Average site perimeter of directed animals on the two-dimensional lattices
We introduce new combinatorial (bijective) methods that enable us to compute the average value of three parameters of directed animals of a given area, including the site perimeter. Our results cover directed animals of any one-line source on the square lattice and its bounded variants, and we give counterparts for most of them in the triangular lattices. We thus prove conjectures by Conway and Le Borgne. The techniques used are based on Viennot’s correspondence between directed animals and heaps of pieces (or elements of a partially commutative monoid).
Let be an oriented graph and a nonempty finite set of vertices of . A directed animal of source on is a finite set of vertices that contains and such that for every vertex of , there exists a vertex of and a path from to going only through vertices of . The vertices of a directed animal are called sites. The area of , denoted by , is the number of sites of .
On Figure 1 are depicted single-source directed animals on the three two-dimensional regular lattices: the square lattice, the triangular lattice, and the honeycomb lattice.
Single-source directed animals constitute a subclass of animals (an animal on a non-oriented graph is simply a finite connected set of vertices of ). While the enumeration of animals on any lattice is an open problem despite extensive research for decades, directed animals are fairly easier to enumerate. As we will not deal with general animals in this paper, we will abusively use the term animal instead of directed animal.
Single-source directed animals on the square and triangular lattices have been enumerated [6, 8, 2]. Specifically, let and be the number of animals of source and area on the square and triangular lattice, respectively. The generating functions of these numbers are:
Even then, much remains unclear. The enumeration of directed animals on the honeycomb lattice is an open problem, and according to , the generating function is probably not D-finite; on the square and triangular lattices, comparatively very little is known when one tries to take into account parameters other than area.
Today, two enumeration methods account for almost every known result on directed animals. One of them is the gas model technique, originally used by Dhar . This technique was further developed by Bousquet-Mélou ; see also [12, 1] for more recent work.
The method used in this paper is the second one, based on a correspondence, due to Viennot , between animals and other objects called heaps of dominoes. The basic idea is to replace each site of an animal by a domino, so that each domino either lies on the ground or sits on one or two other dominoes (Figure 2).
As we will see later, this method works for the triangular lattice as well. However, no simple model of heaps of dominoes has been found to correspond to animals on the honeycomb lattice. This may explain the lack of knowledge on the subject.
The purpose of this paper is to study three other parameters of directed animals, introduced below, and illustrated in Figure 3:
two sites of an animal on the square or triangular lattice are adjacent if they are of the form and . We denote by the number of pairs of adjacent sites of .
a loop consists of two adjacent sites and , along with a third site at . We denote by the number of loops of .
a neighbour of an animal of source is a vertex not in , such that is still a directed animal of source . The number of neighbours of is called the site perimeter of and is denoted by .
Taking, for instance, the site perimeter, we may consider the bivariate generating function counting single-source animals according to both area and perimeter on the square lattice:
This generating function is not known, and is believed not to be D-finite . Instead, we will consider the generating function giving the total number of neighbours in the animals of a fixed area:
By dividing the total site perimeter of the animals of area by the number of these animals, one gets the average site perimeter in animals of a fixed area. Alternatively, this generating function may be seen as counting single-source directed animals with a marked neighbour.
This function, and the ones that similarly give the average number of adjacent sites and loops, turns out to be easier to derive. Specifically, the value of the generating function counting the total number of loops of single-source animals on the square lattice was obtained by Bousquet-Mélou using gas model methods :
As for the total site perimeter on the square lattice, it was the object of a conjecture by Conway in 1996 :
Le Borgne  also conjectured the value of similar generating functions counting the site perimeter of animals on square and triangular lattices of bounded width.
Actually, our results are more general than that: the same methods can be used on different kinds of lattices, obtained by adding one or two vertical walls (the half-lattice, cylindrical lattices and rectangular lattices, defined in Section 3), and on animals with any fixed source.
Knowing, say, the total site perimeter of single-source animals of area on the square lattice, we get their average perimeter by dividing by the number of these animals:
This quantity may thus be computed using (1) and (4). The numbers of adjacent sites and loops are handled similarly. From these generating functions, singularity analysis  yields estimates on these quantities as tends to infinity:
The paper is organized as follows. In Section 2, we introduce in detail the notion of heaps of pieces and give several lemmas useful for animal enumeration. In Section 3, we enumerate directed animals of any source on several kinds of square and triangular lattices, according to area alone. In Section 4, we give a general method to derive the generating functions giving the average number of adjacent sites, number of loops, and site perimeter of directed animals on the square lattice, as well as counterparts of most of these results on the triangular lattice. We derive asymptotic results in Section 5. Finally, we illustrate our formulæ with a few examples in Section 6.
