Séminaire Lotharingien de Combinatoire 59 (2008), Article B59e

SOLUTION TO A COMBINATORIAL PUZZLE ARISING FROM MAYER’S THEORY OF CLUSTER INTEGRALS

OLIVIER BERNARDI

Abstract. Mayer’s theory of cluster integrals allows one to write the partition function of a gas model as a generating function of weighted graphs. Recently, Labelle, Leroux and Ducharme have studied the graph weights arising from the one-dimensional hard-core gas model and noticed that the sum of the weights over all connected graphs with n vertices is (−n)^{n−1}. This is, up to sign, the number of rooted Cayley trees on n vertices and the authors asked for a combinatorial explanation. The main goal of this article is to provide such an explanation.

1. Introduction

In [9], Mayer used an algebraic identity in order to express the partition function of a gas model as a generating function of weighted graphs. By Mayer’s transformation, any choice of an interaction potential between particles in the gas leads to a specific graph weight. For instance, in the case of the one-dimensional hard-core gas, Labelle, Leroux and Ducharme [6] have shown that the Mayer’s weight of a connected graph G having vertex set V (G) = {0, …, n} and edge set E(G) is w(G) = (−1)^{|E(G)|}Vol(Π_{G}) where Vol(Π_{G}) is the volume of the n-dimensional polytope

Π_{G} = {(x_{1}, …, x_{n}) ∈ ℝ^{n}/ x_{0} = 0 and |x_{i} − x_{j}| ≤ 1 for every edge (i, j) ∈ E(G)}.

The pressure in the model is related to Mayer’s weights by

(1)  P = kT ∑_{G connected graph} w(G)\frac{z^{|V (G)|}}{|V (G)|!},

where k is Boltzmann’s constant, T is the temperature and z is the activity.

It is known (see [3]) that the pressure of the hard-core gas is P = kT L(z), where L(z) is the Lambert function defined by the functional equation L(z) = z exp(−L(z)). Comparing this expression of the pressure with (1) and extracting the coefficient of z^{n+1} gives

∑_{G∈C_{n}} w(G) = (−1)^{n}(n + 1)^{n}, (2)

where the sum is over all connected graphs with n + 1 vertices. Labelle et al. observed that the right-hand-side of (2) is, up to sign, the number of rooted Cayley trees with n+1 vertices and asked for a combinatorial explanation [6, Question 1]. The main purpose of this paper is to give such an explanation.

The outline of the paper is as follows. In Section 2, we briefly review Mayer’s theory of cluster integrals following the line of [8]. We illustrate this theory on a very simple model of discrete gas and prove the equivalence of this model with the Potts model on the complete graph. Comparing two expressions of the pressure in the discrete gas leads to a surprising combinatorial identity. In Section 3, we give a combinatorial proof of this identity. In Section 4, we recall Mayer’s setting for the hard-core continuum gas and then give a combinatorial proof of Equation (2), thereby answering the question of Labelle et al.

We close this section with some notations. We denote by ℤ the set of integers and by ℝ the set of real numbers. We denote [n] = {1, …, n} and by 𝔖_{n} the set of permutations of [n]. In this paper, all graphs are simple, undirected and labelled. Let G be a graph. We denote by v(G), e(G) and c(G) respectively the number of vertices, edges and connected components of G. A graph H is a spanning subgraph of G if the vertex sets of H and G are the same while the edge set of H is included in the edge set of G; we denote H ⊆ G in this case. We denote by e = (i, j) the edge with endpoints i an j and write e ∈ G if the edge e belongs to G. For any edge e, we denote by G ⊕ e the graph obtained from G by either adding the edge e if e ∉ G or by deleting this edge if e ∈ G.

