Application of graph combinatorics to rational identities of type A
Abstract.
To a word w, we associate the rational function \Psi_{w}=\prod(x_{w_{i}}x_{w_{i+1}})^{1}. The main object, introduced by C. Greene to generalize identities linked to MurnaghanNakayama rule, is a sum of its images by certain permutations of the variables. The sets of permutations that we consider are the linear extensions of oriented graphs. We explain how to compute this rational function, using the combinatorics of the graph G. We also establish a link between an algebraic property of the rational function (the factorization of the numerator) and a combinatorial property of the graph (the existence of a disconnecting chain).
Key words and phrases:
Rational functions, posets, maps1. Introduction
A partially ordered set (poset) \mathcal{P} is a finite set V endowed with a partial order. By definition, a word w containing exactly once each element of V is called a linear extension if the order of its letters is compatible with \mathcal{P} (if a\leq_{\mathcal{P}}b, then a must be before b in w). To a linear extension w=v_{1}v_{2}\ldots v_{n}, we associate a rational function:
\psi_{w}=\frac{1}{(x_{v_{1}}x_{v_{2}})\cdot(x_{v_{2}}x_{v_{3}})\ldots(x_{v_{% n1}}x_{v_{n}})}. 
We can now introduce the main object of the paper. If we denote by \mathcal{L}(\mathcal{P}) the set of linear extensions of \mathcal{P}, then we define \Psi_{\mathcal{P}} by:
\Psi_{\mathcal{P}}=\sum_{w\in\mathcal{L}(\mathcal{P})}\psi_{w}. 
1.1. Background
The linear extensions of posets contain very interesting subsets of the symmetric group: for example, the linear extensions of the poset considered in the article BUTLER are the permutations smaller than a permutation \pi for the weak Bruhat order. In this case, our construction is close to that of Demazure characters DEMAZURE. S. Butler and M. BousquetMélou characterize the permutations \pi corresponding to acyclic posets, which are exactly the cases where the function we consider is the simplest.
Moreover, linear extensions are hidden in a recent formula for irreducible character values of the symmetric group: if we use the notations of FS, the quantity N^{\lambda}(G) can be seen as a sum over the linear extensions of the bipartite graph G (bipartite graphs are a particular case of oriented graphs). This explains the similarity of the combinatorics in article Fe and in this one.
The function \Psi_{\mathcal{P}} was considered by C. Greene GREENE, who wanted to generalize a rational identity linked to MurnaghanNakayama rule for irreducible character values of the symmetric group. He has given in his article a closed formula for planar posets (\mu_{\mathcal{P}} is the Möbius function of \mathcal{P}):
\Psi_{\mathcal{P}}=\left\{\begin{array}[]{cl}0&\text{if }\mathcal{P}\text{ is % not connected,}\\ \prod\limits_{y,z\in\mathcal{P}}(x_{y}x_{z})^{\mu_{\mathcal{P}}(y,z)}&\text{% if }\mathcal{P}\text{ is connected,}\end{array}\right. 
However, there is no such formula for general posets, only the denominator of the reduced form of \Psi_{\mathcal{P}} is known Bo. In this article, the first author has investigated the effects of elementary transformations of the Hasse diagram of a poset on the numerator of the associated rational function. He has also noticed, that in some case, the numerator is a specialization of a Schur function (Bo, paragraph 4.2) (we can also find multiSchur functions or Schubert polynomials).
In this paper, we obtain some new results on this numerator, thanks to a simple local transformation in the graph algebra, preserving linear extensions.
1.2. Main results
1.2.1. An inductive algorithm
The first main result of this paper is an induction relation on linear extensions (Theorem LABEL:th:boucle). When one applies \Psi on it, it gives an efficient algorithm to compute the numerator of the reduced fraction of \Psi_{\mathcal{P}} (the denominator is already known).
1.2.2. A combinatorial formula
If we iterate our first main result in a clever way, we can describe combinatorially the final result. The consequence is our second main result: if we give to the graph of a poset \mathcal{P} a rooted map structure, we have a combinatorial noninductive formula for the numerator of \Psi_{\mathcal{P}} (Theorem LABEL:th:N_combi).
