A Multiset Version of Even-Odd Permutations Identity

The Ising model [1] is a theoretical physics model of the nearest-neighbor interactions in a crystal structure. In the Ising model, the vertices of a graph $G=(V,E)$ represent particles and the edges describe interactions between pairs of particles. The most common example of a two dimensional Ising model is a planar square lattice where each particle interacts only with its neighbors. A factor (weight) $J_{ij}$ is assigned to each edge $\{i,j\}$, where this factor describes the nature of the interaction between particles $i$ and $j$. A physical state of the system is an assignment of $\sigma_{i}\in\{+1,-1\}$ to each vertex $i$. The Hamiltonian (or energy function) of the system is defined as:

$$H(\sigma)=-\sum_{\{i,j\}\in E}J_{ij}\sigma_{i}\sigma_{j}.$$

The distribution of the physical states over all possible energy levels is encapsulated in the partition function:

$$Z(\beta,G)=\sum_{\sigma}e^{-\beta H(\sigma)},$$

where $\beta$ is changed for $\frac{K}{T}$, in which $K$ is a constant and $T$ is a variable representing the temperature.
Motivated by a generalization of a cycle in a graph, a set A of edges is called even if each vertex of V is incident with an even number of edges of A. The generating function of even subsets denoted by $\mathcal{E}(G,x)$ can be defined as

$$\mathcal{E}(G,x)=\sum_{\text{A: A is even}}\prod_{e\in A}x_{e}.$$

It turns out that the Ising partition function for a graph $G$ may be expressed in terms of the generating function of the even sets of the same graph $G$. More precisely, we have the following Van der Waerden’s formula [2]

$$Z(G,\beta)=2^{|V|}\prod_{\{i,j\}\in E}\cosh(\beta J_{ij})\mathcal{E}(G,x)|_{x^{J_{ij}}=\tanh(\beta J_{ij})}.$$

Now, let $G=(V,E)$ be a planar graph embedded in the plane and for each edge $e$ we associate a formal variable $x_{e}$ which can be seen as a weight of that edge. Let $A=(V,A(G))$ be an arbitrary orientation of $G$. If $e\in E$ then $a_{e}$ will denote the orientation of $e$ in $A(G)$ and $a^{-1}_{e}$ will be the reversed orientation to $a_{e}$. We put $x_{a_{e}}=x_{a^{-1}_{e}}=x_{e}$. A circular sequence $p=v_{1},a_{1},v_{2},a_{2},\ldots,a_{n},(v_{n+1}=v_{1})$ is called non-periodic closed walk if the following conditions are satisfied: $a_{i}\in\{a_{e},a^{-1}_{e}:e\in E\},a_{i}\neq a^{-1}_{i+1}$ and $(a_{1},\ldots,a_{n})\neq Z^{m}$ for some sequence $Z$ and $m>1$. We also let $X(p)=\prod_{i=1}^{n}x_{a_{i}}$. We further let $sign(p)=(-1)^{n(p)}$, where $n(p)$ is a rotation number of $p$; i.e., the number of integral revolutions of the tangent vector. Finally put $W(p)=sign(p)X(p)$.
There is a natural equivalence on non-periodic closed walks; that is, $p$ is equivalent with reversed $p$. Each equivalence class has two elements and will be denoted by $[p]$. We assume $W([p])=W(p)$ and note that this definition is correct since equivalent walks have the same sign.
The following beautiful formula is due to Feynman who conjectured it, but did not gave a proof of it. It was Sherman who gave a proof based on a key combinatorial lemma on coin arrangements [3] .

Here is the original statement of the coin arrangement lemma: Suppose we have a fixed collection of $N$ objects of which $m_{1}$ are of one kind, $m_{2}$ are of second kind, $\ldots$, and $m_{n}$ of $n$-th kind. Let $b_{N,k}$ be the number of exhaustive unordered arrangements of these symbols into $k$ disjoint, nonempty, circularly ordered sets such that no two circular orders are the same and none are periodic. Then, we have

$$\sum_{k=1}^{N}(-1)^{k}b_{N,k}=0,~~~~~~(N>1).$$

It is worth to note that when the collection of objects constitute a set of $n$ elements, then the numbers $b_{n,k}$ are exactly Stirling cycle numbers; that is, the number of permutations of the set $\{1,2,\ldots,n\}$ (or $n$ - permutations) with exactly $k$ cycles in its decompositions into disjoint cycles. It is noteworthy that the coin arrangements lemma in this particular case, can be reformulated as the following well-known identity in combinatorics of permutations.

Our main goal here is to formulate a weighted version of the even-odd permutations identity in the multiset setting.

As Knuth has noted in [7, p.36] , the term multiset was suggested by N.G.de Bruijn in a private communication to him. Roughly speaking, a multiset is an unordered collection of elements in which repetition is allowed.