2. Review of Mayer’s theory of cluster integrals

Consider a gas made of n (indistinguishable) particles in a vessel Ω ⊂ ℝ^{d}. We suppose that the gas is free from outside influence and that interaction between two particles i and j at positions x_{i} and x_{j} is given by the potential ϕ(x_{i}, x_{j}). In the classical Boltzmann setting, the probability measure of a configuration is proportional to exp(−H/kT), where k is Boltzmann’s constant, T is the temperature and H is the Hamiltonian of the system given by

H = ∑_{1≤i≤n} \frac{m_{i}v_{i}^{2}}{2} + ∑_{1≤i<j≤n} ϕ(x_{i}, x_{j}),

where x_{i}, v_{i}, m_{i} and \frac{m_{i}v_{i}^{2}}{2} are respectively the position, velocity, mass and kinetic energy of the i^{th} particle.

The partition function of the gas model is

Z(Ω, T, n) = \frac{1}{h^{dn}n!} ∫∫_{x_{1},...,x_{n}∈Ω, v_{1},...,v_{n}∈ℝ^{d}} exp(−H/kT)dx_{1} … dx_{n}dv_{1} … dv_{n},

where h is Planck’s constant. After integrating over all possible velocity, the partition function becomes

Z(Ω, T, n) = \frac{1}{λ^{n}n!} ∫∫_{Ω^{n}} ∏_{i<j} exp(−\frac{ϕ(x_{i}, x_{j})}{kT}) dx_{1} … dx_{n},

where λ depends on the temperature T.

Mayer noticed that the partition function can be decomposed into a sum over graphs. Indeed, by setting f(x_{i}, x_{j}) = exp(−\frac{ϕ(x_{i},x_{j})}{kT}) − 1, one gets

∏_{i<j} exp(−\frac{ϕ(x_{i}, x_{j})}{kT}) = ∏_{i<j} 1 + f(x_{i}, x_{j}) = ∑_{G⊆K_{n}} ∏_{(i,j)∈G} f(x_{i}, x_{j}),

where the sum is over all graphs on n vertices (equivalently, spanning subgraph of the complete graph K_{n}) and the inner product is over all edges of G. In terms of the partition function, this gives Mayer’s relation:

λ^{n}n!Z(Ω, T, n) = ∫∫_{Ω^{n}} ∏_{i<j} exp(−\frac{ϕ(x_{i}, x_{j})}{kT}) dx_{1} … dx_{n} = ∑_{G⊆K_{n}} W (G),

where W (G) = ∫∫_{Ω^{n}} ∏_{(i,j)∈G} f(x_{i}, x_{j})dx_{1} … dx_{n} is the first Mayer’s weight of the graph G.

Example: the discrete gas. Suppose Ω is made of q distinct boxes B_{1}, …, B_{q} of volume 1 and that the interaction potential ϕ(x_{i}, x_{j}) is equal to α if the particles i and j are in the same box and 0 otherwise. By definition, the Mayer’s weight of a graph G is

W (G) = ∫∫_{Ω^{n}} ∏_{(i,j)∈G} f(x_{i}, x_{j})dx_{1} … dx_{n} = ∑_{c:[n]↦[q]} ∫∫_{x_{1}∈B_{c(1)},...,x_{n}∈B_{c(n)}} ∏_{i<j} f(x_{i}, x_{j})dx_{1} … dx_{n}.

We denote u = exp(−α/kT) and observe that f(x_{i}, x_{j}) = u − 1 if i and j are in the same box and 0 otherwise. Therefore, the product ∏_{(i,j)∈G} f(x_{i}, x_{j}) equals (u − 1)^{e(G)} if the value of c is constant over each connected component of the graph G and 0 otherwise. Summing over all mappings c : [n] ↦ [q] gives

W (G) = q^{c(G)}(u − 1)^{e(G)},

since there are q^{c(G)} mappings c : [n] ↦ [q] which are constant over each connected components of G.

