Cyclic inclusion-exclusion and the kernel of P-partitions

Following the lead of Stanley and Gessel, we consider a linear map which associates to an acyclic directed graph (or a poset) a quasi-symmetric function. The latter is naturally defined as multivariate generating series of non-decreasing functions on the graph (or of P -partitions of the poset). We describe the kernel of this linear map, using a simple combinatorial operation that we call cyclic inclusionexclusion. Our result also holds for the natural non-commutative analog and for the commutative and non-commutative restrictions to bipartite graphs. Résumé. Dans la lignée de Stanley et Gessel, nous considérons une application linéaire qui associe à un graphe dirigé acyclique (ou à un poset) une fonction quasi-symétrique. Celle-ci est définie comme la série génératrice multi-variée des fonction croissantes sur le graphe (ou des P -partitions du poset). Nous décrivons le noyau de cette application linéaire, à l’aide d’une opération combinatoire simple, que nous appelons inclusion-exclusion cyclique. Notre résultat est aussi valable pour l’analogue non-commutatif naturel et les restrictions commutative et non-commutative aux graphes bipartis.


Introduction
Given an acyclic directed graph G = (V, E G ), it is natural to consider the following multivariate generating function where N is the set of positive integers and non-decreasing means that (i, j) ∈ E G implies f (i) ≤ f (j).
An example is given in Section 2.4.A similar definition can be considered for a poset P = (V, < P ), replacing (i, j) ∈ E G by i < P j.
This is a classical object in the algebraic combinatorics literature: using the terminology of the seminal book of Stanley [10], the non-decreasing functions on posets correspond to P -partitions when P has a natural labelling (up to reversing the order of P ).The generating function Γ P has then been considered by Gessel [6], see also Stanley's textbook [11,Section 7.19].While not symmetric in the variables x 1 , x 2 , . .., the function Γ P exhibits some weaker symmetry property and belongs to the now well-studied algebra of quasi-symmetric functions -in fact, the terminology quasi-symmetric function was introduced in [6], precisely to study Γ P .
Although posets are more common objects in the literature, the results of this paper are better formulated in terms of acyclic directed graphs.Obviously the map Γ : G → Γ G defined by (1) can be extended by linearity to the vector space G of formal linear combinations of acyclic directed graphs, that we call here the graph algebra.We refer the reader to [5] for a study of this map (and an extension) from a Hopf algebra point of view.Here, we only focus on the linear structure.
The main result of the present paper is a combinatorial description of the kernel of the map Γ from the graph algebra to quasi-symmetric functions (Theorem 2).This description relies on a simple combinatorial operation, that we call cyclic inclusion-exclusion (the definition and an example are given in Section 3.1).Before giving some background on this operation, let us mention that this description of the kernel of Γ is quite robust.Indeed, we shall prove that cyclic inclusion-exclusion also describes the kernel of some variants of Γ, as follows: • Working with labeled (acyclic directed) graphs, it is natural to associate to them a multivariate generating series in non-commuting variables.This object lives in the algebra of word quasi-symmetric functions [9], sometimes also called quasi-symmetric functions in non-commuting variables; see [1].We give a description of the kernel of this map (denoted Γ nc ) in Theorem 1. • In the long version of this paper [4], we also consider restrictions of the linear maps Γ and Γ nc to bipartite graphs and describe the kernel of these restrictions via cyclic inclusion-exclusion.In all these cases, a byproduct of our proof is the surjectivity of the morphism Γ (respectively Γ nc and their restriction to bipartite graphs).The surjectivity in the commutative non-restricted case was observed by Stanley [12, Note p7], answering a question of Billera and Reiner.
Our proofs use a combination of basic linear algebra, graph combinatorics and (word) quasi-symmetric function manipulations.In the non-commutative/labeled case, we first exhibit a family of graphs so that their images form a Z-basis of word quasi-symmetric functions.Then, we show that these graphs span the quotient of the graph algebra by cyclic inclusion-exclusion relations.With an easy linear algebra argument, this concludes the proof.
The commutative/unlabeled case can be obtained as a corollary of the non-commutative/labeled case.In contrast, restrictions to bipartite graphs must be considered separately from the non-restricted setting.The general structure of the proof is the same in the bipartite setting, although the arguments themselves are quite different.Because of the limited space in this extended abstract, we do not present the bipartite case here.
Along the way, this gives natural bases of the word quasi-symmetric function ring.These new bases are non-commutative lifts of known bases of the quasi-symmetric function ring: in particular, we find natural analogs of Gessel fundamental basis [6] and of two bases introduced respectively by R. Stanley in [12] and by K. Luoto in [8].
Let us now say a word about the cyclic inclusion-exclusion operation and how it has proved useful so far.It has been introduced by the author in the article [3] in the proof of a conjecture of Kerov on irreducible character values of the symmetric group.The question addressed here (whether cyclic inclusion-exclusion relations do or do not span the kernel of Γ) is natural in this context, since it allows to simplify some arguments in the proof.We refer the reader to the long version [4,Section 6] for details.
Remarkably, this operation of cyclic inclusion-exclusion has also been fruitful in a quite different context in [2], where the purpose was to study some rational functions introduced by Greene [7].These functions are indexed by posets and defined as sums over linear extensions of the indexing poset: as such, they automatically verify cyclic inclusion-exclusion relations.This gives an efficient way to compute these rational functions and a powerful tool to study them; see [2].