2 Heaps of pieces
The notion of heaps of pieces is due to Viennot, and this topic is covered in detail in . We repeat the definitions for convenience, and make a few minor additions which we use later.
Intuitively, a heap is a finite set of pieces. It is built by dropping successively the pieces at certain positions, chosen from a given set. When the positions of two pieces overlap, the second piece falls on the first, like in Figure 2. A formal definition is given below.
Let be a set and a reflexive symmetric relation on . A heap of the model is a finite subset of satisfying:
if and with are in , then is not in ;
if is in and , then there exists in such that is in .
The relation is called the concurrency relation, and two positions and are concurrent if is in (in the above intuitive definition, this means that they overlap). The elements of a heap are called pieces. If is a piece of a heap, is called its position and its height.
The pieces of a heap are naturally equipped with a poset structure: define the relation such that whenever and are concurrent and . Let be the reflexive transitive closure of . We say that a piece is below a piece , or is above , if . The pieces and are independent if neither is above the other.
The partial order may be viewed more intuitively: let be a heap and a piece of . If one takes the piece and push it upwards, it pushes along some pieces in the way. If one pushes it high enough, the moved pieces are exactly the pieces above .
The pieces of a heap that are minimal for the order are called minimal pieces; they are exactly the pieces of height 0. The set of their positions is called the base of , and denoted by . Likewise, the pieces that are maximal for are called maximal pieces; we denote by the set of their positions.
We now define generating functions counting the heaps of a model; we denote by the number of pieces of the heap .
Let be a model of heaps and a finite subset of . We denote by and the generating functions (provided they exist) of heaps respectively of base and with base included in :
These two generating functions are obviously linked by:
|Conversely, the inclusion-exclusion principle yields:|
Let us now assume that the set of positions of the model is finite. In this case, the generating functions above may be computed using a result due to Viennot , which we present below.
A heap is called trivial if all its pieces have height 0. This means that a trivial heap may be identified with the set of the positions of its pieces, which must be pairwise nonconcurrent. As is a finite set, there is only a finite number of trivial heaps. We denote by the alternating generating function of trivial heaps included in :
Since is finite, this generating function is actually a polynomial, and is usually relatively easy to compute.
Lemma .3 (Inversion Lemma).
Let be a finite heap model and a subset of . The generating function of heaps of base included in is:
Thanks to this lemma, we see that in a finite model, the generating function is a quotient of two polynomials, and is therefore rational.
2.2 Strict and inflated heaps
In this section, we define families of heaps which we use in the correspondence with directed animals. Let be a heap, and let and be two pieces. We say that sits on if and are concurrent and . Thus, Condition 2 of Definition .1 states that any non-minimal piece must sit on another piece.
The objects obtained by relaxing this condition, keeping only Condition 1, are called pre-heaps.
A heap or pre-heap is strict if it has no piece sitting on another at the same position, i.e. no two pieces and .
An inflated heap is a strict pre-heap , such that for every piece satisfying , at least one of the following pieces is in :
a piece , such that and are concurrent (and );
the piece .
Examples are found in Figure 7.
Let be a heap. A stack of is a maximal set of pieces all at the same position, with consecutive heights. Thus, a heap is strict if all its stacks have only one piece.
Any heap may be built in a unique manner from a strict heap by replacing each piece by a stack consisting of an arbitrary positive number of pieces; in turn, any inflated heap may be built from a heap by “inflating” each stack, pushing along all pieces above it (see Figure 4).
These remarks enable us to derive from Lemma .3 the generating functions of strict and inflated heaps. First, as inflating a stack does not change its base or number of pieces, the generating functions of inflated heaps are the same as those of general heaps.
Let be a model of heaps. We denote by and the generating functions of strict heaps with base and base included in , respectively111For the sake of clarity, all generating functions in this paper follow the same typographical pattern as Definitions .2 and .6: the generating functions of general (or inflated) heaps are denoted by calligraphic letters, while the ones of strict heaps are denoted by standard capital letters. Likewise, a subscript is square brackets always indicates heaps with a base included in , while a subscript denotes heaps of base ..
The construction of Figure 4 translates into a link between the generating functions of strict and general heaps:
These links remain valid between generating functions of heaps with base included in .
2.3 Factorized heaps
We now present a monoid structure, again due to Viennot, on the set of heaps of a given model. Let be a heap and a position. Let be the heap formed by dropping a piece at position on top of ; more formally, let be , where is the largest integer such that this is a heap.