In our discrete gas example, a direct calculation of the partition function gives

λ^{n}n!Z(Ω, T, n) = ∑_{c:[n]↦[q]} ∫∫_{x_{1}∈B_{c(1)}...x_{n}∈B_{c(n)}} ∏_{i<j} exp(−\frac{ϕ(x_{i}, x_{j})}{kT}) dx_{1} … dx_{n} = ∑_{c:[n]↦[q]} u^{δ(c)},

where δ(c) is the number of edges (i, j) ∈ K_{n} such that c(i) = c(j). Hence, Mayer’s relation reads

∑_{c:[n]↦[q]} u^{δ(c)} = ∑_{G⊆K_{n}} q^{c(G)}(u − 1)^{e(G)}. (3)

Equation (3) is a special case of the equivalence established by Fortuin and Kasteleyn [4] between the partition function of the Potts model (see e.g. [1]) and the Tutte polynomial (see e.g. [2]). Indeed, the right-hand-side corresponds to the partition function of the Potts model on the complete graph K_{n} while the left-hand-side corresponds to the subgraph expansion of the Tutte polynomial of K_{n} up to scaling and change of variables. The relation of Fortuin and Kasteleyn is the generalisation of (3) obtained by replacing the complete graph K_{n} by any graph H. This more general case relies on the observation that ∏_{(i,j)∈H} ϕ_{i,j} = ∑_{G⊆H} ∏_{(i,j)∈G} f_{i,j} as soon as ϕ_{i,j} = 1 + f_{i,j} for all (i, j) ∈ H.

We now return to the general theory of Mayer and consider a system with an arbitrary number of particles. The grand canonical partition function is defined by

Z_{gr}(z) ≡ Z_{gr}(Ω, T, z) = ∑_{n≤0} z^{n}λ^{n}Z(Ω, T, n),

where z is the activity of the system. In terms of Mayer’s weights, the grand canonical partition function is the exponential generating functions of graphs weighted by their first Mayer’s weight:

Z_{gr}(z) = ∑_{n≤0} \frac{z^{n}}{n!} ∑_{G⊆K_{n}} W (G) = ∑_{G} W (G)\frac{z^{v(G)}}{v(G)!}.

The macroscopic parameters of the systems, such as the density ρ, or pressure P, can be obtained from Z_{gr}(z) by the relations

P = \frac{kT}{|Ω|} log(Z_{gr}(z)) and ρ = \frac{z}{|Ω|} \frac{∂}{∂z} log(Z_{gr}(z)).

Observe that the first Mayer’s weight is multiplicative over connected components, that is, if a graph G is the disjoint union of two graphs G_{1} and G_{2} then W (G) = W (G_{1})W (G_{2}). This is the key property implying log(Z_{gr}(z)) = ∑_{G connected} W (G)z^{v(G)} (see [6] for a complete proof), or equivalently,

(4) P = \frac{kT}{|Ω|} ∑_{G connected} W (G)\frac{z^{v(G)}}{v(G)!} .

Example: the discrete gas. For the discrete gas model introduced before, Equation (4) gives

(5) \frac{P}{kT} = ∑_{G connected}(u − 1)^{e(G)} \frac{z^{v(G)}}{v(G)!}.

In the special case of an infinite repulsive interaction between particles in the same box, that is, α = ∞ and u = 0, the pressure P can also be computed directly. Indeed, in this case, one gets

λ^{n}n!Z(Ω, T, n) = ∑_{c:[n]↦[q]} u^{δ(c)} = #{c : [n] ↦ [q] injective} = q(q − 1) ⋯ (q − n + 1),

and

Z_{gr}(z) = ∑_{n≥0} z^{n}λ^{n}Z(Ω, T, n) = ∑_{n≥0} \binom{q}{n}z^{n} = (1 + z)^{q}.

This expression for the grand canonical partition function comes to no surprise since each of the q boxes contains either nothing (activity 1) or one particle (activity z). Now,

\frac{P}{kT} = \frac{1}{q} log(Z_{gr}(z)) = log(1 + z) = ∑_{n>0} \frac{(−1)^{n−1}}{n} z^{n},

and extracting the coefficient of z^{n} in both side of (5) gives

(6)  (−1)^{n−1} ∑_{G⊆K_{n} connected}(−1)^{e(G)} = (n − 1)!.

