Chip-Firing on Infinite $k$-ary Trees

In 1975, Arthur Engel [3] introduced the probabilistic abacus, an early version of chip-firing. In 1986, Joel Spencer [9] independently discovered one-dimensional chip-firing. In 1991, Björner, Lovász, and Shor introduced a chip-firing game on graphs.

The chip-firing game is played on a graph and is an important part of structural combinatorics. In this game, each vertex has a certain number of chips and can fire if the number of chips is greater than or equal to its degree. When fired, a vertex sends a single chip to each neighbor. The game ends when we can no longer fire any vertices. The resulting configuration is called a stable configuration.

Chip-firing is similar to Abelian sandpiles, which were introduced in 1987 by Bak, Tang, and Wiesenfeld [1]. In Abelian sandpiles, each place on a finite grid has a slope. Each slope eventually increases as grains of sand are randomly placed on the grid. When the slope reaches a certain height, they collapse and split the sand grains evenly among their neighbors on the grid. Abelian sandpiles were the first dynamical system discovered to have self-organized criticality.

Chip-firing has deep connections to the Tutte polynomial and group theory. More details can be found in Merino’s survey from 2005 [6].

More recently, Musiker and Nguyen [7] have discovered many of the properties of chip-firing on infinite binary trees when starting with $2^{n}-1$ chips at the root in both labeled and unlabeled chip-firing. Inagaki, Khovanova, and Luo [4] have also examined infinite binary trees. They found formulae for the total number of times each vertex fires, the total number of times the root fires, and the total number of fires.

Continuing the work of Inagaki, Khovanova, and Luo, we generalize some of the formulae they found to infinite $k$-ary trees. We calculate the number of times each vertex fires, the number of times the tree’s root fires, and the total number of fires in a $k$-ary tree. We also calculate and study many old and new integer sequences.

In Section 2, we provide preliminary definitions and results that are important for the rest of the paper. The graph we use is a rooted, undirected infinite $k$-ary tree with a self-loop at the root, and our chip-firing game starts with $N$ chips at the root.

In Section 3, we show that if we start with $N$ chips at the root of our tree, the stable configuration will form a finite tree of height $n$, where $n$ is the integer such that $\frac{k^{n}-1}{k-1}\leq N\leq\frac{k^{n+1}-k}{k-1}$. The number of chips on a vertex in the stable configuration is dependent only on its layer and is a function of the $k$-ary expansion of $N-\frac{k^{n}-1}{k-1}$.

In Section 4, we calculate the number of times each vertex fires, given the total number of chips ($N$). In Section 4.2, we use the formula from the previous section to explore the number of root fires in more detail. Given $k$, the number of root fires as a function of $N$—the total number of chips—has an interesting structure. Its first and second difference sequences exhibit fractal properties, which we study in more detail.

In Section 5, we calculate the total number of fires and the corresponding difference sequences. We consider a case for $N=\frac{k^{n}-1}{k-1}$, which plays a special role in our paper. We explain the connection between our sequences and schizophrenic numbers, which are numbers combining properties of rational and irrational numbers.

In a rooted tree, we distinguish one vertex as the root. Every other vertex $v$ has exactly one parent vertex $v_{p}$, defined as the first vertex after $v$ in the shortest path from $v$ to the root. The vertex $v$ is said to be a child of the vertex $v_{p}$. Any vertex other than vertex $v$ on the shortest path from $v$ to root is an ancestor of $v$, while $v$ is a descendant of each of its ancestors.

We define a vertex $v$ as being on layer $i+1$ if the shortest path from the root to $v$ traverses $i$ edges. By this definition, the root $r$ is on layer 1.

Sometimes, we need to refer to a subgraph of our underlying graph. For this, we need some more definitions.

The height of a tree is the length of the longest path from the root to a leaf. In a perfect $k$-ary tree, every non-leaf vertex has $k$ children, and all leaves are the same distance from the root. In other words, all leaves are located on the same layer $h+1$, where $h$ is the height of such a tree.

We consider a standard unlabeled chip-firing game [5], where some number of indistinguishable chips are placed at each vertex. A vertex can fire if it has at least the same number of chips as its degree. In our case, a vertex can fire when it has at least $k+1$ chips. When a vertex fires, it sends one chip along each edge. We consider the chip-firing game where the initial state has $N$ chips at the root and no chips anywhere else. The vertices fire until no firing is possible.

A stable configuration is the distribution of chips on the graph where no vertex can fire; in other words, in a stable configuration, every vertex has fewer chips than its degree. In our case, we reach a stable configuration when every vertex has at most $k$ chips.

The following theorem, proved in [2], describes when the chip-firing game may stabilize. It applies to any connected undirected graph $G=(V,E)$, where $V$ is the set of vertices and $E$ is the set of edges. We denote the number of vertices and edges by $|V|$ and $|E|$, respectively.

In our case, we have an infinite $k$-ary tree with a finite number of chips; therefore, we have $N<|E|$, implying that the configuration leads to a finite game.

Klivans [5] studied many properties of unlabeled chip-firing. In particular, we use the following property from this work called the Global Confluence.

This theorem implies that the order of fires does not matter. In our case, the game is finite, implying that we always reach a unique stable configuration.

Our formulae related to $k$-ary trees can be better expressed in base $k$. Hence, we use the notation $(m)_{k}$ or $m_{k}$ to indicate that a string of digits $m$ is written in base $k$. For example, $111_{2}=(111)_{2}=7$. Correspondingly, we will write $[n]_{k}$ to denote the string of digits representing a decimal integer $n$ in base $k$. Thus, $[n]_{k}=m$ is equivalent to $n=(m)_{k}$.

