parking functions on nonsingular M -matrices Jun Ma a,1 , Yeong-Nan Yeh b,∗,2 a b
Department of Mathematics, Shanghai Jiaotong University, Shanghai, China Institute of Mathematics, Academia Sinica, Taipei, Taiwan
a r t i c l e
i n f o
Article history: Received 24 September 2014 Accepted 28 September 2015 Submitted by R. Brualdi MSC: 05C30 05C05
a b s t r a c t In this paper, let Δ be a nonsingular M -matrix. A generalization of G-parking functions, which is called Δ-parking functions, is studied. An explicit characterization for Δ-parking functions is given. It is shown that Δ-parking functions can be obtained by a simple way from recurrent configurations on the nonsingular M -matrix Δ. It is proved that the number of Δ-parking functions is equal to the determinant of Δ. © 2015 Elsevier Inc. All rights reserved.
Keywords: Chip-firing game Nonsingular M -matrix Parking function Sandpile model
1. Introduction In 1966, Konheim and Weiss [8] introduced the conception of parking functions in the study of the linear probes of random hashing function. Many generalizations of parking functions were studied. Please refer to [4,9,12,14–17]. In 2004, Postnikov and Shapiro [13] introduced a new generalization, the G-parking functions, in the study of * Corresponding author. 1 2
E-mail address: [email protected] (Y.-N. Yeh). Partially supported by SRFDP 20110073120068 and NSFC 11571235. Partially supported by NSC 101-2115-M-001-013-MY3.
http://dx.doi.org/10.1016/j.laa.2015.09.054 0024-3795/© 2015 Elsevier Inc. All rights reserved.
J. Ma, Y.-N. Yeh / Linear Algebra and its Applications 489 (2016) 1–14
2
certain quotients of the polynomial ring. Let G be a digraph with vertex set V (G) = {0, 1, 2, . . . , n} and edge set E(G). We allow G to have multiple edges and loops. For any I ⊆ V (G) \ {0} and v ∈ I, define outdegI,G (v) to be the number of edges directed from the vertex 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) \ {0} → {0, 1, 2, . . .}, such that for every I ⊆ V (G) \ {0} there exists a vertex v ∈ I such that 0 ≤ f (v) < outdeg I,G (v). For the complete graph G = Kn+1 on n + 1 vertices, Kn+1 -parking functions are exactly the classical parking functions. G-parking functions are an important tool for the determination of the rank defined by Baker and Norine in [2] on a Riemann Roch theorem for graphs. In 1990, Dhar [5] introduced the abelian sandpile model, which is also known as the chip-firing game, and showed that the number of recurrent configurations on a toppling matrix equals the determinant of the matrix. In 1993, Gabrielov [6] studied the sandpile model for a class of toppling matrices, which is more general than in [5]. An explicit characterization for recurrent configurations on a toppling-matrix appeared originally in [11] and later in [1,10,7]. Throughout the paper, we always let Δ be an integer matrix. We state the definition of toppling matrices as follows. Definition 1.1. A Z-matrix is an n × n matrix Δ = (Δij )1≤i,j≤n such that Δij ≤ 0 for all i = j. Let Δ be a Z-matrix. We say that Δ is a toppling matrix if there exists an integer vector h of length n with h ≥ 0 such that Δh > 0. Here 0 denotes a vector of length n in which all coordinates have value 0, and h ≥ h (h > h resp.) means that hi ≥ hi (hi > hi resp.) for every i. Toppling matrices are simply nonsingular M -matrices which have been extensively studied in the matrix theory literature, see also Berman and Plemmons’s book [3]. In this book, the 50 conditions which are equivalent to the statement: “Δ is a nonsingular M -matrix” are given. Following [3], we define nonsingular M -matrices as follows: Definition 1.2. Let Δ be a Z-matrix. If any of the following equivalent conditions hold then Δ is called a non-singular M -matrix: (1) (2) (3) (4)
There exists a vector h > 0 such that Δh > 0. Δ is nonsingular and all principal minors of Δ are positive. There exists an integer vector h of length n with h ≥ 0 such that Δh > 0. There exists a positive diagonal matrix H such that ΔH has all positive row sums.
Clearly, Condition (2) implies that a Z-matrix Δ is a nonsingular M -matrix if and only if its transposed matrix ΔT of Δ is a nonsingular M -matrix, moreover, we have that
J. Ma, Y.-N. Yeh / Linear Algebra and its Applications 489 (2016) 1–14
3
a Z-matrix Δ is a nonsingular M -matrix if and only if each principal submatrix of Δ is a nonsingular M -matrix. The main objective of the present paper is to generalize the G-parking functions associated to a nonsingular M -matrix. Let Δ be a nonsingular M -matrix. In this paper, the generalization for G-parking functions is called Δ-parking functions. We give an explicit characterization for Δ-parking functions and show that Δ-parking functions can be obtained by a simple way from recurrent configurations on the nonsingular M -matrix Δ. It is proved that the number of Δ-parking functions is equal to the determinant of Δ. The rest of this paper is organized as follows. In Section 2, we consider an equivalent definition for nonsingular M -matrices. In Section 3, we define Δ-parking functions and give their explicit characterization. In Section 4, we define Δ-recurrent configurations, show that Δ-parking functions can be obtained by a simple way from Δ-recurrent configurations, and prove that the number of Δ-parking functions is equal to the determinant of Δ. 2. An equivalent definition for nonsingular M -matrices In this section, we study an equivalent definition for nonsingular M -matrices. Let Δ be a Z-matrix. In [6], Gabrielov gave a sufficient condition for nonsingular M -matrices. Proposition 2.1 (Gabrielov). (See [6].) Let Δ be a Z-matrix. Suppose that Δ is nonsingular and has all nonnegative column sums. Then Δ is a nonsingular M -matrix. The condition, Δ has all nonnegative column sums, is not necessary for nonsingular M -matrices. For example, let us consider the matrix Δ=
2 −3
−1 4
.