Labelled and unlabeled graphs
Throughout the paper we work with simple directed graphs, both labeled and unlabeled.
An unlabeled graph is an equivalence class of labeled graphs, with respect to the relabeling operation.

Quasi-symmetric functions
The ring of quasi-symmetric functions was introduced by I. Gessel [6] and may be seen as an extension of symmetric functions.A modern introduction can be found in [11,Section 7.19].
Let n be a non-negative integer.A composition (or integer composition) of n is a list I = (i 1 , i 2 , . . ., i r ) of positive integers, whose sum is equal to n.The notation I n means that I is a composition of n and (I) denotes the number of parts of I.In numerical examples, it is customary to omit parentheses and commas.For example, 212 is a composition of 5.
Consider the algebra C[[X]] of formal power series in a totally ordered countable set of commutative variables X = {x 1 , x 2 , . . .}.
. .with finitely many non-zero entries.For such a sequence, we denote by v ← the finite list obtained by omitting the zeroes in v.

Definition 2.2 A formal power series f ∈ C[[X]] is said to be quasi-symmetric if and only if
• there exists m such that each monomial with a nonzero coefficient in f has degree at most m; • and, for any v and w such that v ← = w ← , the coefficients of X v and X w in P are equal.
One can easily prove that the set of quasi-symmetric functions is a subalgebra of C[[X]], called the quasi-symmetric function ring and denoted QSym.
It should be clear that any symmetric function is quasi-symmetric.The algebra QSym of quasisymmetric functions has a basis of monomial quasi-symmetric functions (M I ) indexed by compositions I = (i 1 , . . ., i r ), where

Word quasi-symmetric functions
The natural non-commutative analog of QSym is the algebra of word quasi symmetric functions, denoted by WQSym.We recall here its construction, following the presentation of Bergeron and Zabrocki [1, Section 5.2].An equivalent, but slightly different presentation, using packed words instead of setcompositions, can be found in a paper of Novelli and Thibon [9, Section 2.1].Consider a totally ordered countable alphabet of non-commuting variables A = {a 1 , a 2 , . . .}.We denote by C A the algebra of formal power series in the set of variables A. Monomials in A are canonically indexed by finite words w on the alphabet N as follows a w = a w1 a w2 . . .a w |w| .The evaluation eval(w) of a word w is the integer sequence v = (v 1 , v 2 , . . .), where v i is the number of letters i in w.Then the commutative image of a w is X eval(w) .
To a word w on the (ordered) alphabet N of length , we associate the set-composition I = ∆(w) such that j ∈ I |{wr:wr≤wj }| (for every j in [ ]).In other words, I 1 contains the positions of the smallest letter in w, I 2 the positions of the second smallest, and so on.For example, ∆(275525) = 15|346|2.Definition 2.4 A formal power series f in C A is a word quasi symmetric function if and only if • there exists m such that each monomial with a nonzero coefficient in f has degree at most m; • and the coefficients of a v and a w in f are equal as soon as ∆(v) and ∆(w) coincide.
One can easily prove that the set WQSym of word quasi symmetric functions is an algebra.A linear basis of WQSym is given as follows: Note that sending the variables a 1 , a 2 , . . . to their commutative analogs x 1 , x 2 , . . .defines an algebra morphism from WQSym to QSym.Clearly, its commutative image is M 212 (given in Example 2.3).It is clear that Γ nc (G) is a word quasi-symmetric function.Therefore, Γ nc extends to a linear map from G to WQSym.The map Γ defined by Eq. ( 1) in the introduction and introduced by Gessel in [6] is a quotient of Γ nc , replacing the variables a 1 , a 2 , . . .by their commutative analogs x 1 , x 2 , . . .