In this way, a heap may be built one piece at a time. This may be done by adding the pieces in any order compatible with the partial order . This idea is used to define the product of two heaps.
Let and be two heaps. The product is built by letting all pieces of fall on , in any order compatible with .
A factorized heap is a heap , with a distinguished factorization . We denote such a heap or . A factorized heap is strict if is strict; it is almost strict if both and are strict.
We now give a way to compute the base of a factorized heap. If is a heap, let the neighbourhood of , denoted , be the set of positions concurrent to at least one piece of .
Let be a factorized heap. The base of the heap is given by:
As the heap is built by dropping all pieces of , then all pieces of , no piece of can be above a piece of . Therefore, all minimal pieces of are also minimal pieces in .
A minimal piece of is minimal in if and only if it is not above a piece of . This happens if and only if its position is not in the neighbourhood of , hence the given formula. ∎
Given a heap with base , we may factorize as . Lemma .8 asserts that the base of is included in . This yields:
2.4 Heaps marked with a set of pieces
A number of our problems in animal enumeration can be seen as enumeration of heaps marked with a set of pieces; for example, computing the average number of adjacent sites in animals is linked to enumerating animals with two adjacent sites marked, which is in turn linked to enumerating heaps with some pieces marked. Here, we give a means to link such marked heaps to factorized heaps, which prove to be more manageable.
A marked heap is a heap , marked with a set of pairwise independent pieces.
Let be a marked heap. Let be the factorized heap where consists of all pieces below at least a piece of . Let be the factorized heap where consists of all pieces above at least a piece of .
An almost strict marked heap is a marked heap such that no piece of sits on an unmarked piece at the same position.
We call marked stack a stack of an almost strict marked heap containing a marked piece; such a stack may have one or two pieces, and the marked piece is always the lowest piece of the stack.
Let be an almost strict marked heap. Let be the set consisting of the highest piece of each marked stack. Define the following factorized heaps:
The mappings and (resp. and ) are bijections from the set of marked heaps to the set of factorized heaps (resp. almost strict marked heaps to almost strict factorized heaps).
Their inverse bijections are as follows. Let be a factorized heap, let be the set of maximal pieces of and the set of minimal pieces of . We have:
If is an almost strict factorized heap, let be the set consisting of the lowest piece of each stack of containing a piece of . We have:
Let us first do the case of marked heaps. This fact easily stems from the poset structure on the pieces of a heap: the set of maximal pieces of is the only set of pairwise independent pieces such that every piece of is below a piece of . The case of is identical.
We handle almost strict marked heaps similarly. Pulling downwards the lowest piece of each marked stack, or pushing upwards the highest piece, ensures that and are strict in the resulting factorized heap; conversely, as the pieces of and are the lowest of their stacks, and are almost strict marked heaps. ∎
As a first application, we give a means to compute the generating functions of heaps marked with one piece at a fixed position.
Let be a position and a set of positions. We denote by the generating function of heaps with base included in , marked with a piece at position . We denote by the set of heaps with base included in avoiding , i.e. such that no piece is concurrent to . As usual, we use analogous notations for strict heaps.
The generating functions counting heaps of base included in marked with a piece at position is given by:
Let be a heap with a marked piece at position . We use the bijection to turn it into a factorized heap . We know that has base , and that has base included in . According to Lemma .8, we have:
If is in , this simply means that is included in ; if not, it means that must be in as well, so that must not avoid . We thus get the result for general heaps.
Let be the generating function of almost strict marked heaps, with exactly one marked piece at position . The bijection turns these heaps into almost strict factorized heaps, on which we apply the same reasoning. Moreover, as only one piece may be duplicated, we have the identity:
This yields the second formula. ∎
3 Directed animals and heaps of dominoes
Let be either a subset of or of the form with an even number. The square lattice over , denoted by , is the oriented graph with vertices such that is even, and edges and . The triangular lattice over , denoted by , is with additionnal edges .
Let be a subset or quotient of in the same conditions as above. Let be the relation defined by if and only if . The model of heaps is called the model of heaps of dominoes with set of positions .
Up to a translation, there are four kinds of models and associated lattices:
the full model is the model ;
the half model is the model ;
the cylindrical model of width is the model , with even;
the rectangular model of width is the model .
As seen in Figure 7, directed animals of source on the square lattice are identical to strict heaps of dominoes of base in the model , while directed animals on the triangular lattice are inflated heaps.
We say that a heap of dominoes is aligned if all its pieces are such that is even. In particular, all heaps and inflated heaps corresponding to directed animals are aligned.