It is easy to check that Δ
1 1 = , 1 1
and so Δ is a nonsingular M -matrix. But Δ has a negative column row. We are interested in the following equivalent definition for nonsingular M -matrices. Proposition 2.2. Let Δ be a Z-matrix. Δ is a nonsingular M -matrix if and only if Δ is nonsingular and there exists a vector r = (r1 , . . . , rn ) > 0 such that rΔ ≥ 0. Proof. Suppose that Δ is nonsingular and there exists a vector r = (r1 , . . . , rn ) > 0 such that rΔ ≥ 0. Let
J. Ma, Y.-N. Yeh / Linear Algebra and its Applications 489 (2016) 1–14
4
⎛
r1 ⎜ 0 ˜ ˜ Δ = Δ(r) = ⎝ ··· 0
0 r2 ··· 0
··· ··· ··· ···
⎞ ⎛ r1 Δ11 0 0 ⎟ ⎜ r2 Δ21 Δ=⎝ · · ·⎠ ··· rn rn Δn1
r1 Δ12 r2 Δ22 ··· rn Δn2
··· ··· ··· ···
⎞ r1 Δ1n r2 Δ2n ⎟ ··· ⎠ rn Δnn
˜ has all nonnegative column sums. Δ ˜ is nonsingular since Since rΔ ≥ 0, the matrix Δ ˜ is a nonsingular M -matrix. ˜ = r1 · · · rn · det Δ = 0. By Proposition 2.1, we have Δ det Δ ˜ > 0. Suppose By Definition 1.2, there exists a column vector h > 0 such that Δh T v = (v1 , v2 , . . . , vn ) = Δh. We have ˜ = (r1 v1 , r2 v2 , . . . , rn vn )T > 0. Δh This implies Δh > 0 and Δ is a nonsingular M -matrix. Conversely, suppose that Δ is a nonsingular M -matrix. We have det Δ = 0 and the transposed matrix ΔT of Δ is a nonsingular M -matrix. By Definition 1.2, there exists a column vector h > 0 such that ΔT h > 0. So hT Δ > 0. 2 3. Δ-parking functions For any nonsingular M -matrix Δ, let R (Δ) = {r ∈ Zn | rΔ ≥ 0 and r > 0}
where Z is the set of integers. For any r ∈ R (Δ), denote by Ω(r) the set of nonzero integer vectors χ = (χ(1), · · · , χ(n)) such that 0 ≤ χ(i) ≤ ri for every i. Let Δj = (Δ1j , . . . , Δnj )T be the j-th column of Δ. For any two vectors X, Y of n
length n, we consider the standard inner product given by X, Y = Xi Yi . We define i=1
(Δ, r)-parking functions as follows: Definition 3.1. Let r ∈ R (Δ). A (Δ, r)-parking function is a function f : {1, 2, . . . , n} → {0, 1, 2, . . .} such that for any χ ∈ Ω(r) there exists an integer j ∈ {1, 2, . . . , n} with χ(j) ≥ 1 such that 0 ≤ f (j) < χ, Δj . Denote by P(Δ, r) the set of (Δ, r)-parking functions.
J. Ma, Y.-N. Yeh / Linear Algebra and its Applications 489 (2016) 1–14
5
Example 3.2. Let us consider a digraph G with vertex set {0, 1, . . . , n} and at least one 0-sink spanning tree. Let LG be the Laplace matrix that corresponds to the digraph G and L0 the truncated Laplace matrix obtained from the matrix LG by deleting the rows and columns indexed by 0. The transposed matrix LT0 of L0 satisfies the conditions in (1) and the vector 1 ∈ R (LT0 ), where 1 denotes a vector of length n in which all coordinates have value 1. (LT0 , 1)-parking functions are exactly G-parking functions. Example 3.3. The matrix Δ and the vector r are given as follows: Δ=
2 −1 −3 4
, r = (2, 1).
Since the sum of entries of the first column of Δ is less than 0, the transposed matrix of Δ is not a truncated Laplace matrix of any digraph. We have Ω(r) = {(1, 0), (2, 0), (0, 1), (1, 1), (2, 1)} and P(Δ, r) = {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1)}. For any r ∈ R (Δ), let m = m(r) =
n
ri and denote by V (r) a multiset in which the
i=1
multiplicity of the integer i is ri for every i ∈ {1, 2, . . . , n}. For any submultiset W of V (r), let χ(i) be the multiplicity of the integer i in W for every i ∈ {1, 2, · · · , n}. Then χ = (χ(1), . . . , χ(n)) ∈ Ω(r). We also call χ the characteristic function of W . Example 3.4. Let us consider the matrix Δ and the vector r in the Example 3.3. Then V (r) = {1, 1, 2} and m(r) = 3. Take W = {1, 1}. Then the characteristic function χ of W is (2, 0) ∈ Ω(r). Lemma 3.5. For any r ∈ R (Δ), let m = m(r) =
n
ri . Then f is a (Δ, r)-parking
i=1
function if and only if there is a sequence of integers in the multiset V (r) π(1), . . . , π(m) such that for every i ∈ {1, 2, · · · , m}, 0 ≤ f (π(i)) < χi , Δπ(i) where χi is the characteristic function of the multiset {π(i), π(i + 1), . . . , π(m)}.
6
J. Ma, Y.-N. Yeh / Linear Algebra and its Applications 489 (2016) 1–14
Proof. Suppose that f is a (Δ, r)-parking function. We construct a sequence π(1), π(2), . . . , π(m) of integers in V (r) by the following algorithm. Algorithm A. • Step 1. Let W1 = V (r), χ1 the characteristic function of W1 and U1 = j ∈ W1 | 0 ≤ f (j) < χ1 , Δj . Set π(1) ∈ U1 by choosing π(1) as any integer in U1 . • Step 2. At time i ≥ 2, suppose π(1), . . . , π(i − 1) are determined. Let Wi = V (r) \ {π(1), . . . , π(i − 1)}, χi the characteristic function of Wi and Ui = j ∈ Wi | 0 ≤ f (j) < χi , Δj . Set π(i) ∈ Ui by choosing π(i) as any integer in Ui . By Algorithm A, iterating Step 2 until i = m, we obtain the sequence of integers as desired. Conversely, suppose that there is a sequence of integers in V (r) π(1), . . . , π(m) such that for every i ∈ {1, 2, . . . , m}, 0 ≤ f (π(i)) < χi , Δπ(i) , where χi is the characteristic function of {π(i), π(i + 1), . . . , π(m)}. For any χ ∈ Ω(r), let k be the largest index i ∈ {1, 2, . . . , m} such that χi ≥ χ. Let j = π(k). Then χ(j) = χk (j) and χk+1 (j) = χk (j) − 1. The Algorithm A tells us that 0 ≤ f (j) < χk , Δπ(k) = χk , Δj . By the choice of k we have χ, Δj = χ(j)Δjj −
i=j
χ(i)(−Δij ) ≥ χ(j)Δjj −
i=j
χk (i)(−Δij ).
J. Ma, Y.-N. Yeh / Linear Algebra and its Applications 489 (2016) 1–14
7
Table 1 (Δ, r)-parking functions f and their corresponding sequences π. f
π
f
π
f
π
(0, 0) (1, 0)
1, 2, 1 2, 1, 1
(0, 1) (1, 1)
2, 1, 1 2, 1, 1
(0, 2)
1, 2, 1
Moreover χ(j) = χk (j) gives: χ, Δj ≥ χk , Δj . Proving that for any χ ∈ Ω(r) there exists an integer j such that f (j) < χ, Δj ; hence f is a (Δ, r)-parking function. 2 Example 3.6. Let us consider the matrix Δ and the vector r in the Example 3.3. We have V (r) = {1, 1, 2} and P(Δ, r) = {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1)}. In Table 1, we list all (Δ, r)-parking functions f and their corresponding sequences π of integers in V (r) obtained by Algorithm A. Note that we can also obtain the sequence 2, 1, 1 of integers in V (r) by the Algorithm A for the Δ-parking function (0, 0). So, in general, for a Δ-parking function f , the sequence of integers in V (r) obtained by the Algorithm A is not unique. Lemma 3.7. Suppose that r, r ∈ R (Δ) and r ≤ r . Then P(Δ, r ) ⊆ P(Δ, r). Proof. Note that Ω(r) ⊆ Ω(r ) since r ≤ r . So f is a (Δ, r)-parking function if it is a (Δ, r )-parking function. Hence we have P(Δ, r ) ⊆ P(Δ, r). 2 Lemma 3.8. Suppose that r, r ∈ R (Δ). Then P(Δ, r + r ) = P(Δ, r) ∩ P(Δ, r ). Proof. By Lemma 3.7, we have P(Δ, r + r ) ⊆ P(Δ, r) and P(Δ, r + r ) ⊆ P(Δ, r ). So, P(Δ, r + r ) ⊆ P(Δ, r) ∩ P(Δ, r ). n n
Conversely, let m = ri and m = ri . For any f ∈ P(Δ, r) ∩ P(Δ, r ), by i=1
i=1
Lemma 3.5, there is a sequence π(1), . . . , π(m) of integers in V (r) such that 0 ≤ f (π(i)) < χi , Δπ(i)
J. Ma, Y.-N. Yeh / Linear Algebra and its Applications 489 (2016) 1–14
8
for every i ∈ {1, 2, · · · , m} and a sequence π (1), . . . , π (m ) of integers in V (r ) such that
0 ≤ f (π (i)) < χi , Δπ (i) for every i ∈ {1, 2, · · · , m } where χi and χi 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 π(i) π (i − m)
σ(i) =
if if
1≤i≤m 1 + m ≤ i ≤ m + m
For every i = 1, 2, · · · , m + m , let χ ˆi be the characteristic functions of {σ(i), · · · , σ(m + m )}. Then we have f (σ(i)) =
f (π(i)) f (π (i − m))
if if
1≤i≤m 1 + m ≤ i ≤ m + m
and χ ˆi , Δ
σ(i)
=
χi + r , Δσ(i) = χi , Δσ(i) + r , Δσ(i) χ i , Δσ(i)
if if
1≤i≤m 1 + m ≤ i ≤ m + m
Since r Δ ≥ 0, we have f (σ(i)) < χ ˆi , Δσ(i) for every i = 1, 2, . . . , m + m . By Lemma 3.5, f is a (Δ, r + r )-parking function. Hence, P(Δ, r + r ) = P(Δ, r) ∩ P(Δ, r ). 2 Corollary 3.9. (1) Suppose that r ∈ R (Δ) and b is a positive integer. Then P(Δ, br) = P(Δ, r).