Identity (6) is quite surprising at first sight but can be understood by recognising in the left-hand-side the evaluation of the Tutte polynomial of K_{n} counting the root-connected acyclic orientations (acyclic orientation in which the vertex 1 is the only source) [5]. Indeed, in the case of the complete graph K_{n}, root-connected acyclic orientations are linear orderings of [n] in which 1 is the least element, or equivalently, permutations of {2, …, n}. In the next section, we give a combinatorial proof of Equation (6) which avoids introducing the whole theory of the Tutte polynomial (though it is based on it) and prepares for the more evolved proof of Equation (2).

3. Pressure in the hard-core discrete gas and increasing trees

In this section, we give a combinatorial proof of (6) by exhibiting an involution Φ on connected graphs which cancels the contribution of almost all graphs in the sum

∑_{G⊆K_{n} connected}(−1)^{e(G)}.

We consider the lexicographic order on the edges of K_{n} defined by (i, j) < (k, l) if either min(i, j) < min(k, l) or min(i, j) = min(k, l) and max(i, j) < max(k, l). For a graph G and an edge e = (i, j) (not necessarily in G), we denote by G^{>e} the spanning subgraph of G made of the edges which are greater than e. We say that e = (i, j) is G-active if there is a path in G^{>e} connecting i and j and we denote by e_{G}^{∗} the least G-active edge (if there are some). We then define a mapping Ψ on the set of connected graphs by setting: Ψ(G) = G if there is no G-active edge and Ψ(G) = G ⊕ e_{G}^{∗} otherwise.

Lemma 1. The mapping Ψ is an involution on connected graphs.

Proof. • First observe that the image of a connected graph is connected. Indeed, if the edge e_{G}^{∗} exists and belongs to G, then it is in a cycle of G and deleting it does not disconnect G.

• We now prove that any edge is G-active if and only if it is Ψ(G)-active. Suppose that the edge e = (i, j) is G-active and let P be a path of G^{>e} connecting i and j. Since e_{G}^{∗} ≤ e the path P does not contain e_{G}^{∗}, hence P ⊆ Ψ(G)^{>e} and e is Ψ(G)-active. Suppose conversely that e = (i, j) is Ψ(G)-active and let P be a path of Ψ(G)^{>e} connecting i and j. If P does not contain e_{G}^{∗}, then P ⊆ G^{>e} and e is G-active. Otherwise, e_{G}^{∗} > e and there is a path Q of G^{>e_{G}^{∗}} ⊆ G^{>e} connecting the endpoints of e_{G}^{∗}. Thus, there is a path contained in (P − e_{G}^{∗}) ∪ Q ⊆ G^{>e} connecting i and j and again e is G-active.

• By the preceding point, there is a G-active edge if and only if there is a Ψ(G)-active edge and in this case e_{Ψ(G)}^{∗} = e_{G}^{∗}. Thus, Ψ(Ψ(G)) = G. □

The mapping Ψ is an involution and (−1)^{e(G)}+(−1)^{e(Ψ(G))} = 0 whenever G ≠ Ψ(G), hence

(7)  ∑_{G⊆K_{n} connected}(−1)^{e(G)} = ∑_{G⊆K_{n} connected,Ψ(G)=G}(−1)^{e(G)}.

We now characterise the fixed points of the involution Ψ. A tree on {1, …, n} is said increasing if the labels of the vertices are increasing along any simple path starting from the vertex 1.

Lemma 2. A connected graph G has no G-active edge if and only if it is an increasing tree.