Remark 2.8
The map Γ nc , and hence its quotient Γ, are Hopf algebra morphisms (with a suitable Hopf algebra structure on G).Indeed, this is the restriction to posets of the map Γ (1,1,1) introduced by Foissy and Malvenuto in [5].

Definition and example
Let G be a directed graph.Assume that, as an undirected graph, it contains a cycle C.Formally, such a cycle C is a list C = (x 1 , . . ., x k ) such that, for 1 ≤ i ≤ k, either (x i , x i+1 ) is an edge of G, or (x i+1 , x i ) is an edge of G (where, by convention, x k+1 := x 1 ).In the first case, we say that (x i , x i+1 ) is in C + .In the second case, we say that Another description of the sets C + and C − is the following.Edges of C have two orientations: their orientation in the cycle C and their orientation as edges of G.We denote C + (respectively C − ) the set of edges of C, for which these two orientations coincide (respectively do not coincide).
Finally, for a subset D of edges of G, we denote by G \ D the (directed acyclic) graph obtained from G by erasing the edges in D (and keeping the same set of vertices).Then, in the graph algebra G , we set Example 3.1 Consider the graph G ex from Figure 1.The non-oriented version of G ex contains several cycles, among them C ex = (6, 2, 3, 5, 1).This cycle is represented as a subgraph of G ex in Figure 1 with the two edge orientations described above.Then the set C + ex is equal to {(6, 2), (2, 3), (3, 5)} and CIE Gex,Cex is given in Figure 1.

Cyclic inclusion-exclusion relations
Proof: Let n be the size of G. From the definitions, one has: where [condition] is 1 if the condition is fulfilled and 0 else.
The idea of the proof is to show that for any function f : [n] → N, one has If f is not a G\C + non-decreasing function, then each summand of ( 3) is zero and the conclusion holds trivially in this case.Consider now a G\C + non-decreasing function f : [n] → N.For such a function f , we define Hence the left-hand side of (3) can be rewritten as D f ⊆D⊆C + (−1) |D| , which is equal to zero if and only if D f = C + .Therefore, we need to show that, for any G\C + non-decreasing function, D f is strictly included in C + .
We proceed by contradiction.Suppose that we can find a G\C + non-decreasing function f for which D f = C + .This means that, for each (x, y) in C + , one has f (x) > f (y).Besides, since f is a G\C + non-decreasing function, one has f (x) ≤ f (y) for any edge (x, y) of G which is not in C + .Recall now that C is a cycle in the undirected version of G. Formally, C is a list (x 1 , . . ., x k ) such that, for 1 ≤ i ≤ k, (by convention, x k+1 = x 1 ) • either (x i , x i+1 ) is an edge of G and (x i , x i+1 ) ∈ C + ; • or (x i+1 , x i ) is an edge of G and (x i , x i+1 ) ∈ C − .Using the remarks above, we can conclude in both cases that f (x i ) ≥ f (x i+1 ).Bringing everything together, As C + cannot be empty (otherwise, (x k , . . ., x 1 ) would be a directed cycle), at least one of these inequalities should be strict.We have reached a contradiction and D f must be strictly included in C + , which ends the proof of the proposition.2 Proposition 3.2 gives some relations between the word quasi-symmetric functions Γ nc (G).We call these relations cyclic inclusion-exclusion relations (CIE relations for short).Formally, the elements (CIE G,C ) span a subspace of G , that we shall denote C , which is included in the kernel of Γ nc .
We shall prove in the next section that any relation among the Γ nc (G) can be deduced from CIE relations.In other words, the space C is exactly the kernel of Γ nc .We will also prove a similar result for its commutative quotient Γ. Remark 3.3 With the definition above, a cycle can go several times through the same vertex, or even contain several copies of the same edge.We could also forbid this in the definition of cycles, the results of the paper would be identical.