J. Ma, Y.-N. Yeh / Linear Algebra and its Applications 489 (2016) 1–14
9
(2) Suppose that r1 , r2 , · · · , rk ∈ R (Δ) and b1 , b2 , · · · , bk are k positive integers. Then k
P(Δ, b1 r1 + b2 r2 + · · · + bk rk ) =
P(Δ, ri ).
i=1
Theorem 3.10. For any r, r ∈ R (Δ), P(Δ, r) = P(Δ, r ). Proof. Note that there is a positive b such that br ≥ r since r > 0. By Lemma 3.7 and Corollary 3.9(1), we have P(Δ, r) = P(Δ, br) ⊆ P(Δ, r ). Similarly, we have P(Δ, r ) ⊆ P(Δ, r). Hence, P(Δ, r) = P(Δ, r ).
2
Theorem 3.10 tells us that the set of (Δ, r)-parking functions is independent of r for any r ∈ R (Δ). So, (Δ, r)-parking functions are simply called Δ-parking functions and let P(Δ) be the set of Δ-parking functions. Let Δ = ZΔ1 ⊕ ZΔ2 ⊕ · · · ⊕ ZΔn be the sublattice in Zn spanned by the vectors Δi , where Δi = (Δi1 , . . . , Δin ) is the i-th row of Δ. We define an equivalence relation ∼ on Zn by declaring that f ∼ f if and only if f − f ∈ Δ . Lemma 3.11. Let r ∈ R (Δ). Suppose f and f are two (Δ, r)-parking functions. If f − f ∈ Δ , then f = f . Proof. Assume that f = f . Then f −f = xΔ and x = 0. By symmetry, we may suppose that xj > 0 for some j ∈ {1, 2, . . . , n}. Let b be a positive integer such that min{bri | i = 1, 2, · · · , n} ≥ max{xj | xj > 0 and 1 ≤ j ≤ n}. Let χ(j) =
xj 0
if if
xj > 0 xj ≤ 0
for each j = 1, 2, · · · , n. Then χ ∈ Ω(Δ, br) and for any j with χ(j) > 0
0 ≤ f (j) = f (j) −
n
xk Δk,j
k=1
≤ f (j) −
n
xk Δk,j
k=1 xk >0
= f (j) − χ, Δj .
J. Ma, Y.-N. Yeh / Linear Algebra and its Applications 489 (2016) 1–14
10
So, for any j with χ(j) > 0, we have f (j) ≥ χ, Δj . Hence f is not a (Δ, br)-parking function since χ ∈ Ω(br). But Corollary 3.9(1) implies that f is a (Δ, br)-parking function since f is a (Δ, r)-parking function, a contradiction. 2 Lemma 3.11 implies that distinct Δ-parking functions cannot be equivalent and every equivalent class of Zn contains at most one Δ-parking function. So we obtain the following corollary. Corollary 3.12. The number of Δ-parking functions is less than or equal to det Δ. Proof. Since the order of the quotient of the integer lattice Zn /Δ is det Δ, it follows from Lemma 3.11 and Theorem 3.10 that |P(Δ)| ≤ det Δ. 2 4. Δ-recurrent configurations For a nonsingular M -matrix Δ, let Δi = (Δi1 , . . . , Δin ) be the i-th row of Δ. A row vector u = (u1 , . . . , un ) is called a configuration if ui ≥ 0 for all i. In the sandpile model, the number ui is interpreted as the number of particles, or grains of sand, at site i = 1, . . . , n. For any site i, if ui ≥ Δii , we say that the site i is critical. A configuration u is called stable if no site is critical, i.e., 0 ≤ ui < Δii for all sites i. A critical site i is toppled, that is a subtraction the vector Δi from the vector u. the vertex i is toppled
u
u − Δi
Furthermore, a sequence of topplings is a sequence of sites i1 , i2 , . . . , ik such that ij is a critical site of u − Δi1 − · · · − Δij−1 for any 1 ≤ j ≤ k.
u
i1 ,i2 ,...,ik are toppled
u−
k
Δij
j=1
A representation vector for the sequence of topplings is a vector r = (r1 , . . . , rn ) with rs = |{j | ij = s, 1 ≤ j ≤ k}|. Clearly, u −
k
Δij = u − rΔ.
j=1
Remark 4.1. In [5], it is proved that every configuration can be transformed into a stable configuration by a sequence of topplings and the stable configuration does not depend on the order in which topplings are performed. For any r ∈ R (Δ), we define (Δ, r)-recurrent configurations as follows.
J. Ma, Y.-N. Yeh / Linear Algebra and its Applications 489 (2016) 1–14
11
Definition 4.2. Let u be a configuration and r ∈ R (Δ). We say that u is a (Δ, r)-recurrent configuration if u is stable and the configuration u + rΔ can be transformed into u by a sequence of topplings. Denote by R(Δ, r) the set of (Δ, r)-recurrent configurations. Example 4.3. The matrix Δ and the vector r are given as those in Example 3.3. Then R(Δ, r) = {(1, 3), (1, 2), (1, 1), (0, 3), (0, 2)}. Lemma 4.4. Let d = d(Δ) = (Δ11 − 1, Δ22 − 1, . . . , Δnn − 1). For any r ∈ R (Δ), a configuration u is a (Δ, r)-recurrent configuration if and only if d − u is a (Δ, r)-parking function. Proof. Let m = m(r) =
n
rj . Suppose that u is a (Δ, r)-recurrent configuration. By
j=1
Definition 4.2, the configuration u + rΔ can be transformed into u by a sequence i1 , i2 , . . . , im of topplings. Note that r is the representation vector for the sequence i1 , i2 , . . . , im . For every j ∈ {1, 2, . . . , m}, let χj be the characteristic function of the multiset {ij , ij+1 , . . . , im }. Then we have
uij +
m
Δik ,ij ≥ Δij ,ij
k=j
and (d − u)ij = Δij ,ij − 1 − uij ≤
m
Δik ,ij − 1 = χj , Δij − 1 < χj , Δij .