Proof. • We suppose first that G is an increasing tree and want to prove that no edge is Gactive. Since G has no cycle, no edge in G is G-active. Consider now an edge e = (i, j) ∉ G and the nearest common ancestor k of i and j (the root vertex of G being the vertex 1). There is an edge e′ = (k, l) containing k on the path of G connecting i and j. Since G is an increasing tree, k ≤ min(i, j) and l ≤ max(i, j). Thus, e′ = (k, l) < e = (i, j) and e is not G-active.

• Suppose now that there is no G-active edge. First observe that G is a tree since if G had a cycle then the minimal edge in this cycle would be active. We now want to prove that the tree G is increasing. Suppose the contrary and consider a sequence of labels 1 = i_{1} < i_{2} < … < i_{r} > i_{r+1} on a path of G starting from the vertex i_{1} = 1. Then, the edge (i_{r−1}, i_{r+1}) is G-active and we reach a contradiction. □

By Lemma 2, the fixed points of the involution Ψ are the increasing trees. The increasing trees on {1, …, n} are known to be in bijection with the permutations of {2, …, n} [10]. Hence, there are (n − 1)! increasing trees on [n] and continuing Equation (7) gives

(−1)^{n−1} ∑_{G⊆K_{n} connected} (−1)^{e(G)} = _{∑_{G⊆K_{n} connected} Ψ(G)=G} (−1)^{e(G)+n−1} = #{increasing trees on [n]} = (n − 1)!.

This completes the proof of Equation (6).

4. Pressure in the hard-core continuum gas and Cayley trees

In the 1-dimensional hard-core continuum gas, the vessel is an interval Ω = [−q/2, q/2] and the potential of interaction between two particles i and j is ϕ(x_{i}, x_{j}) = ∞ if |x_{i} −x_{j}| ≤ 1 and 0 otherwise. By definition, f(x_{i}, x_{j}) ≡ exp(−ϕ(x_{i}, x_{j})/kT) − 1 is equal to -1 if |x_{i} − x_{j}| ≤ 1 and 0 otherwise. Thus, the first Mayer weight of a graph G on n vertices, is

W (q, T, G) = ∫∫_{[− \frac{q}{2} , \frac{q}{2} ]^{n}} ∏_{(i,j)∈G} f(x_{i}, x_{j})dx_{1} … dx_{n} = (−1)^{e(G)} ∫∫_{[− \frac{q}{2} , \frac{q}{2} ]^{n}} ∏_{(i,j)∈G} 𝟙_{|x_{i}−x_{j}|≤1}.

In the thermodynamic limit where the volume q of the Vessel tends to infinity, it becomes interesting to consider the second Mayer’s weight of connected graphs defined by w(G) = lim_{q→∞} \frac{W (q,T,G)}{q} and related to the pressure by P = kT ∑_{G connected} w(G)z^{G}. In [6], it is shown that for any connected graph G on {0, …, n}, the second Mayer weight w(G) equals (−1)^{e(G)}Vol(Π_{G}), where Vol(Π_{G}) is the volume of the n-dimensional polytope

Π_{G} = {(x_{1}, …, x_{n}) ∈ ℝ^{n}/ x_{0} = 0 and |x_{i} − x_{j}| ≤ 1 for all edges (i, j) ∈ G}.

For instance, the polytope Π_{K_{3}} is represented in Figure 1. The rest of this paper is devoted to the proof of Equation (2) given in the introduction.

x_{2}  x_{1} π_{2} π_{5} π_{1}  π_{6} π_{3} π_{4}

Figure 1. The polytope Π_{K_{3}} (dashed) and its decomposition into subpolytopes.

