Order polynomials and P\'olya's enumeration theorem

P\'olya's enumeration theorem is concerned with counting labeled sets up to symmetry. Given a finite group acting on a finite set of labeled elements it states that the number of labeled sets up to symmetry is given by a polynomial in the number of labels. We give a new perspective on this theorem by generalizing it to partially ordered sets and order preserving maps. Further we prove a reciprocity statement in terms of strictly order preserving maps generalizing a classical result by Stanley (1970). We apply our results to counting graph colorings up to symmetry.


Introduction
Counting objects up to symmetry is a basic problem of enumerative combinatorics. A fundamental result in this context is Pólya's enumeration theorem which is concerned with counting labelings of a set of objects modulo symmetry. Here a labeling of a set X is defined as a map f : X → Y where Y is the set of labels. If G is a group acting on X then G also acts on the set of labelings Y X := {f : X → Y }. Pólya's enumeration theorem now states: Theorem 1.1 (Pólya's enumeration theorem [3]). Let G be a finite group acting on a finite set X and let Y be a finite set of n = |Y | labels. Then where Y X /G is the collection of orbits of Y X and c(g) is the number of cycles of g as permutation of X.
We give a new perspective on this theorem by generalizing it in terms of partially ordered sets, or posets for short, and order preserving maps. More precisely, we consider a finite poset P and a group G acting on P by automorphisms. Then G acts in a natural way also on the set of all order preserving maps Hom(P, [n]) from P into the n-chain [n] = {1 < · · · < n}. We show that the number of orbits of Hom(P, [n]) is given by a polynomial Ω P,G (n) which we call the orbital order polynomial. Pólya's enumeration theorem then follows by specializing this result to antichains. Further we give a combinatorial interpretation for Ω P,G (−n) in terms of orbits of strictly order preserving maps. This naturally generalizes the classical polynomiality and reciprocity theorems for order preserving maps due to Stanley [4].
The results can be applied to graph colorings. We consider a finite group G acting by automorphisms on a finite simple graph Γ = (V, E) and the function χ Γ,G (n) counting proper colorings c : V → [n] up to group action. Cameron and Kayibi [1] seem to be the first who considered this function which they called the orbital chromatic polynomial. Previously, Hanlon [2] treated the case of G being the automorphism group of Γ. It is easy to see that χ Γ,G (n) indeed agrees with a polynomial for all n ≥ 1. We further give a representation as a sum of order polynomials.
We also give a combinatorial interpretation for evaluating this polynomial at negative integers in terms of acyclic orientations and compatible colorings. This naturally generalizes Stanley's reciprocity for graph colorings [5].
2. Order preserving Pólya-enumeration 2.1. Groups. Let G be a finite group with identity element e and X be a finite set. A group operation of G on X is a map · : G × X → X such that g · (h · x) = (gh) · x and e · x = x for all g, h ∈ G and x ∈ X. We say that G operates or acts on X. For every g ∈ G we denote by X g the fixpoints of g, i.e., X g = {x ∈ X : g · x = x}. For an element x ∈ X we denote by Gx = {g · x : g ∈ G} the orbit of x. The set of all orbits partitions X and is called X/G.
Burnside's lemma (see e.g. [8, Theorem 10.5]) gives a formula for the number of orbits in terms of fixpoints: Theorem 2.1 (Burnside's lemma). Let G be a finite group acting on a finite set X. Then For an element x ∈ X the stabilizer of x is Stab(x) = {g ∈ G : g · x = x}. The operation can be restricted to any subgroup H ⊆ G. For g ∈ G we denote by g = {e, g, g 2 , . . .} the cyclic subgroup generated by g. The orbit of x under the action of g is denoted by [x] g and is called a cycle of g. In particular, c(g) = |X/ g | is the number of cycles of g. Identifying g with the corresponding permutation on X gives the usual notion of cycles. If a set X has additional structure we say that a group G acts by automorphisms on X if the group operation respects the structure, that is, for all g ∈ G the map x → g · x is a structure preserving bijection.
Let P be a finite poset. Then a group G acts on P by automorphisms if for all g ∈ G and for all p and q in P we have g · p ≺ g · q whenever p ≺ q.
For a finite simple graph Γ = (V, E) an action of a group G on V respects the structure of Γ if for all edges uv ∈ E we have that there is an edge between g · u and g · v for all g ∈ G.
The operation of G on X induces an operation on Y X . This induced operation is defined by