k=j
It follows from Lemma 3.5 that d − u is a (Δ, r)-parking function. Conversely, suppose f = d − u is a (Δ, r)-parking function. By Proposition 3.5, there is a sequence of integers in V (r) π(1), . . . , π(m) such that for every i ∈ {1, 2, . . . , m} 0 ≤ f (π(i)) < χi , Δπ(i) , where χi is the characteristic function of {π(i), π(i + 1), . . . , π(m)}. So,
12
J. Ma, Y.-N. Yeh / Linear Algebra and its Applications 489 (2016) 1–14
Table 2 (Δ, r)-recurrent configurations u and the sequences of topplings for u + rΔ. u
d−u
u + rΔ
a sequence of topplings for u + rΔ
(1, 3) (1, 2) (1, 1) (0, 3) (0, 2)
(0, 0) (0, 1) (0, 2) (1, 0) (1, 1)
(2, 5) (2, 4) (2, 3) (1, 5) (1, 4)
1, 2, 1 2, 1, 1 1, 2, 1 2, 1, 1 2, 1, 1
uπ(i) = Δπ(i),π(i) − 1 − f (π(i)) > Δπ(i),π(i) − 1 − χi , Δπ(i) = Δπ(i),π(i) − 1 −
m
Δπ(k),π(i)
k=i
and uπ(i) +
m
Δπ(k),π(i) ≥ Δπ(i),π(i) .
k=i
This implies that u + rΔ can be transformed into u by the sequence π(1), π(2), . . . , π(m) of topplings. 2 Example 4.5. Let us consider the matrix Δ and the vector r in the Example 3.3. We have d = (1, 3), rΔ = (1, 2), V (r) = {1, 1, 2} and R(Δ, r) = {(1, 3), (1, 2), (1, 1), (0, 3), (0, 2)}. In Table 2, we list all (Δ, r)-recurrent configurations u, their corresponding (Δ, r)-parking functions d − u, the configurations u + rΔ and sequences of topplings for u + rΔ. Theorem 4.6. For any r ∈ R (Δ), R(Δ, r) = R(Δ, r ). Proof. The required results follow from Lemma 4.4 and Theorem 3.10.
2
Theorem 4.6 tells us that the set of (Δ, r)-recurrent configurations is independent of r for any r ∈ R (Δ). So, (Δ, r)-parking functions are simply called Δ-recurrent configurations and let R(Δ) be the set of Δ-recurrent configurations. Lemma 4.7. Let r ∈ R (Δ). For any integer vector v = (v1 , . . . , vn ), there exists a (Δ, r)-recurrent configuration u such that v − u ∈ Δ .
J. Ma, Y.-N. Yeh / Linear Algebra and its Applications 489 (2016) 1–14
13
Proof. Note that det Δ > 0 and (det Δ)1 = (1adj(Δ))Δ ∈ Δ . For any integer vector v = (v1 , . . . , vn ), there exists a positive integer k such that v + k(det Δ)1 > 0. It is sufficient to prove for any configuration v = (v1 , . . . , vn ), there exists a (Δ, r)-recurrent configuration u such that v − u ∈ Δ . We now suppose v is a configuration. We start from v, increase vi by (rΔ)i for all i ∈ {1, 2, · · · , n} and then transform v + rΔ into a stable configuration by a sequence 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 configurations is finite. So at least one of them must recur. This means that there exists a stable configuration u for which u+b ·rΔ can be transformed into u by a sequence of topplings. Hence, u is a (Δ, br)-recurrent configuration. By Corollary 3.9 and Lemma 4.4, we have u is a (Δ, r)-recurrent configuration and u − v ∈ Δ . 2 Lemma 4.7 implies that every equivalent class of Zn contains at least one Δ-recurrent configuration. So we have the following corollary. Corollary 4.8. The number of Δ-recurrent configurations is larger than or equal to det Δ. Proof. Since the order of the quotient of the integer lattice Zn /Δ is det Δ, it follows from Lemma 4.7 and Theorem 4.6 that |R(Δ)| ≥ det Δ. 2 Theorem 4.9. |P(Δ)| = |R(Δ)| = det Δ. Proof. Combining Corollaries 3.12, 4.8, Lemma 4.4 and Theorem 4.6, we have |P(Δ)| = |R(Δ)| = det Δ. 2 Acknowledgements The authors are thankful to the referees for their helpful comments to improve the paper. References [1] A. Asadi, S. Backman, Chip-firing and Riemann–Roch theory for directed graphs, Electron. Notes Discrete Math. 38 (2011) 63–68. [2] M. Baker, S. Norine, Riemann–Roch theorem and Abel–Jacobi theory on a finite graph, Adv. Math. 215 (2007) 766–788. [3] A. Berman, J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1976. [4] R. Cori, D. Poulalhon, Enumeration of (p, q)-parking functions, Discrete Math. 256 (2002) 609–623. [5] D. Dhar, Self-organised critical state of the sandpile automaton models, Phys. Rev. Lett. 64 (14) (1990) 1613–1616. [6] A. Gabrielov, Asymmetric abelian avalanches and sandpile, MSI, Cornell University, 1993, preprint 93-65. [7] J. Guzmán, C. Klivans, Chip-firing and energy minimization on M -matrices, J. Combin. Theory Ser. A 132 (2015) 14–31.
14
J. Ma, Y.-N. Yeh / Linear Algebra and its Applications 489 (2016) 1–14
[8] A.G. Konheim, B. Weiss, An occupancy discipline and applications, SIAM J. Appl. Math. 14 (1966) 1266–1274. [9] J.P. Kung, C.H. Yan, Gon˘ carov polynomials and parking functions, J. Combin. Theory Ser. A 102 (2003) 16–37. [10] D. Perkinson, J. Perlman, J. Wilmes, Primer for the algebraic geometry of sandpiles, in: Tropical and Non-Archimedean Geometry, in: Contemp. Math., vol. 605, Amer. Math. Soc., Providence, RI, 2013. [11] J. Perlman, Sandpiles: a bridge between graphs and toric ideals, Thesis, Reed College, 2009. [12] J. Pitman, R.P. Stanley, A polytope related to empirical distributions, plane trees, parking functions, and the associahedron, Discrete Comput. Geom. 27 (4) (2002) 603–634. [13] A. Postnikov, B. Shapiro, Trees, parking functions, syzygies, and deformations of monomial ideals, Trans. Amer. Math. Soc. 356 (2004) 3109–3142. [14] J. Riordan, Ballots and trees, J. Combin. Theory 6 (1969) 408–411. [15] R.P. Stanley, Hyperplane arrangements, parking functions, and tree inversions, in: B. Sagan, R. Stanley (Eds.), Mathematical Essays in Honor of Gian-Carlo Rota, Birkhäuser, Boston and Basel, 1998, pp. 359–375. [16] C.H. Yan, Generalized tree inversions and k-parking functions, J. Combin. Theory Ser. A 79 (1997) 268–280. [17] C.H. Yan, Generalized parking functions, tree inversions, and multicolored graphs, Adv. in Appl. Math. 27 (2001) 641–670.