Subpolytopes. As observed by Bodo Lass [7], it is possible to decompose the polytope Π_{G} into subpolytopes of volume 1/n!. Each subpolytope is defined by fixing the integral parts and the relative order of the fractional parts of the coordinates x_{1}, …, x_{n}. Let us first recall some definitions: for any real number x we write x = h(x) + ϵ(x) where h(x) ∈ ℤ is the integral part and 0 ≤ ϵ(x) < 1 is the fractional part. Given a vector of integers h = (h_{1}, …, h_{n}) ∈ ℤ^{n} and a permutation σ ∈ 𝔖_{n}, we denote by π(h, σ) the polytope whose interior is made of the points (x_{1}, …, x_{n}) such that h(x_{i}) = h_{i} for all i = 1 … n and 0 < ϵ(x_{σ^{−1}(1)}) < ⋯ < ϵ(x_{σ^{−1}(n)}) < 1. In particular, the polytope π(h, σ) contains the point (h_{1} + \frac{σ(1)}{n+1}, …, h_{n} + \frac{σ(n)}{n+1},) in its interior.

Observe that the condition |x_{i} − x_{j}| < 1 is equivalent to h(x_{i}) − h(x_{j}) ∈ {0, sign(ϵ(x_{j}) − ϵ(x_{i}))} where the value of sign(x) is -1 if x < 0, +1 if x > 0 and 0 if x = 0. Therefore, a point (x_{1}, …, x_{n}) in the interior of the polytope π(h, σ) is in Π_{G} if and only if (i, j) ∈ G implies h_{i} − h_{j} ∈ {0, sign(σ(j) − σ(i))} with the convention that h_{0} = 0 and σ(0) = 0. This condition only depends on the pair (h, σ), therefore either the polytope π(h, σ) is included in the polytope Π_{G} or the interiors of the two polytopes are disjoints. Moreover, a simple calculation shows that the volume of the polytope π(h, σ) is 1/n! for all h ∈ ℤ^{n}, σ ∈ 𝔖_{n}. This proves the following lemma.

Lemma 3. For any connected graph G on {0, …, n}, the value n!Vol(Π_{G}) counts the pairs h ∈ ℤ^{n}, σ ∈ 𝔖_{n} such that π(h, σ) is a subpolytope of Π_{G}.

For example, the polytope Π_{K_{3}} represented in Figure 1 contains 6 subpolytopes π_{1} = π((−1, −1), 12), π_{2} = π((−1, −1), 21), π_{3} = π((0, −1), 12), π_{4} = π((−1, 0), 21), π_{5} = π((0, 0), 12), π_{6} = π((0, 0), 21) each having volume 1/2.

Rearrangement. Summing the second Mayer’s weights over connected graphs and using Lemma 3 gives

∑_{G∈C_{n}} w(G) = ∑_{G∈C_{n}}(−1)^{e(G)}Vol(Π_{G}) = \frac{1}{n!} _{∑_{h∈ℤ^{n},σ∈𝔖_{n},G∈C_{n}} ; π(h,σ)⊆Π_{G}}(−1)^{e(G)}. (8)

Let σ be a permutation of [n]. For any vector h = (h_{1}, …, h_{n}) in ℤ^{n}, we denote σ(h) = (h_{σ(1)}, …, h_{σ(n)}). For any graph G labelled on {0, …, n}, we denote by σ(G) the graph where the label i is replaced by σ(i) for all i = 1, …, n.

Lemma 4. Let h be a vector in ℤ^{n}, let σ be a permutation of [n] and let G be a graph. Then, π(h, σ) ⊆ Π_{G} if and only if π(σ^{−1}(h), Id) ⊆ Π_{σ(G)}, where Id is the identity permutation.

We omit the proof of Lemma 4 which is straightforward. From this lemma, one gets for any permutation σ of [n],

∑_{h∈ℤ^{n}, G∈C_{n}} ; π(h,σ)⊆Π_{G} (−1)^{e(G)} =_{∑_{h∈ℤ^{n}, G∈C_{n}} ; π(σ^{−1}(h),Id)⊆Π_{σ(G)}} (−1)^{e(G)} =_{∑_{h∈ℤ^{n}, G∈C_{n}} ; π(h,Id)⊆Π_{G}} (−1)^{e(σ^{−1}(G))} =_{∑_{h∈ℤ^{n}, G∈C_{n}} ; π(h,Id)⊆Π_{G}} (−1)^{e(G)}.