Let be a square lattice and its associated triangular lattice; let be a one-line source, i.e., a set of vertices of the form . We denote by and the generating functions of animals of source in the lattices and , respectively. These two generating functions also count strict and inflated heaps of base in the model of heaps of dominoes .
In this section, we give means to compute ; the generating function is then given by performing the substitution . Of course, we compute in the same way the generating functions and of animals with a source included in .
3.2 Bounded lattices
The bounded lattices correspond to finite models of heaps of dominoes, that is, the cylindrical models and the rectangular models. Let be a finite model and be a set of positions. The identity (6) and Lemma .3 give the value of the generating function of heaps of base :
All we need is therefore to compute the generating functions of trivial heaps. Do do this, we define two sequences of polynomials.
Define the sequences and of polynomials by , , and for all :
The polynomials are often called the Fibonacci polynomials. With these polynomials, we can compute the generating function , in any finite model and for any set , using the two lemmas below. We state them without proof, and refer to  for more detail. Examples are also given in Section 6.
Let . The generating function of trivial heaps in the rectangular model of width is . If is even, the generating function of trivial heaps in the cylindrical model of width is .
Let be a finite set of positions; write , where the are intervals of , with minimal. The generating function of trivial heaps included in is:
We now give explicit formulæ for the special case . In the following, we call zero-source animals the animals of source in any model. Let and be the generating functions of zero-source animals in the cylindrical and rectangular triangular lattices of width , respectively. We have:
The generating functions and counting zero-source animals on the square lattices are derived by performing the substitution .
3.3 Unbounded lattices
We now address the unbounded lattices, i.e. the full and half lattices. We start with zero-source animals, defined above, which are simplest. In the rectangular and half models, such animals are also called half-animals.
Let and be the generating functions of zero-source animals on the full square lattice and the half square lattice, respectively. Let and be their counterparts on the triangular lattices.
The generating functions of zero-source animals on the infinite models are:
Of course, as usual, the generating functions counting animals on the square lattices can be obtained by performing the substitution in the ones counting animals on the triangular lattices.
In the full model, a compact source is a finite set of consecutive even positions. In the half model, a compact source is a finite set of consecutive even positions, starting at 0.
The next result gives the generating function of animals with a given compact source. The proof may be found in .
Let be a compact source with sites. The generating function of animals of source on the full triangular lattice is:
|On the half lattice, this generating function is:|
Finally, we are able to compute the generating function of animals with an arbitrary source .
Let or . Let be a one-line source and be the smallest compact source containing . The generating function of animals of source in is:
Let us address the full lattice. Let be a cylindrical model large enough so that . The generating function of heaps of base in the model is:
As is the smallest compact source containing , is also the smallest interval containing , which ensures that no position of can be concurrent to . Therefore, Lemma .20 entails that:
We conclude by letting tend to infinity.
In the case of the half-lattice, we repeat the same reasoning, this time taking for a rectangular model large enough so that . ∎
4 Average number of adjacent sites, loops and neighbours of directed animals
4.1 Notations and results
In this section, is a square lattice (full, half, cylindrical or rectangular), and is a one-line source of . In Section 3, we have computed the generating functions and counting directed animals on of source and with a source included in .
We are now interested in three parameters of the directed animals: number of adjacent sites, number of loops, and site perimeter. The first two are defined in Section 1; we denote by and the number of pairs of adjacent sites and number of loops of the animal , respectively.
Let be an animal on with a source included in . An -neighbour of is a vertex of not in such that is still a directed animal with a source included in .
Assume now that the graph is embedded in a larger graph . An internal -neighbour of is an -neighbour of seen as an animal on . An external -neighbour of is an -neighbour of seen as an animal on .
Finally, a vertex of is at the edge of the lattice if it has outdegree 1.
For the purpose of this definition, we regard the half-lattice and the rectangular lattices as embedded in the full lattice. The full and cylindrical lattices are not naturally embedded in any larger graph, so in these lattices internal and external neighbours are identical. Moreover, if is the source of , the -neighbours of coincide with the usual neighbours of , as defined in Section 1.
Assuming no ambiguity on the set , we denote by the number of internal -neighbours of (or internal site perimeter) and by its number of external -neighbours (or external site perimeter). We also denote by the number of sites of at the edge of the lattice .
The generating functions defined below are linked to the average value of each parameter in animals of a given area.
Define the following generating functions, counting animals with a source included in :
Let be the generating function of animals marked with two adjacent sites.
Let be the generating function of animals marked with a loop.
Let be the generating function of animals marked with an external -neighbour.
Let be the generating function of animals marked with an internal -neighbour.