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
Parking functions on nonsingular M -matrices Jun Ma a,1 , Yeong-Nan Yeh b,∗,2 a b
Department of Mathematics, Shanghai Jiaotong University, Shanghai, China Institute of Mathematics, Academia Sinica, Taipei, Taiwan
a r t i c l e
i n f o
Article history: Received 24 September 2014 Accepted 28 September 2015 Submitted by R. Brualdi MSC: 05C30 05C05
a b s t r a c t In this paper, let Δ be a nonsingular M -matrix. A generalization of G-parking functions, which is called Δ-parking functions, is studied. An explicit characterization for Δ-parking functions is given. It is shown that Δ-parking functions can be obtained by a simple way from recurrent configurations on the nonsingular M -matrix Δ. It is proved that the number of Δ-parking functions is equal to the determinant of Δ. © 2015 Elsevier Inc. All rights reserved.
Keywords: Chip-firing game Nonsingular M -matrix Parking function Sandpile model
1. Introduction In 1966, Konheim and Weiss [8] introduced the conception of parking functions in the study of the linear probes of random hashing function. Many generalizations of parking functions were studied. Please refer to [4,9,12,14–17]. In 2004, Postnikov and Shapiro [13] introduced a new generalization, the G-parking functions, in the study of * Corresponding author. 1 2
E-mail address: [email protected] (Y.-N. Yeh). Partially supported by SRFDP 20110073120068 and NSFC 11571235. Partially supported by NSC 101-2115-M-001-013-MY3.
http://dx.doi.org/10.1016/j.laa.2015.09.054 0024-3795/© 2015 Elsevier Inc. All rights reserved.
J. Ma, Y.-N. Yeh / Linear Algebra and its Applications 489 (2016) 1–14
2
certain quotients of the polynomial ring. Let G be a digraph with vertex set V (G) = {0, 1, 2, . . . , n} and edge set E(G). We allow G to have multiple edges and loops. For any I ⊆ V (G) \ {0} and v ∈ I, define outdegI,G (v) to be the number of edges directed from the vertex 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) \ {0} → {0, 1, 2, . . .}, such that for every I ⊆ V (G) \ {0} there exists a vertex v ∈ I such that 0 ≤ f (v) < outdeg I,G (v). For the complete graph G = Kn+1 on n + 1 vertices, Kn+1 -parking functions are exactly the classical parking functions. G-parking functions are an important tool for the determination of the rank defined by Baker and Norine in [2] on a Riemann Roch theorem for graphs. In 1990, Dhar [5] introduced the abelian sandpile model, which is also known as the chip-firing game, and showed that the number of recurrent configurations on a toppling matrix equals the determinant of the matrix. In 1993, Gabrielov [6] studied the sandpile model for a class of toppling matrices, which is more general than in [5]. An explicit characterization for recurrent configurations on a toppling-matrix appeared originally in [11] and later in [1,10,7]. Throughout the paper, we always let Δ be an integer matrix. We state the definition of toppling matrices as follows. Definition 1.1. A Z-matrix is an n × n matrix Δ = (Δij )1≤i,j≤n such that Δij ≤ 0 for all i = j. Let Δ be a Z-matrix. We say that Δ is a toppling matrix if there exists an integer vector h of length n with h ≥ 0 such that Δh > 0. Here 0 denotes a vector of length n in which all coordinates have value 0, and h ≥ h (h > h resp.) means that hi ≥ hi (hi > hi resp.) for every i. Toppling matrices are simply nonsingular M -matrices which have been extensively studied in the matrix theory literature, see also Berman and Plemmons’s book [3]. In this book, the 50 conditions which are equivalent to the statement: “Δ is a nonsingular M -matrix” are given. Following [3], we define nonsingular M -matrices as follows: Definition 1.2. Let Δ be a Z-matrix. If any of the following equivalent conditions hold then Δ is called a non-singular M -matrix: (1) (2) (3) (4)
There exists a vector h > 0 such that Δh > 0. Δ is nonsingular and all principal minors of Δ are positive. There exists an integer vector h of length n with h ≥ 0 such that Δh > 0. There exists a positive diagonal matrix H such that ΔH has all positive row sums.
Clearly, Condition (2) implies that a Z-matrix Δ is a nonsingular M -matrix if and only if its transposed matrix ΔT of Δ is a nonsingular M -matrix, moreover, we have that
J. Ma, Y.-N. Yeh / Linear Algebra and its Applications 489 (2016) 1–14
3
a Z-matrix Δ is a nonsingular M -matrix if and only if each principal submatrix of Δ is a nonsingular M -matrix. The main objective of the present paper is to generalize the G-parking functions associated to a nonsingular M -matrix. Let Δ be a nonsingular M -matrix. In this paper, the generalization for G-parking functions is called Δ-parking functions. We give an explicit characterization for Δ-parking functions and show that Δ-parking functions can be obtained by a simple way from recurrent configurations on the nonsingular M -matrix Δ. It is proved that the number of Δ-parking functions is equal to the determinant of Δ. The rest of this paper is organized as follows. In Section 2, we consider an equivalent definition for nonsingular M -matrices. In Section 3, we define Δ-parking functions and give their explicit characterization. In Section 4, we define Δ-recurrent configurations, show that Δ-parking functions can be obtained by a simple way from Δ-recurrent configurations, and prove that the number of Δ-parking functions is equal to the determinant of Δ. 2. An equivalent definition for nonsingular M -matrices In this section, we study an equivalent definition for nonsingular M -matrices. Let Δ be a Z-matrix. In [6], Gabrielov gave a sufficient condition for nonsingular M -matrices. Proposition 2.1 (Gabrielov). (See [6].) Let Δ be a Z-matrix. Suppose that Δ is nonsingular and has all nonnegative column sums. Then Δ is a nonsingular M -matrix. The condition, Δ has all nonnegative column sums, is not necessary for nonsingular M -matrices. For example, let us consider the matrix Δ=
2 −3
−1 4
.