where the second equality is obtained by changing the order of summations on the graphs G and on the vectors h. Therefore, continuing Equation (8) gives

(9)  ∑_{G∈C_{n}} w(G) = \frac{1}{n!} ∑_{σ∈𝔖_{n}} _{∑_{h∈ℤ^{n},G∈C_{n}} ; π(h,σ)⊆Π_{G}}(−1)^{e(G)} = _{∑_{h∈ℤ^{n},G∈C_{n}} ; π(h,Id)⊆Π_{G}}(−1)^{e(G)}.

Killing involution. Let h be a vector in ℤ^{n}. We will now evaluate the sum

∑_{G∈C_{n}} ; π(h,Id)⊆Π_{G}(−1)^{e(G)}

thanks to a killing involution similar to the one defined in Section 3. To the vector h = (h_{1}, …, h_{n}) we associate the centroid \overline{h} = (\overline{h}_{0}, \overline{h}_{1}, …, \overline{h}_{n}), where \overline{h}_{i} = h_{i}+\frac{i}{n+1} for i = 0, …, n with the convention that h_{0} = 0. We also denotes by G_{h} the graph on {0, …, n} whose edges are the pairs (i, j) such that |\overline{h}_{i} − \overline{h}_{j}| < 1. Observe that for any graph G, π(h, Id) ⊆ Π_{G} if and only if (\overline{h}_{1}, …, \overline{h}_{n}) ∈ Π_{G} if and only if G ⊆ G_{h}.

We order the edges e = (i, j) of the graph G_{h} by the lexicographic order on the corresponding pairs (\overline{h}_{i}, \overline{h}_{j}), that is, (i, j) < (k, l) if either min(\overline{h}_{i}, \overline{h}_{j}) < min(\overline{h}_{k}, \overline{h}_{l}) or min(\overline{h}_{i}, \overline{h}_{j}) = min(\overline{h}_{k}, \overline{h}_{l}) and max(\overline{h}_{i}, \overline{h}_{j}) < max(\overline{h}_{k}, \overline{h}_{l}). For a graph G ⊆ G_{h} and an edge e = (i, j) in G_{h}, we denote by G^{>e} the set of edges in G that are greater than e and we say that e is (G, h)-active if there is a path in G^{>e} connecting i and j. We also denote by e_{G,h}^{∗} the least (G, h)-active edge if there is any. We then define a mapping Ψ_{h} on the set of connected spanning subgraphs of G_{h} by: Ψ_{h}(G) = G if there is no G-active edges and Ψ_{h}(G) = G ⊕ e_{G,h}^{∗} otherwise.

Lemma 5. For any vector h in ℤ^{n}, the mapping Ψ_{h} is an involution on the connected subgraphs of G_{h}.

The proof of Lemma 5 is identical to the proof of Lemma 1. As a consequence, one gets

∑_{G⊆G_{h} connected}(−1)^{e(G)} = ∑_{G⊆G_{h} connected, Ψ_{h}(G)=G}(−1)^{e(G)}.

We now characterise the fixed points of Ψ_{h}. Let i_{0} be the index of the least coordinate of the centroid \overline{h} (that is, \overline{h}_{i_{0}} = min_{i∈\{0,...,n\}}(\overline{h}_{i})). A tree on \{0, …, n\} is said h-increasing if the labels i_{0}, i_{1}, …, i_{k} of the vertices along any simple path starting at vertex i_{0} are such that \overline{h}_{i_{0}} < \overline{h}_{i_{1}} < ⋯ < \overline{h}_{i_{k}}.

Lemma 6. Let h be a vector in ℤ^{n}. A connected graph G ⊆ G_{h} is an h-increasing tree if and only if there is no (G, h)-active edge.