2.2.
Order preserving maps. Let P be a finite poset. A map φ : P → [n] is called order preserving if φ(p) ≤ φ(q) whenever p ≺ q and equality is forbidden for strictly order preserving maps. We denote the set of all order preserving maps by Hom(P, [n]), and the set of strictly order preserving maps by Hom • (P, [n]). Their cardinalities are given by the order polynomials Ω P (n) and Ω • P (n) respectively. The following classical result is due to Stanley [4]. Theorem 2.3 (Stanley [4]). For a finite poset P the function Ω P (n) agrees with a polynomial of degree |P | for all n ≥ 1, and For every finite group G acting on P by automorphisms and for every g ∈ G we define a partial order on P g = P/ g by defining [x] g ≺ [y] g whenever there arex ∈ [x] g andỹ ∈ [y] g such that x ≺ỹ. This, in fact, yields a poset, the quotient poset (see e.g. [6]).
Lemma 2.4. Let P be a finite poset and G a finite group acting by automorphisms on P . Then P g is a poset with c(g) elements for all g ∈ G.
Proof. First we observe that every cycle [x] g is an antichain. For that suppose g l x ≺ g k x for some k, l ≥ 0 and consequently x ≺ g k−l x. From that we get x ≺ g k−l x ≺ g 2(k−l) x ≺ · · · ≺ x as g k−l has finite order which is a contradiction.
An order preserving action of G on P induces an action on Hom(P, [n]) and Hom • (P, [n]) as subsets of [n] P . For n ≥ 1 we define the orbital order polynomials Ω P,G (n) = |Hom(P, [n])/G| and Ω • P,G (n) = |Hom • (P, [n])/G|. The following main theorem states that Ω P,G (n) and Ω • P,G (n) are indeed polynomials for n ≥ 1, and gives formulas in terms of order polynomials: Theorem 2.5. Let G be a finite group acting by automorphisms on a finite poset P . Then for n ≥ 1. In particular, Ω P,G (n) and Ω • P,G (n) agree with polynomials of degree |P | for n ≥ 1.
Proof. We only show equation (1) as the argument for equation (2) is a one-to-one correspondence. Further, observe that deg Ω Pg (n) = c(g) = |P | if and only if g acts trivially on P , and c(e) = |P |.
By applying Theorem 2.5 to antichains, we get Pólya's enumeration theorem in the language of posets: Corollary 2.6. Let G be a finite group acting by automorphisms on a finite antichain A, and let Y = [n]. Then Proof. As A is an antichain we have Hom(A, [n]) = Y A . The result follows by observing that A g is an antichain with c(g) elements for all g ∈ G and Ω Ag (n) = n |Ag| .
is a set of representatives of Hom(A, Y )/S k and therefore Ω A,S k (n) = Ω [k] (n). By Corollary 2.6 we conclude that

Combinatorial reciprocity.
Let G be a finite group acting on a finite poset P by automorphisms. As the number of orbits |Hom(P, [n])/G| agrees with a polynomial by Theorem 2.5 we ponder the question if there is a combinatorial interpretation for evaluating this polynomial at negative integers. For that we have to consider a certain class of order preserving maps. The sign sgn(g) of an element g ∈ G is defined as the sign of g as a permutation of P and is equal to (−1) |P |+c(g) . An order preserving map φ ∈ Hom(P, [n]) is called even if for all g ∈ Stab(φ) we have sgn(g) = 1. The set of all even order preserving maps is denoted by Hom + (P, [n]), and we define Hom Proof. Again, we only show equation (3) as equation (4) follows by analogous arguments. By equation (1) and Theorem 2.3 we have We observe that |P g | = c(g) is the number of orbits under the action of g . Therefore equation (5) becomes  Hence, g∈Stab(φ) sgn(g) = 0 whenever φ is not even. Therefore the right-hand side of equation (5)  Proof. This follows from the fact that every order preserving map from an antichain is automatically strictly order preserving.
Example 2.10. In Example 2.7 we have φ ∈ Hom + (A, [n]) if and only if φ is injective. Therefore Remark 1. An alternative, geometric route is by way of Ehrhart theory of order polytopes. Geometrically the setting can be translated into counting lattice points in order polytopes where the action of the group is given by permuting coordinates. This complements results by Stapledon [7] who considers lattice preserving group actions and counts lattice points inside stable rational polytopes.