It is easy to check that Δ
1 1 = , 1 1
and so Δ is a nonsingular M -matrix. But Δ has a negative column row. We are interested in the following equivalent definition for nonsingular M -matrices. Proposition 2.2. Let Δ be a Z-matrix. Δ is a nonsingular M -matrix if and only if Δ is nonsingular and there exists a vector r = (r1 , . . . , rn ) > 0 such that rΔ ≥ 0. Proof. Suppose that Δ is nonsingular and there exists a vector r = (r1 , . . . , rn ) > 0 such that rΔ ≥ 0. Let
J. Ma, Y.-N. Yeh / Linear Algebra and its Applications 489 (2016) 1–14
4
⎛
r1 ⎜ 0 ˜ ˜ Δ = Δ(r) = ⎝ ··· 0
0 r2 ··· 0
··· ··· ··· ···
⎞ ⎛ r1 Δ11 0 0 ⎟ ⎜ r2 Δ21 Δ=⎝ · · ·⎠ ··· rn rn Δn1
r1 Δ12 r2 Δ22 ··· rn Δn2
··· ··· ··· ···
⎞ r1 Δ1n r2 Δ2n ⎟ ··· ⎠ rn Δnn
˜ has all nonnegative column sums. Δ ˜ is nonsingular since Since rΔ ≥ 0, the matrix Δ ˜ is a nonsingular M -matrix. ˜ = r1 · · · rn · det Δ = 0. By Proposition 2.1, we have Δ det Δ ˜ > 0. Suppose By Definition 1.2, there exists a column vector h > 0 such that Δh T v = (v1 , v2 , . . . , vn ) = Δh. We have ˜ = (r1 v1 , r2 v2 , . . . , rn vn )T > 0. Δh This implies Δh > 0 and Δ is a nonsingular M -matrix. Conversely, suppose that Δ is a nonsingular M -matrix. We have det Δ = 0 and the transposed matrix ΔT of Δ is a nonsingular M -matrix. By Definition 1.2, there exists a column vector h > 0 such that ΔT h > 0. So hT Δ > 0. 2 3. Δ-parking functions For any nonsingular M -matrix Δ, let R (Δ) = {r ∈ Zn | rΔ ≥ 0 and r > 0}
where Z is the set of integers. For any r ∈ R (Δ), denote by Ω(r) the set of nonzero integer vectors χ = (χ(1), · · · , χ(n)) such that 0 ≤ χ(i) ≤ ri for every i. Let Δj = (Δ1j , . . . , Δnj )T be the j-th column of Δ. For any two vectors X, Y of n
length n, we consider the standard inner product given by X, Y = Xi Yi . We define i=1
(Δ, r)-parking functions as follows: Definition 3.1. Let r ∈ R (Δ). A (Δ, r)-parking function is a function f : {1, 2, . . . , n} → {0, 1, 2, . . .} such that for any χ ∈ Ω(r) there exists an integer j ∈ {1, 2, . . . , n} with χ(j) ≥ 1 such that 0 ≤ f (j) < χ, Δj . Denote by P(Δ, r) the set of (Δ, r)-parking functions.
J. Ma, Y.-N. Yeh / Linear Algebra and its Applications 489 (2016) 1–14
5
Example 3.2. Let us consider a digraph G with vertex set {0, 1, . . . , n} and at least one 0-sink spanning tree. Let LG be the Laplace matrix that corresponds to the digraph G and L0 the truncated Laplace matrix obtained from the matrix LG by deleting the rows and columns indexed by 0. The transposed matrix LT0 of L0 satisfies the conditions in (1) and the vector 1 ∈ R (LT0 ), where 1 denotes a vector of length n in which all coordinates have value 1. (LT0 , 1)-parking functions are exactly G-parking functions. Example 3.3. The matrix Δ and the vector r are given as follows: Δ=
2 −1 −3 4
, r = (2, 1).
Since the sum of entries of the first column of Δ is less than 0, the transposed matrix of Δ is not a truncated Laplace matrix of any digraph. We have Ω(r) = {(1, 0), (2, 0), (0, 1), (1, 1), (2, 1)} and P(Δ, r) = {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1)}. For any r ∈ R (Δ), let m = m(r) =
n
ri and denote by V (r) a multiset in which the
i=1
multiplicity of the integer i is ri for every i ∈ {1, 2, . . . , n}. For any submultiset W of V (r), let χ(i) be the multiplicity of the integer i in W for every i ∈ {1, 2, · · · , n}. Then χ = (χ(1), . . . , χ(n)) ∈ Ω(r). We also call χ the characteristic function of W . Example 3.4. Let us consider the matrix Δ and the vector r in the Example 3.3. Then V (r) = {1, 1, 2} and m(r) = 3. Take W = {1, 1}. Then the characteristic function χ of W is (2, 0) ∈ Ω(r). Lemma 3.5. For any r ∈ R (Δ), let m = m(r) =
n
ri . Then f is a (Δ, r)-parking
i=1
function if and only if there is a sequence of integers in the multiset V (r) π(1), . . . , π(m) such that for every i ∈ {1, 2, · · · , m}, 0 ≤ f (π(i)) < χi , Δπ(i) where χi is the characteristic function of the multiset {π(i), π(i + 1), . . . , π(m)}.
6
J. Ma, Y.-N. Yeh / Linear Algebra and its Applications 489 (2016) 1–14
Proof. Suppose that f is a (Δ, r)-parking function. We construct a sequence π(1), π(2), . . . , π(m) of integers in V (r) by the following algorithm. Algorithm A. • Step 1. Let W1 = V (r), χ1 the characteristic function of W1 and U1 = j ∈ W1 | 0 ≤ f (j) < χ1 , Δj . Set π(1) ∈ U1 by choosing π(1) as any integer in U1 . • Step 2. At time i ≥ 2, suppose π(1), . . . , π(i − 1) are determined. Let Wi = V (r) \ {π(1), . . . , π(i − 1)}, χi the characteristic function of Wi and Ui = j ∈ Wi | 0 ≤ f (j) < χi , Δj . Set π(i) ∈ Ui by choosing π(i) as any integer in Ui . By Algorithm A, iterating Step 2 until i = m, we obtain the sequence of integers as desired. Conversely, suppose that there is a sequence of integers in V (r) π(1), . . . , π(m) such that for every i ∈ {1, 2, . . . , m}, 0 ≤ f (π(i)) < χi , Δπ(i) , where χi is the characteristic function of {π(i), π(i + 1), . . . , π(m)}. For any χ ∈ Ω(r), let k be the largest index i ∈ {1, 2, . . . , m} such that χi ≥ χ. Let j = π(k). Then χ(j) = χk (j) and χk+1 (j) = χk (j) − 1. The Algorithm A tells us that 0 ≤ f (j) < χk , Δπ(k) = χk , Δj . By the choice of k we have χ, Δj = χ(j)Δjj −
i=j
χ(i)(−Δij ) ≥ χ(j)Δjj −
i=j
χk (i)(−Δij ).
J. Ma, Y.-N. Yeh / Linear Algebra and its Applications 489 (2016) 1–14
7
Table 1 (Δ, r)-parking functions f and their corresponding sequences π. f
π
f
π
f
π
(0, 0) (1, 0)
1, 2, 1 2, 1, 1
(0, 1) (1, 1)
2, 1, 1 2, 1, 1
(0, 2)
1, 2, 1
Moreover χ(j) = χk (j) gives: χ, Δj ≥ χk , Δj . Proving that for any χ ∈ Ω(r) there exists an integer j such that f (j) < χ, Δj ; hence f is a (Δ, r)-parking function. 2 Example 3.6. Let us consider the matrix Δ and the vector r in the Example 3.3. We have V (r) = {1, 1, 2} and P(Δ, r) = {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1)}. In Table 1, we list all (Δ, r)-parking functions f and their corresponding sequences π of integers in V (r) obtained by Algorithm A. Note that we can also obtain the sequence 2, 1, 1 of integers in V (r) by the Algorithm A for the Δ-parking function (0, 0). So, in general, for a Δ-parking function f , the sequence of integers in V (r) obtained by the Algorithm A is not unique. Lemma 3.7. Suppose that r, r ∈ R (Δ) and r ≤ r . Then P(Δ, r ) ⊆ P(Δ, r). Proof. Note that Ω(r) ⊆ Ω(r ) since r ≤ r . So f is a (Δ, r)-parking function if it is a (Δ, r )-parking function. Hence we have P(Δ, r ) ⊆ P(Δ, r). 2 Lemma 3.8. Suppose that r, r ∈ R (Δ). Then P(Δ, r + r ) = P(Δ, r) ∩ P(Δ, r ). Proof. By Lemma 3.7, we have P(Δ, r + r ) ⊆ P(Δ, r) and P(Δ, r + r ) ⊆ P(Δ, r ). So, P(Δ, r + r ) ⊆ P(Δ, r) ∩ P(Δ, r ). n n
Conversely, let m = ri and m = ri . For any f ∈ P(Δ, r) ∩ P(Δ, r ), by i=1
i=1
Lemma 3.5, there is a sequence π(1), . . . , π(m) of integers in V (r) such that 0 ≤ f (π(i)) < χi , Δπ(i)
J. Ma, Y.-N. Yeh / Linear Algebra and its Applications 489 (2016) 1–14
8
for every i ∈ {1, 2, · · · , m} and a sequence π (1), . . . , π (m ) of integers in V (r ) such that
0 ≤ f (π (i)) < χi , Δπ (i) for every i ∈ {1, 2, · · · , m } where χi and χi 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 π(i) π (i − m)
σ(i) =
if if
1≤i≤m 1 + m ≤ i ≤ m + m
For every i = 1, 2, · · · , m + m , let χ ˆi be the characteristic functions of {σ(i), · · · , σ(m + m )}. Then we have f (σ(i)) =
f (π(i)) f (π (i − m))
if if
1≤i≤m 1 + m ≤ i ≤ m + m
and χ ˆi , Δ
σ(i)
=
χi + r , Δσ(i) = χi , Δσ(i) + r , Δσ(i) χ i , Δσ(i)
if if
1≤i≤m 1 + m ≤ i ≤ m + m
Since r Δ ≥ 0, we have f (σ(i)) < χ ˆi , Δσ(i) for every i = 1, 2, . . . , m + m . By Lemma 3.5, f is a (Δ, r + r )-parking function. Hence, P(Δ, r + r ) = P(Δ, r) ∩ P(Δ, r ). 2 Corollary 3.9. (1) Suppose that r ∈ R (Δ) and b is a positive integer. Then P(Δ, br) = P(Δ, r).