Proof. • We suppose first that G is an h-increasing tree. Since G has no cycle, no edge in G is active. Consider now an edge e = (i, j) ∉ G and the nearest common ancestor k of i and j (the root vertex of G being the vertex i_{0}). Let also e′ = (k, l) be an edge containing k on the path in G between i and j. Since G is an h-increasing tree, \overline{h}_{k} ≤ min(\overline{h}_{i}, \overline{h}_{j}) and \overline{h}_{l} ≤ max(\overline{h}_{i}, \overline{h}_{j}). Thus, e′ = (k, l) < e = (i, j) and e is not (G, h)-active.

• Suppose now that there is no (G, h)-active edge. First observe that G is a tree since if G had a cycle then the minimal edge in this cycle would be active. We now want to prove that the tree G is h-increasing. Suppose the contrary and consider a sequence of labels i_{0}, i_{1}, …, i_{r}, i_{r+1} such that \overline{h}_{i_{0}} < ⋯ < \overline{h}_{i_{r−1}} < \overline{h}_{i_{r}}, \overline{h}_{i_{r+1}} on a path of G starting from the vertex i_{0}. Then, the edge (i_{r−1}, i_{r+1}) belongs to G_{h} (since |\overline{h}_{i_{r−1}} − \overline{h}_{i_{r+1}}| < max(|\overline{h}_{i_{r−1}} − \overline{h}_{i_{r}}|, |\overline{h}_{i_{r}} − \overline{h}_{i_{r+1}}|) < 1) and is (G, h)-active. We reach a contradiction. □

From Lemma 6, one gets for any vector h ∈ ℤ^{n},

∑_{G⊆G_{h} connected}(−1)^{e(G)} = (−1)^{n} #{h-increasing trees}.

Therefore, continuing Equation (9) gives

(10)  ∑_{G∈C_{n}} w(G) = ∑_{h∈ℤ^{n}}  ∑_{G⊆G_{h} connected}(−1)^{e(G)} = (−1)^{n} ∑_{h∈ℤ^{n}} #{h-increasing trees}.

Cayley trees. We now relate h-increasing trees and Cayley trees.

Lemma 7. Any rooted Cayley tree on {0, …, n} with root i_{0} is h-increasing for exactly one vector h in ℤ^{n} such that \overline{h}_{i_{0}} = min_{i∈\{0,...,n\}}(\overline{h}_{i}).

Proof. Let T be a Cayley tree rooted on i_{0}. The tree T is h-increasing with

\overline{h}_{i_{0}} = min_{i∈\{0,...,n\}}(\overline{h}_{i})

if and only if any vertex j ≠ i_{0} satisfies \overline{h}_{i} < \overline{h}_{j} < \overline{h}_{i} + 1, where i is the father of j. The condition \overline{h}_{i} < \overline{h}_{j} < \overline{h}_{i} + 1 holds if and only if either i < j and h_{j} = h_{i} or j < i and h_{j} = h_{i} + 1. Therefore, tree T is h-increasing with \overline{h}_{i_{0}} = min(\overline{h}_{i}) if and only if for every index i = 0, …, n, the difference h_{i} − h_{i_{0}} is the number of descents in the sequence of labels i_{0}, i_{1}, …, i_{s} = i along the path of T from i_{0} to i (a descent is an index r < s such that i_{r+1} < i_{r}). Knowing that h_{0} = 0 completes the proof. □

It is well known that the number of rooted Cayley trees on {0, …, n} is (n + 1)^{n}. Thus, Lemma 7 gives

(11)  ∑_{G∈C_{n}} w(G) = (−1)^{n} ∑_{h∈ℤ^{n}} #h-increasing trees = (−1)^{n}(n + 1)^{n}.

This completes the proof of Equation (2) and answers a question of Labelle et al. [6].

Acknowledgements: I would like to thank Pierre Leroux who warmly encouraged me to work on the combinatorial interpretations of identities arising from Mayer’s theory of cluster integrals.

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