A Z-basis of WQSym
The purpose of this Section is to prove that Γ nc (G I ) is a Z-basis of WQSym.The proof requires to consider two additional bases of WQSym and to observe that three change of basis matrices are unitriangular (with respect to different orders of the basis elements).It will be more convenient for us to work with descent-starred permutations, instead of set-compositions.

Definition 4.3
We define a descent-starred permutation as a couple (σ, D) such that D is a subset of the descent set {i; σ(i) > σ(i + 1)} of the permutation σ.The descents in D are termed starred.
In numerical example, we represent a descent-starred permutation (σ, D) by the word notation of σ in which the elements of index in D are followed by a star.For example the descent-starred permutation (3142, {3}) will be denoted 314 2. Proof: From the numerical notation of a set-composition I, we sort each part in decreasing order and remove vertical bars to get the word notation of σ.Then mark with a star the descents inside the same part of I.For example, the descent-starred permutation associated to 15|346|2 is 5 16 4 32.This is clearly a bijection.2 Let us define two families of word quasi-symmetric functions indexed by descent-starred permutations: M (σ,D) and F (σ,D) .Both are defined as a sum Let us focus on M (σ,D) .We require that k σ(x) = k σ(x+1) for x ∈ D, which implies that the function x → k x should be constant on the parts of the associated set-composition I.Moreover, the conditions Remark 4.5 The commutative projection of F (σ,D) is F J , where F is the fundamental basis of QSym and J the (integer) composition associated with the set D (we use here the terminology of [11,Section 7.19]).
Proof: Omitted: the argument relies on triangular changes of basis, introducing an intermediate family (L (σ,D) ).See the long version [4] for details. 2 We now explain how Γ nc (G I ) expands on the F basis.If I = (I 1 , . . ., I r ) is a set-composition, we consider the following set DSP(I) of descent-starred permutations: • As a word σ = w 1 • • • w r , where w m is a permutation of I m ; • The descent in position x is starred if σ x and σ x+1 are in the same part of I.In other words, for each m, we mark the descents in w m , but not the potential descent created by concatenating w m and w m+1 .For example, take I ex = 15|346|2, then DSP(I ex ) contains the following 12 descent-starred permutations:  Proof: Let (σ, D) be a descent-starred permutation in DSP(I) and a k1 • • • a kn a monomial in F (σ,D) .Consider j 1 in I m and j 2 in I m+1 for some m.By definition of DSP(I), clearly, j 1 appears before j 2 in σ, which implies that k j1 ≤ k j2 .Therefore j → k j is a G I non-increasing function and the monomial a k1 • • • a kn also appears in Γ nc (G I ).
Conversely, we have to prove that any monomial in Γ nc (G I ) appears in exactly one of the functions F (σ,D) , for (σ, D) in DSP(I).Let f be a G I non-decreasing function from [n] to N. We want to construct (σ, D) in DSP(I) such that, for all x between 1 and n − 1, one has f σ(x) ≤ f σ(x + 1) , with strict inequalities for x in D.
By definition of DSP(I), σ should be a concatenation w 1 • • • w r , where each w m is a permutation of I m .Moreover, the potential descent between w m and w m+1 is not starred.Since f is G I non-decreasing, the inequality (6) holds when σ(x) and σ(x + 1) lie respectively in I m and I m+1 for some m (by definition of DSP(I), in this case, x / ∈ D).Therefore, we focus on the case where σ(x) and σ(x + 1) are in the same part I m of I.
Let us consider the restriction f m of f to I m .We claim that there exists a unique word w m which is a permutation of I m and such that f m (w m i ) ≤ f m (w m i+1 ), with strict inequality whenever i is a descent of w m .Indeed this word is obtained by ordering lexicographically the pairs (f m (y), y) y∈Im and keeping only the second element of each pair.We mark the descents in w m and by concatenating all the words w m (for 1 ≤ m ≤ r), we get a descent-starred permutation (σ, D) in DSP(I).By construction, this descent-starred permutation is the unique one in DSP(I) such that a f (1) • • • a f (n) appears in F (σ,D) , which ends the proof. 2 Example 4.8 Take I ex as in Example 4.2, Γ nc (G Iex ) is given by Eq. (5).The summation set can be split as follows: • either k 1 ≤ k 5 or k 5 < k 1 ; • besides, the integers k 3 , k 4 and k 6 fulfill exactly one of the 6 following inequalities: Combining both case distinctions yield 12 different cases, and Γ nc (G Iex ) is a sum of 12 different terms which are the F functions indexed by the 12 descent-starred permutations in DSP(I ex ) (which are listed above).