J. Ma, Y.-N. Yeh / Linear Algebra and its Applications 489 (2016) 1–14
9
(2) Suppose that r1 , r2 , · · · , rk ∈ R (Δ) and b1 , b2 , · · · , bk are k positive integers. Then k
P(Δ, b1 r1 + b2 r2 + · · · + bk rk ) =
P(Δ, ri ).
i=1
Theorem 3.10. For any r, r ∈ R (Δ), P(Δ, r) = P(Δ, r ). Proof. Note that there is a positive b such that br ≥ r since r > 0. By Lemma 3.7 and Corollary 3.9(1), we have P(Δ, r) = P(Δ, br) ⊆ P(Δ, r ). Similarly, we have P(Δ, r ) ⊆ P(Δ, r). Hence, P(Δ, r) = P(Δ, r ).
2
Theorem 3.10 tells us that the set of (Δ, r)-parking functions is independent of r for any r ∈ R (Δ). So, (Δ, r)-parking functions are simply called Δ-parking functions and let P(Δ) be the set of Δ-parking functions. Let Δ = ZΔ1 ⊕ ZΔ2 ⊕ · · · ⊕ ZΔn be the sublattice in Zn spanned by the vectors Δi , where Δi = (Δi1 , . . . , Δin ) is the i-th row of Δ. We define an equivalence relation ∼ on Zn by declaring that f ∼ f if and only if f − f ∈ Δ . Lemma 3.11. Let r ∈ R (Δ). Suppose f and f are two (Δ, r)-parking functions. If f − f ∈ Δ , then f = f . Proof. Assume that f = f . Then f −f = xΔ and x = 0. By symmetry, we may suppose that xj > 0 for some j ∈ {1, 2, . . . , n}. Let b be a positive integer such that min{bri | i = 1, 2, · · · , n} ≥ max{xj | xj > 0 and 1 ≤ j ≤ n}. Let χ(j) =
xj 0
if if
xj > 0 xj ≤ 0
for each j = 1, 2, · · · , n. Then χ ∈ Ω(Δ, br) and for any j with χ(j) > 0
0 ≤ f (j) = f (j) −
n
xk Δk,j
k=1
≤ f (j) −
n
xk Δk,j
k=1 xk >0
= f (j) − χ, Δj .
J. Ma, Y.-N. Yeh / Linear Algebra and its Applications 489 (2016) 1–14
10
So, for any j with χ(j) > 0, we have f (j) ≥ χ, Δj . Hence f is not a (Δ, br)-parking function since χ ∈ Ω(br). But Corollary 3.9(1) implies that f is a (Δ, br)-parking function since f is a (Δ, r)-parking function, a contradiction. 2 Lemma 3.11 implies that distinct Δ-parking functions cannot be equivalent and every equivalent class of Zn contains at most one Δ-parking function. So we obtain the following corollary. Corollary 3.12. The number of Δ-parking functions is less than or equal to det Δ. Proof. Since the order of the quotient of the integer lattice Zn /Δ is det Δ, it follows from Lemma 3.11 and Theorem 3.10 that |P(Δ)| ≤ det Δ. 2 4. Δ-recurrent configurations For a nonsingular M -matrix Δ, let Δi = (Δi1 , . . . , Δin ) be the i-th row of Δ. A row vector u = (u1 , . . . , un ) is called a configuration if ui ≥ 0 for all i. In the sandpile model, the number ui is interpreted as the number of particles, or grains of sand, at site i = 1, . . . , n. For any site i, if ui ≥ Δii , we say that the site i is critical. A configuration u is called stable if no site is critical, i.e., 0 ≤ ui < Δii for all sites i. A critical site i is toppled, that is a subtraction the vector Δi from the vector u. the vertex i is toppled
u
u − Δi
Furthermore, a sequence of topplings is a sequence of sites i1 , i2 , . . . , ik such that ij is a critical site of u − Δi1 − · · · − Δij−1 for any 1 ≤ j ≤ k.
u
i1 ,i2 ,...,ik are toppled
u−
k
Δij
j=1
A representation vector for the sequence of topplings is a vector r = (r1 , . . . , rn ) with rs = |{j | ij = s, 1 ≤ j ≤ k}|. Clearly, u −
k
Δij = u − rΔ.
j=1
Remark 4.1. In [5], it is proved that every configuration can be transformed into a stable configuration by a sequence of topplings and the stable configuration does not depend on the order in which topplings are performed. For any r ∈ R (Δ), we define (Δ, r)-recurrent configurations as follows.
J. Ma, Y.-N. Yeh / Linear Algebra and its Applications 489 (2016) 1–14
11
Definition 4.2. Let u be a configuration and r ∈ R (Δ). We say that u is a (Δ, r)-recurrent configuration if u is stable and the configuration u + rΔ can be transformed into u by a sequence of topplings. Denote by R(Δ, r) the set of (Δ, r)-recurrent configurations. Example 4.3. The matrix Δ and the vector r are given as those in Example 3.3. Then R(Δ, r) = {(1, 3), (1, 2), (1, 1), (0, 3), (0, 2)}. Lemma 4.4. Let d = d(Δ) = (Δ11 − 1, Δ22 − 1, . . . , Δnn − 1). For any r ∈ R (Δ), a configuration u is a (Δ, r)-recurrent configuration if and only if d − u is a (Δ, r)-parking function. Proof. Let m = m(r) =
n
rj . Suppose that u is a (Δ, r)-recurrent configuration. By
j=1
Definition 4.2, the configuration u + rΔ can be transformed into u by a sequence i1 , i2 , . . . , im of topplings. Note that r is the representation vector for the sequence i1 , i2 , . . . , im . For every j ∈ {1, 2, . . . , m}, let χj be the characteristic function of the multiset {ij , ij+1 , . . . , im }. Then we have
uij +
m
Δik ,ij ≥ Δij ,ij
k=j
and (d − u)ij = Δij ,ij − 1 − uij ≤
m
Δik ,ij − 1 = χj , Δij − 1 < χj , Δij .