1.2.3. A condition for \Psi_{\mathcal{P}} to factorize
Greene’s formula for the function associated to a planar poset is a quotient of products of polynomials of degree 1. In the nonplanar case, the denominator is still a product of degree 1 terms, but not the numerator. So we may wonder when the numerator N(\mathcal{P}) can be factorized.
Our third main result is a partial answer (a sufficient but not necessary condition) to this question: the numerator N(\mathcal{P}) factorizes if there is a chain disconnecting the Hasse diagram of \mathcal{P} (see Theorem LABEL:factorization_chaine for a precise statement). An example is drawn on figure 1 (the disconnecting chain is (2,5)). Note that we use here and in the whole paper a unusual convention: we draw the posets from left (minimal elements) to right (maximal elements).
1.3. Open problems
1.3.1. Around the map structure
Theorem LABEL:th:N_combi is a cominatorial formula for the numerator of \Psi_{\mathcal{P}} involving a map structure on the corresponding graph. Can we find a formula, which does not depend any additional structure on the graph?
Furthermore if we use orderedembeddings of graphs in \mathbb{R}\times\mathbb{R} (see definition LABEL:def:planar), the map structure is not independant from the poset structure. Is there a way to use this link?
1.3.2. Necessary condition for factorization
The conclusion of the factorization Theorem LABEL:factorization_chaine is sometimes true, even when the separating path is not a chain: see for example Figure 2 (the path (5,6,3) disconnects the Hasse diagram, but is not a chain).
This equality, and many more, can be easily proved using the same method as Theorem LABEL:factorization_chaine. Can we give a necessary (and sufficient) condition for the numerator of a poset to factorize into a product of numerators of subposets? Are all factorizations of this kind?
1.3.3. Characterisation of the numerator
Let us consider a bipartite poset \mathcal{P} (which has only minimal and maximal elements, respectively a_{1},\ldots,a_{l} and b_{1},\ldots,b_{r}). The numerator N(\mathcal{P}) of \Psi_{\mathcal{P}} is a polynomial in b_{1},\ldots,b_{r} which degree in each variable can be easily bounded (Bo, Proposition 3.1). Moreover, we know, by Corollary LABEL:corol:annulation, that N(\mathcal{P})=0 on some affine subspaces of the space of variables. Unfortunately, these vanishing relations and its degree do not characterize N(\mathcal{P}) up to a multiplicative factor. Is there a bigger family of vanishing relations, linked to the combinatorics of the Hasse diagram of the poset, which characterizes N(\mathcal{P})?
This question comes from the following observation: for some particular posets, the numerator is a Schubert polynomial and Schubert polynomials are known to be easily defined by vanishing conditions LASCOUX2008.
1.4. Outline of the paper
In section 2, we present some basic definitions on graphs and posets.
In section 3, we introduce our main object and its basic properties.
In section LABEL:sect_oper, we state our first main result: an inductive relation for linear extensions. The next section (LABEL:sect_exemples) is devoted to some explicit computations using this result.
Section LABEL:sect:combi gives a combinatorial description of the result of the iteration of our inductive relation: we derive from it our second main result, a combinatorial formula for the numerator of \Psi_{\mathcal{P}}.
The last Section (LABEL:sectchainfact) is devoted to our third main result: a sufficient condition of factorization.
2. Graphs and posets
Oriented graphs are a natural way to encode information of posets. To avoid confusions, we recall all necessary definitions in paragraph 2.1. The definition of linear extensions can be easily formulated directly in terms of graphs (paragraph 2.2).
We will also define some elementary removal operations on graphs (paragraph 2.3), which will be used in the next section. Due to transitivity relations, it is not equivalent to perform these operations on the Hasse diagram or on the complete graph of a poset, that’s why we prefer to formulate everything in terms of graphs.
2.1. Definitions and notations on graphs
In this paper, we deal with finite directed graphs. So we will use the following definition of a graph G:

A finite set of vertices V_{G}.

A set of edges E_{G} defined by E_{G}\subset V_{G}\times V_{G}.
If e\in E_{G}, we will note by \alpha(e)\in V_{G} the first component of e (called origin of e) and \omega(e)\in V_{G} its second component (called end of e). This means that each edge has an orientation.
Let e=(v_{1},v_{2}) be an element of V_{G}\times V_{G}. Then we denote by \overline{e} the pair (v_{2},v_{1}).
With this definition of graphs, we have four definitions of injective walks on the graph.
\begin{array}[]{ccc}&\begin{array}[]{c}\text{can not go }\text{backwards}% \end{array}&\begin{array}[]{c}\text{can go }\text{backwards}\end{array}\\ \hline\text{closed}&\text{circuit}&\text{cycle}\\ \hline\text{nonclosed}&\text{chain}&\text{path}\end{array} 
More precisely,
Definition 2.1.
Let G be a graph and E its set of edges.
 chain:

A chain is a sequence of edges c=(e_{1},\ldots,e_{k}) of G such that \omega(e_{1})=\alpha(e_{2}), \omega(e_{2})=\alpha(e_{3}), \ldots and \omega(e_{k1})=\alpha(e_{k}).
 circuit:

A circuit is a chain (e_{1},\ldots,e_{k}) of G such that \omega(e_{k})=\alpha(e_{1}).
 path:

A path is a sequence (e_{1},\ldots,e_{h}) of elements of E\cup\overline{E} such that \omega(e_{1})=\alpha(e_{2}), \omega(e_{2})=\alpha(e_{3}), \ldots and \omega(e_{k1})=\alpha(e_{k}).
 cycle:

A cycle C is a path with the additional property that \omega(e_{k})=\alpha(e_{1}). If C is a cycle, then we denote by HE(C) the set C\cap E.
In all these definitions, we add the condition that all edges and vertices are different (except of course, the equalities in the definition).
Remark 1.
The difference between a cycle and a circuit (respectively a path and a chain) is that, in a cycle (respectively in a path), an edge can appear in both directions (not only in the direction given by the graph structure). The edges, which appear in a cycle C with the same orientation than their orientation in the graph, are exactly the elements of HE(C).
To make the figures easier to read, \alpha(e) is always the leftmost extremity of e and \omega(e) its rightmost one. Such drawing construction is not possible if the graph contains a circuit. But its case will not be very interesting for our purpose.
Example 1.
An example of graph is drawn on figure 3. In the lefthand side, the nondotted edges form a chain c, whereas, in the righthand side, they form a cycle C, such that HE(C) contains 3 edges: (1,6),(6,8) and (5,7).
\psfig{width=48.0pt}\hskip 28.452756pt\psfig{width=48.0pt} 
The cyclomatic number of a graph G is E_{G}V_{G}+c_{G}, where c_{G} is the number of connected components of G. A graph contains a cycle if and only if its cyclomatic number is not 0 (see GRAPH). If it is not the case, the graph is called forest. A connected forest is, by definition, a tree. Beware that, in this context, there are no rules for the orientation of the edges of a tree (often, in the literature, an oriented tree is a tree which edges are oriented from the root to the leaves, but we do not consider such objects here).
2.2. Posets, graphs, Hasse diagrams and linear extensions
In this paragraph, we recall the link between graphs and posets.
Given a graph G, we can consider the binary relation on the set V_{G} of vertices of G:
x\leq y\lx@stackrel{{\scriptstyle\text{\tiny def}}}{{\Longleftrightarrow}}% \left(x=y\text{ or }\exists\ e\in E_{G}\text{ such that }\left\{\begin{array}[% ]{c}\alpha(e)=x\\ \omega(e)=y\end{array}\right.\right) 
This binary relation can be completed by transitivity. If the graph has no circuit, the resulting relation \leq is antisymmetric and, hence, endows the set V_{G} with a poset structure, which will be denoted {\tt poset}(G).
The application {\tt poset} is not injective. Among the preimages of a given poset \mathcal{P}, there is a minimum one (for the inclusion of edge set), which is called Hasse diagram of \mathcal{P} (see figure 4 for an example).
The definition of linear extensions given in the introduction can be formulated in terms of graphs:
Definition 2.2.
A linear extension of a graph G is a total order \leq_{w} on the set of vertices V such that,
for each edge e of G, one has {\alpha(e)}\leq_{w}{\omega(e)}.
The set of linear extensions of G is denoted \mathcal{L}(G). Let us also define the formal sum \varphi(G)=\sum\limits_{w\in\mathcal{L}(G)}w.
We will often see a total order \leq_{w} defined by v_{i_{1}}\leq_{w}v_{i_{2}}\leq_{w}\ldots\leq_{w}v_{i_{n}} as a word w=v_{i_{1}}v_{i_{2}}\ldots v_{i_{n}}.
For example, the linear extensions of the poset drawn in the figure 4 are 1234 and 1324.
Remark 2.
If G contains a circuit, then it has no linear extensions. Else, its linear extensions are the linear extensions of {\tt poset}(G). Thus considering graphs instead of posets does not give more general results.
The following lemma comes straight forward from the definition:
Lemma 2.1.
Let G and G^{\prime} be two graphs with the same set of vertices. Then one has:
E(G)\subset E(G^{\prime})\text{ and }w\in\mathcal{L}(G^{\prime})% \Longrightarrow w\in\mathcal{L}(G); 
w\in\mathcal{L}(G)\text{ and }w\in\mathcal{L}(G^{\prime})\Longleftrightarrow w% \in\mathcal{L}(G\vee G^{\prime}), 
where G\vee G^{\prime} is defined by \left\{\begin{array}[]{l}V(G\vee G^{\prime})=V(G)=V(G^{\prime});\\ E(G\vee G^{\prime})=E(G)\cup E(G^{\prime}).\end{array}\right.
2.3. Elementary operations on graphs
The main tool of this paper consists in removing some edges of a graph G.
Definition 2.3.
Let G be a graph and E^{\prime} a subset of its set of edges E_{G}. We will denote by G\backslash E^{\prime} the graph G^{\prime} with