We denote by $\nu_{k}(n)$ the largest power of $k$ that divides $n$. In other words, $\nu_{k}(n)$ is the number of trailing zeros in $[n]_{k}$. When $k$ is prime, the value $\nu_{k}(n)$ is called the $k$-adic valuation of $n$. We will extend the terminology to composite numbers as well. In other words,

Let $f_{i}(N,k)$ denote the total number of fires of each vertex on layer $i+1$. When $N$ and $k$ are fixed, we will sometimes use the notation $f_{i}$ to simplify the calculations. We first look at the difference in the number of fires between layers $i-1$ and $i$.

Now, we can calculate the number of fires for each vertex on layer $i+1$.

The following corollary allows us to express the number of fires of each vertex on layer $i+1$ through the number of fires of the root.

The corollary above implies that we can focus only on the number of root fires to understand the properties of the number of vertex fires on other layers.

As a particular case of Theorem 6, we get the number of root fires.

As in [4], we are looking for a recursive formula.

The case of $N=\frac{k^{n}-1}{k-1}$ is important throughout our paper, as the stable configuration reaches layer $n$ for the first time when we start with $\frac{k^{n}-1}{k-1}$ chips in this form placed at the root. Moreover, if we start with $\frac{k^{n}-1}{k-1}$ chips at the root, then $c_{i}=1$ for $0\leq i<n$. Thus, the number of fires for a vertex on level $i+1$ is

Plugging in $k=2$, we get

matching the answer from [7].

The total number of root fires when $N=\frac{k^{n}-1}{k-1}$ is

Plugging in $k=2$, we see that for $2^{n}-1$ chips on a binary tree, the root fires $2^{n}-n-1$ times, which is the same result as in [7]. These numbers form sequence A000295 in the OEIS [8]. Starting from index 1, the first few terms are

When we plug in $k=3$, we get that there are $\frac{3^{n}-2n-1}{4}$ root fires, which corresponds to sequence A000340 in the OEIS with a shifted index. Similarly, when we plug in $k=4$ and $k=5$, we get $\frac{4^{n}-3n-1}{9}$ and $\frac{5^{n}-4n-1}{16}$ root fires, which correspond to sequences A014825 and A014827 in the OEIS [8]. These sequences appear later in our paper in Table 4.

Notice that for all $N\in\{ak+1,ak+2,\ldots,(a+1)k\}$, the same vertex of the tree fires the same number of times. Thus, it makes sense to consider the number of fires as a function of $\left\lceil\frac{N}{k}\right\rceil$. Therefore, we introduce a new set of functions:

Table 1 shows values of $g_{0}(m,k)$ for $1\leq m\leq 10$ and $2\leq k\leq 6$.

The values for $g_{0}(m,2)$ were calculated in [4], and they form sequence A376116 in the OEIS [8].

The sequence $g_{0}(m,3)$ has been submitted to the OEIS as sequence A378724 and starts as

Consider the difference sequence $d_{i}(m,k)=g_{i}(m+1,k)-g_{i}(m,k)$. In this section, we are interested in the case $i=0$. The following proposition describes the recursion for $d_{0}(m,k)$.

Now, we can state the formula.

In other words, excluding the first digit, if $[m]_{k}$ ends with $i$ ones, we have $d_{0}(m,k)=g_{0}(m+1,k)-g_{0}(m,k)=i+1$.

One can notice that these sequences all have a similar structure due to a similar repetition pattern. Each sequence begins with $k$ ones. Afterward, there are $k-1$ ones between every two numbers that are not equal to 1. Furthermore, for any $m>1$, if we remove all numbers less than $m$, the resulting sequence starts with $k$ occurrences of $m$, followed by $k-1$ occurrences of $m$ between every two numbers that are not equal to $m$.

In other words, we can construct the difference sequence $d_{0}(m,k)$ in the following way. Start by creating a sequence of all $1$’s. Then, replace every $k$th $1$ with a $2$ beginning with the $(k+1)$th $1$. Next, replace every $k$th $2$ with a $3$, starting with the $(k+1)$th $2$. Continue this process indefinitely, replacing every $k$th occurrence of $x-1$ with $x$, beginning with the $(k+1)$th $x-1$. This fractal structure of sequences $d_{0}$ follows from Proposition 10.

We can also calculate the total number of fires on an infinite undirected $k$-ary tree with a self-loop at the root. Let $F(N,k)$ denote the total number of fires in a $k$-ary tree when we start with $N$ chips placed at the root. As we have done previously, we sometimes remove the parameters in the function $F$ when $N$ and $k$ are fixed.

For $k=2$, the sequence of the total number of fires was calculated in [4]. It is sequence A376131 in the OEIS. The sequence starts from index 1 as

For $k=3$, the sequence of the total number of fires starts from index 1 as

This sequence has been submitted to the OEIS as sequence A378726. We have a recursive formula for function $F$, similar to the formula for $f_{0}$ found in Corollary 9.

If we start with $N=\frac{k^{n}-1}{k-1}$ chips at the root, we have $c_{i}=1$ for $0\leq i<n$. This case is special, so we want to look at the total number of fires for such $N$ separately.

For $k=2$, we get

This matches Corollary 3.7 in [7], and it is sequence A045618 in the OEIS, with an index shift.

For $k=3$, we get

which is sequence A212337 in the OEIS, with an index shift.

As is often the case in this paper, the difference sequences of our sequences are easier to analyze. For example, for $k=2$, our sequence A045618$(n+1)$ is the sum of the first $n$ terms of sequence A000337, which is a much older sequence that is simpler. Similarly, for $k=3$, the sequence A212337$(n+1)$ is the sum of the first $n$ terms of A014915.