k=j
It follows from Lemma 3.5 that d − u is a (Δ, r)-parking function. Conversely, suppose f = d − u is a (Δ, r)-parking function. By Proposition 3.5, there is a sequence of integers in V (r) π(1), . . . , π(m) such that for every i ∈ {1, 2, . . . , m} 0 ≤ f (π(i)) < χi , Δπ(i) , where χi is the characteristic function of {π(i), π(i + 1), . . . , π(m)}. So,
12
J. Ma, Y.-N. Yeh / Linear Algebra and its Applications 489 (2016) 1–14
Table 2 (Δ, r)-recurrent configurations u and the sequences of topplings for u + rΔ. u
d−u
u + rΔ
a sequence of topplings for u + rΔ
(1, 3) (1, 2) (1, 1) (0, 3) (0, 2)
(0, 0) (0, 1) (0, 2) (1, 0) (1, 1)
(2, 5) (2, 4) (2, 3) (1, 5) (1, 4)
1, 2, 1 2, 1, 1 1, 2, 1 2, 1, 1 2, 1, 1
uπ(i) = Δπ(i),π(i) − 1 − f (π(i)) > Δπ(i),π(i) − 1 − χi , Δπ(i) = Δπ(i),π(i) − 1 −
m
Δπ(k),π(i)
k=i
and uπ(i) +
m
Δπ(k),π(i) ≥ Δπ(i),π(i) .
k=i
This implies that u + rΔ can be transformed into u by the sequence π(1), π(2), . . . , π(m) of topplings. 2 Example 4.5. Let us consider the matrix Δ and the vector r in the Example 3.3. We have d = (1, 3), rΔ = (1, 2), V (r) = {1, 1, 2} and R(Δ, r) = {(1, 3), (1, 2), (1, 1), (0, 3), (0, 2)}. In Table 2, we list all (Δ, r)-recurrent configurations u, their corresponding (Δ, r)-parking functions d − u, the configurations u + rΔ and sequences of topplings for u + rΔ. Theorem 4.6. For any r ∈ R (Δ), R(Δ, r) = R(Δ, r ). Proof. The required results follow from Lemma 4.4 and Theorem 3.10.
2
Theorem 4.6 tells us that the set of (Δ, r)-recurrent configurations is independent of r for any r ∈ R (Δ). So, (Δ, r)-parking functions are simply called Δ-recurrent configurations and let R(Δ) be the set of Δ-recurrent configurations. Lemma 4.7. Let r ∈ R (Δ). For any integer vector v = (v1 , . . . , vn ), there exists a (Δ, r)-recurrent configuration u such that v − u ∈ Δ .
J. Ma, Y.-N. Yeh / Linear Algebra and its Applications 489 (2016) 1–14
13
Proof. Note that det Δ > 0 and (det Δ)1 = (1adj(Δ))Δ ∈ Δ . For any integer vector v = (v1 , . . . , vn ), there exists a positive integer k such that v + k(det Δ)1 > 0. It is sufficient to prove for any configuration v = (v1 , . . . , vn ), there exists a (Δ, r)-recurrent configuration u such that v − u ∈ Δ . We now suppose v is a configuration. We start from v, increase vi by (rΔ)i for all i ∈ {1, 2, · · · , n} and then transform v + rΔ into a stable configuration by a sequence 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 configurations is finite. So at least one of them must recur. This means that there exists a stable configuration u for which u+b ·rΔ can be transformed into u by a sequence of topplings. Hence, u is a (Δ, br)-recurrent configuration. By Corollary 3.9 and Lemma 4.4, we have u is a (Δ, r)-recurrent configuration and u − v ∈ Δ . 2 Lemma 4.7 implies that every equivalent class of Zn contains at least one Δ-recurrent configuration. So we have the following corollary. Corollary 4.8. The number of Δ-recurrent configurations is larger than or equal to det Δ. Proof. Since the order of the quotient of the integer lattice Zn /Δ is det Δ, it follows from Lemma 4.7 and Theorem 4.6 that |R(Δ)| ≥ det Δ. 2 Theorem 4.9. |P(Δ)| = |R(Δ)| = det Δ. Proof. Combining Corollaries 3.12, 4.8, Lemma 4.4 and Theorem 4.6, we have |P(Δ)| = |R(Δ)| = det Δ. 2 Acknowledgements The authors are thankful to the referees for their helpful comments to improve the paper. References [1] A. Asadi, S. Backman, Chip-firing and Riemann–Roch theory for directed graphs, Electron. Notes Discrete Math. 38 (2011) 63–68. [2] M. Baker, S. Norine, Riemann–Roch theorem and Abel–Jacobi theory on a finite graph, Adv. Math. 215 (2007) 766–788. [3] A. Berman, J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1976. [4] R. Cori, D. Poulalhon, Enumeration of (p, q)-parking functions, Discrete Math. 256 (2002) 609–623. [5] D. Dhar, Self-organised critical state of the sandpile automaton models, Phys. Rev. Lett. 64 (14) (1990) 1613–1616. [6] A. Gabrielov, Asymmetric abelian avalanches and sandpile, MSI, Cornell University, 1993, preprint 93-65. [7] J. Guzmán, C. Klivans, Chip-firing and energy minimization on M -matrices, J. Combin. Theory Ser. A 132 (2015) 14–31.
14
J. Ma, Y.-N. Yeh / Linear Algebra and its Applications 489 (2016) 1–14
[8] A.G. Konheim, B. Weiss, An occupancy discipline and applications, SIAM J. Appl. Math. 14 (1966) 1266–1274. [9] J.P. Kung, C.H. Yan, Gon˘ carov polynomials and parking functions, J. Combin. Theory Ser. A 102 (2003) 16–37. [10] D. Perkinson, J. Perlman, J. Wilmes, Primer for the algebraic geometry of sandpiles, in: Tropical and Non-Archimedean Geometry, in: Contemp. Math., vol. 605, Amer. Math. Soc., Providence, RI, 2013. [11] J. Perlman, Sandpiles: a bridge between graphs and toric ideals, Thesis, Reed College, 2009. [12] J. Pitman, R.P. Stanley, A polytope related to empirical distributions, plane trees, parking functions, and the associahedron, Discrete Comput. Geom. 27 (4) (2002) 603–634. [13] A. Postnikov, B. Shapiro, Trees, parking functions, syzygies, and deformations of monomial ideals, Trans. Amer. Math. Soc. 356 (2004) 3109–3142. [14] J. Riordan, Ballots and trees, J. Combin. Theory 6 (1969) 408–411. [15] R.P. Stanley, Hyperplane arrangements, parking functions, and tree inversions, in: B. Sagan, R. Stanley (Eds.), Mathematical Essays in Honor of Gian-Carlo Rota, Birkhäuser, Boston and Basel, 1998, pp. 359–375. [16] C.H. Yan, Generalized tree inversions and k-parking functions, J. Combin. Theory Ser. A 79 (1997) 268–280. [17] C.H. Yan, Generalized parking functions, tree inversions, and multicolored graphs, Adv. in Appl. Math. 27 (2001) 641–670.