A generating family for the quotient
We will now show that (G I ), where I runs over all set-compositions, spans the quotient G /C .Here is the key combinatorial lemma in this section.Proof: Let G be an acyclic directed graph with vertex set [n] and edge set E G .Throughout the proof, we denote ∼ the following symmetric relation: x ∼ y if, in G, there is no directed path from x to y, nor from y to x.When x ∼ y, the directed graphs G (x,y) and G (y,x) obtained from G by adding respectively an edge from x to y or from y to x are still acyclic.We distinguish three cases.
Case 1: G is not the graph of a transitive relation.
In other words, there exist x, y and z such that • there is an edge from x to y and from y to z in G; • there is no edge from x to z.We consider G 0 = G (x,z) the graph obtained from G by adding an edge between x and z.As a directed graph, G 0 is acyclic: otherwise, there would be a path from z to x in G and, together with (x, y) and (y, z), this path would be a directed cycle in G.But the non-oriented version of G 0 contains a cycle C = (x, z, y).Using the notation of Section 3.1, one has C + = {(x, z)} and the corresponding cyclic inclusion-exclusion element is CIE G0,C = G 0 − G. Hence, in G /C , one has G = G 0 and the statement is true in this case.
This case is illustrated on the left-hand side of Figure 2 with examples of graphs G and G 0 .Dashed edges are edges of G and G 0 that do not play a role in the proof.Case 2: the relation ∼ is not an equivalence relation.
By assumption, there exist vertices x, y, z such that • there is a path (x, v 1 , • • • , v k , z) from x to z in G; • one has x ∼ y and y ∼ z.By definition of ∼, the graph G (x,y) is acyclic.Moreover, it does not contain a path from z to y.Indeed, as y ∼ z in G, such a path should use the edge (x, y) and thus be the concatenation of a path from z to x with the edge (x, y).But G does not contain a path from z to x (indeed, it contains a path from x to z and no directed cycles).
But G 0 \ {(x, y), (y, z)} is G, so, in the quotient G /C , one has and the statement is proved in this case.
This case is illustrated on the right-hand side of Figure 2 with examples of graphs G and G 0 .Here, the dashed edge illustrates the fact that the length of the path P from x to z can be arbitrary.Potential extra edges and vertices of G and G 0 have not been represented for more readability.

Example 2 . 5
Consider the set-composition I = 25|4|13.Then the associated basis element of WQSym is M I = k<l<m a m a k a m a l a k .

Proposition 3 . 2
For any graph G and cycle C of G, one has:

Remark 3 . 4 A
weaker form of Proposition 3.2 (in the commutative setting) has been established in [2, Theorem 4.1].The structure of the proof is identical.4 The kernel of Gessel's morphism 4.1 The graphs G I Definition 4.1 Let I = (I 1 , . . ., I r ) be a set-composition of [n].We consider the directed graph G I with vertex set [n] and edge set j<k I j × I k .In other words, there is an edge between x and y if the index of the set of I containing x is smaller than the one of the set-containing y.Example 4.2 Take I ex = 15|346|2.Then G Iex and the associated word quasi symmetric function are

Lemma 4 . 4
Descent-starred permutations of n are in bijection with set-compositions of [n].

Corollary 4 . 9
The family Γ nc (G I ) is a Z-basis of WQSym.Proof: If (σ, D) is the descent-starred permutation associated by Lemma 4.4 to a set-composition I of n of length r, then the size of D is n − r.Besides, for each element (σ , D ) ∈ DSP(I), the size of D is smaller than n − r, unless (σ , D ) = (σ, D).Hence Proposition 4.7 implies that the matrix of Γ nc (G I ) in the basis F (σ,D) is unitriangular with respect to the order (σ , D ) < 2 (σ, D) ⇔ |D | < |D| and Γ nc (G I ) is a Z-basis of WQSym.

Lemma 4 .Fig. 2 :
Fig. 2: On the left (resp.right): graphs G and G0 in the first (resp.second) case of the proof of Lemma 4.11.