It is worth to note that by a simple counting argument, one can obtain that the number of permutations of the multiset $M=[a_{1}^{m_{1}},a_{2}^{m_{2}},\ldots,a_{n}^{m_{n}}]$ of cardinality $N$ is equal to $\frac{N!}{m_{1}!m_{2}!\cdots m_{n}!}$ .
In the rest of this section, we quickly review the basics of the combinatorics of words. The reader can consult the reference [5].
Let $\Sigma$ be a finite alphabet. The elements of $\Sigma$ are called letters. A finite sequence of elements of $\Sigma$ is called a word ( or string ) over the alphabet $\Sigma$. An empty sequence of letters is called an empty word and is denoted by $\lambda$.
The set of all words over the alphabet $\Sigma$ will be denoted by $\Sigma^{\star}$. We also denote the set of non-empty words by $\Sigma^{+}$. A word $u$ is called a factor ( resp. a prefix, resp. a suffix) of a word $w$, if there exists words $w_{1}$ and $w_{2}$ such that $w=w_{1}uw_{2}$ (resp. $w=uw_{2}$, resp. $w=w_{1}u$).
The $k$-th power of a word $w$ is defined by $w^{k}=ww^{k-1}$ with the convention that $w^{0}=\lambda$. A word $w\in\Sigma^{+}$ is called primitive if the equation $w=u^{n}$ ($u\in\Sigma^{+}$) implies $n=1$. Two words $w$ and $u$ are conjugate if there exist two words $w_{1}$ and $w_{2}$ such that $w=w_{1}w_{2}$ and $u=w_{2}w_{1}$. It is easy to see that the conjugacy relation is an equivalence relation. A conjugacy class (or necklace) is a class of this equivalence relation.
For an ordered alphabet $(\Sigma,<)$, the lexicographic order $\trianglelefteq$ on $(\Sigma^{\star},<)$ is defined by letting $w_{1}\trianglelefteq w_{2}$ if

• •

  $w_{1}=uw_{2},\hskip 11.38092pt(u\in\Sigma^{\star})\hskip 11.38092pt\text{or}$

• •

  $w_{1}=ras,\hskip 8.5359ptw_{2}=rbt\hskip 8.5359pta<b,\hskip 8.5359pt\text{for}\hskip 5.69046pta,b\in\Sigma\hskip 8.5359pt\text{and}\hskip 5.69046ptr,s,t\in\Sigma^{\star}.$

In particular, if $w_{1}\trianglelefteq w_{2}$ and $w_{1}$ is not a proper prefix of $w_{2}$, we write $w_{1}\vartriangleleft w_{2}$.

A word is called a Lyndon word if it is primitive and the smallest word with respect to the lexicographic order in it’s conjugacy class.

The following factorization of the words as a non-increasing product of Lyndon words is of fundamental importance in the combinatorics of words. From now on, we will denote the set of all Lyndon words by $L$.

One of the important results about the characterization of Lydon words is the following.

In this section, we first briefly review basics of the combinatorics of permutations. For more detailed introduction see [6]. From now on, we will denote the set $\{1,2,\ldots,n\}$ by $[n]$.
Recall that a permutation $\tau$ of a set $[n]$ (or simply an $n$-permutation) is a bijective function $\tau:[n]\mapsto[n]$. A one-line representation of $\tau$ is denoted by $\tau=\tau(1)\tau(2)\cdots\tau(n)$ .
Recall that from abstract algebra, we know that any permutation can be written as a product of disjoint cycles. Hence, a representation of a permutation in terms of disjoint cycles is called cycle representation.

The set of all permutations of the set $[n]$ will be denoted by $S_{n}$.

We will denote the number of inversions of a permutation $\tau$ with $inv(\tau)$.

We recall the well-known fact due to Cauchy [8] that for any permutation $\tau\in S_{n}$, the parity of $inv(\tau)$ and $ind_{c}(\tau)$ are the same. Therefore, we can divide the class of all permutations $S_{n}$ into two important subclasses.

Considering the above discussions, the coin arrangements lemma in the case that there exists exactly one coin of each type can be restate as follows.

In the rest of this section, we attempt to formulate a multiset version of the above well-known result in combinatorics of permutations.
For finding the right formulation of the coin arrangement lemma for multisets, we have to first replace permutations of the set $[n]$ with words of length $N$ defined on the multiset $M=[1^{m_{1}},2^{m_{2}},\ldots,n^{m_{n}}]$ of cardinality $N$ . The next step is to find the analogue of the cyclic decomposition of permutations into disjoint cycles. It seems that the Lyndon factorization of a word in which all factors are distinct is the suitable candidate. Hence, we come up with the following analogue of cycle index.

Thus, we finally get the following reformulation of the Sherman’s original coin arrangements lemma.

In the next section, we will give a bijective proof a weighted version of the above coin arrangements lemma.

In this section, we will first give a weighted reformulation of the coin arrangements lemma. Then, we present a bijective proof of our main result by constructing a weight-preserving involution on the set of words. But before doing it, for the sake of completeness, we present the original proof of Sherman based on the so called Witt identity in the context of combinatorial group theory [4] .

Now, considering the Witt identity, the proof of the coin arrangements lemma can be simply obtained by equating the coefficients of monomials of the same degree in both sides of the identity.
To obtain a weighted generalization of the coin arrangements lemma, we first associate a formal variable $u_{a}$ with each letter $a$ of alphabet $\Sigma$ which can be viewed as a weight of that letter. For any Lyndon word $l=i_{1}i_{2}\cdots i_{h}$ , we define the weight $wt(l)$ of the Lyndon word $l\in L$ as the product of weights of it’s letters. That is, $wt(l)=u_{i_{1}}u_{i_{2}}\cdots u_{i_{h}}$ . The weight of an $N$-word $w\in\Sigma^{\star}$, is defined as $wt(w)=\prod_{l\in tup(w)}wt(l)$ . From now on, we will denote the set of all even (resp. odd) $N$-words over $M$ by $E$ (resp. $O$). Thus, a weighted version of the coin arrangement lemma can be read as follows.

The following lemma is the key in the proof of the above theorem.