Also define , , and the analogous generating functions of animals of source .
To compute these generating functions, we again use the correspondence between animals and heaps of dominoes. We denote by the model of heaps of dominoes associated with the lattice ; we also denote by the aligned set of positions associated to the source .
We now define some generating functions counting heaps. As usual, a generating function with a subscript counts heaps with a base included in , and one with a subscript counts heaps with base , so this precision will often be omitted from the definition.
In some cases, we consider heaps having a minimal piece outside ; we call such a piece an illegal minimal piece. Note that an illegal minimal piece does not have to be aligned with .
Define the following generating functions:
Let be the generating function of strict heaps marked with a piece at a position , such that is in .
If is in , let be the generating function of strict heaps with a minimal piece at position and an illegal minimal piece at . Let be the sum of over all such that .
Let be the generating function of strict heaps marked with a piece at a position , such that either or is not in .
Also define the analogues , and ; thus, is the generating functions of strict heaps of base , such that is in but is not.
We now show that these generating functions can be computed using previous results. First, we have:
where counts strict heaps with base , and is computed using the results of Section 3. Next, the generating function of strict heaps with base included in marked with a single piece is . Thus, we have:
where counts strict heaps with base included in marked with a piece at position and is computed using Lemma .15. In all three equations, the sum goes over a finite number of positions , which ensures that all three generating functions can be computed. In this regard, the full and cylindrical models are the simplest, as is equal to and is zero.
In square lattices, the generating functions counting the total number of adjacent pieces, loops and site perimeters of the animals with source included in are given by:
Moreover, the corresponding generating functions for animals of source are:
where denotes the number of pairs of adjacent sites in the source .
Applications of this theorem are given in Section 6.
4.2 Site perimeters
First, we remark that a vertex of has outdegree 1 if and only if either or is not in . Thus, the generating function satisfies:
where the sum goes over all animals of source included in . The same goes for the generating function .
The number of external and internal -neighbours of every directed animal with source included in satisfy:
By summing the identities of this lemma over all animals of source included in , and using (24), we prove the identities (18) and (19). By summing them over all animals of source , we get (22) and (23).
When dealing with the external site perimeter, the lattice is embedded in a lattice which is either the full lattice or a cylindrical lattice. Let be the number of pairs of vertices such that is a site of and is a child of (i.e., is an edge of ), whether in or not. As every vertex has outdegree 2, we have
Now, as is a directed animal, a child of a site of is either a site of or an external -neighbour of . The only sites and -neighbours of not counted are the ones in ; moreover, two sites have a child in common if and only if they are adjacent. Hence:
which yields the announced formula for .
If is either the half-lattice or a rectangular lattice, then each site on the edge of the lattice has one external neighbour not in . Thus, we have:
4.3 Average number of adjacent sites and loops
To prove the remaining identities of Theorem .29, dealing with the number of adjacent sites and loops, we use several bijections between various sets of heaps marked with certain pieces. Rather than strict marked heaps, it is convenient here to use almost strict marked heaps (see Definition .11).
Although the proof seems complicated due to the high number of sets and associated generating functions that we must consider, each bijection is actually very simple, consisting in adding/removing a single piece.
Define the following sets of almost strict marked heaps, assumed to have a base included in :
Let be the set of almost strict heaps marked with two adjacent pieces.
Let be the set of almost strict heaps marked with the top piece of a loop.
Let be the set of almost strict heaps marked with a piece at a position , such that is in .
Let be the set of almost strict heaps marked with a minimal piece at a position , and having an illegal minimal piece at position .
Let (resp. ) be the set of almost strict heaps marked with two independent pieces at positions and (resp. ).
Let (resp. ) be the set of almost strict heaps marked with a piece at a position , and having an illegal minimal piece at position (resp. ) independent from .
Let , , , , , , , be the generating functions of these sets.
The asterisks are used above to denote sets of almost strict marked heaps; we link these heaps to strict marked heaps later. Note that we have the inclusions and .
The following identity holds:
To prove this result, we use a first bijection, which removes exactly one piece:
Let be a heap of . Pull the marked piece downwards to form the factorized heap , such that is the only maximal piece of . Let and . As is a loop, the heap has two maximal pieces and , which are adjacent (see Figure 8, left). Moreover, as and have same base and neighbourhood, Lemma .8 guarantees that and have the same base. We may thus set:
This operation is easily reversible, by putting back the piece . ∎
The following identity holds:
We again prove this result with a bijection removing one piece:
Let be a heap of . We set , pulling the pieces and downwards.
As the pieces and are not adjacent, one of them (say, ) is higher than the other. Let