the same set of vertices as G ;

the set of edges E_{G^{\prime}} defined by E_{G^{\prime}}:=E_{G}\backslash E^{\prime}.
Definition 2.4.
If G is a graph and V^{\prime} a subset of its set of vertices V, V^{\prime} has an induced graph structure: its edges are exactly the edges of G, which have both their extremities in V^{\prime}.
If V\backslash V^{\prime}=\{v_{1},\ldots,v_{l}\}, the graph induced by V^{\prime} will be denoted by G\backslash\{v_{1},\ldots,v_{l}\}. The symbol is the same than in definition 2.3, but it should not be confusing.
Definition 2.5 (Contraction).
We denote by G/e the graph (here, the set of edges can be a multiset) obtained by contracting the edge e (i.e. in G/e, there is only one vertex v instead of v_{1} and v_{2}, the edges of G different from e are edges of G/e: if their origin and/or end in G is v_{1} or v_{2}, it is v in G/e).
Then, if \alpha(e)\neq\omega(e), G/e is a graph with the same number of connected components and the same cyclomatic number as G.
3. Rational functions on graphs
3.1. Definition
Given a graph G with n vertices v_{1},\ldots,v_{n}, we are interested in the following rational function \Psi(G) in the variables (x_{v_{i}})_{i=1\ldots n}:
\Psi(G)=\sum_{w\in\mathcal{L}(G)}\frac{1}{(x_{w_{1}}x_{w_{2}})\ldots(x_{w_{n% 1}}x_{w_{n}})}. 
We also consider the renormalization:
N(G):=\Psi(G)\cdot\prod_{e\in E_{G}}(x_{\alpha(e)}x_{\omega(e)}). 
In fact, we will see later that it is a polynomial. Moreover, if G is the Hasse diagram of a poset, \displaystyle\Psi(G)=\frac{N(G)}{\prod\limits_{e\in E_{G}}(x_{\alpha(e)}x_{% \omega(e)})} is a reduced fraction.
3.2. Pruning invariance
Thanks to the following lemma, it will be easy to compute N on forests (note that these results have already been proved in Bo, but the following demonstrations are simpler and make this article selfcontained).
Lemma 3.1.
Let G be a graph with a vertex v of valence 1 and e the edge of extremity (origin or end) v. Then one has
N(G)=N\big{(}G\backslash\{v\}\big{)}. 
For example,
N\left(\begin{array}[]{c}\mbox{\psfig{width=48.0pt}}\end{array}\right)=N\left(% \begin{array}[]{c}\mbox{\psfig{width=24.0pt}}\end{array}\right)=x_{1}+x_{2}x_% {3}x_{4}. 
Proof.
One wants to prove that:
(x_{\alpha(e)}x_{\omega(e)})\cdot\left(\sum_{w^{\prime}\in\mathcal{L}(G)}\psi% _{w^{\prime}}\right)=\sum_{w\in\mathcal{L}(G\setminus\{v\})}\psi_{w}. 
But one has a map \text{er}_{v}:\mathcal{L}(G)\rightarrow\mathcal{L}(G\setminus\{v\}) which sends a word w^{\prime} to the word w obtained from w^{\prime} by erasing the letter v (see figure LABEL:fig_example_er). So it is enough to prove that, for each w\in\mathcal{L}(G\setminus\{v\}), one has :
(x_{\alpha(e)}x_{\omega(e)})\cdot\left(\sum_{w^{\prime}\in\text{er}_{v}^{1}(% w)}\psi_{w^{\prime}}\right)=\psi_{w}. 
Let us assume that v is the end of e and w=w_{1}\ldots w_{n1}\in\mathcal{L}(G\setminus\{v\}). We denote by k the index in w of the origin of e. The set \text{er}_{v}^{1}(w) is:
\big{\{}w_{1}\ldots w_{i}vw_{i+1}\ldots w_{n1},i\geq k\big{\}} 
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