Graphs
Let Γ = (V, E) be a finite simple graph and let G be a finite group acting on Γ by automorphisms.
A n-coloring of Γ is a map c : V → [n]. The coloring is called proper if c(v) = c(w) whenever there is an edge between v and w. The action of G on Γ induces an action on the set of all colorings, and also on the set of all proper colorings which we denote by Col n (Γ). The orbital chromatic polynomial χ Γ,G is defined by χ Γ,G (n) = |Col n (Γ)/G| for all n ≥ 1. An orientation σ : E → V of Γ assigns to every edge e a vertex of e called its head. An orientation is acyclic if there are no directed cycles. Every acyclic orientation σ induces a partial order on the vertex set of Γ by defining v ≺ σ w if there is a directed path from v to w. For the corresponding poset we write Γ σ . G acts on the set Σ of all acyclic orientations of Γ: For an edge uv we define (g · σ)(uv) = g · σ(g −1 · uv). The next theorem gives us an expression of χ Γ,G (n) in terms of order polynomials. In particular, χ Γ,G (n) is a polynomial for all n ≥ 1.
Theorem 3.1. Let Γ be a graph and let G be a group acting on Γ. Then G acts on Col n (Γ) and we have for all n ≥ 1. In particular, χ Γ,G (n) agrees with a polynomial of degree |Γ| for all n ≥ 1.
Proof. By Theorem 2.1 we have Let φ be an element of Col n (Γ) g and let σ be the acyclic orientation induced by the coloring φ, i.e. an edge e = uv is oriented from u to v whenever φ(u) < φ(v). Then φ is a strictly order preserving map from Γ σ into [n] and σ is fixed by g, because for every edge vw ∈ E we have v ≺ σ w by definition if and only if φ(v) < φ(w), and as φ is fixed by g, this implies φ(gv) < φ(gw) which is equivalent to gv ≺ σ gw, i.e. σ ∈ Σ g .
Example 3.2. Let k > 2. We consider a cycle C k on k vertices {x i } i∈Z k . Then its symmetry group is the dihedral group which acts on C k by Col n (C k ) sr l .
Let c ∈ Col n (C k ). If l = 2q is even, then sr 2q · c = c ⇔ (r q · c) = s · (r q · c) and therefore Col n (C k ) sr l = Col n (C k ) s = Col n ([ k 2 + 1]) if k is even, 0 otherwise.
If k > 1 is odd, we therefore get with Z k := Z/kZ = r ⊂ D k . If k = p > 2 is a prime this simplifies even further: This example is reminiscient of counting necklaces with colored beads (see e.g. We observe that if G acts on Γ by automorphisms it also acts on Σ n (Γ) by g·(c, σ) = (g·c, g·σ) for all (c, σ) ∈ Σ n (Γ) and g ∈ G. An element (c, σ) ∈ Σ n (Γ) is called even if for all g ∈ Stab((c, σ)) we have sgn(g) = 1 as permutation of the vertices. We denote the set of all even elements of Σ n (Γ) by Σ n,+ (Γ). The action of G restricts to an action on Σ n,+ (Γ). We get the following reciprocity statement: Ω Γ σ g (n).
For G acting trivially on Γ we recover a well-known theorem by Stanley: Corollary 3.5 ([5, Thm. 1.2]). Let Γ be a graph and χ Γ its chromatic polynomial. Then |χ Γ (−n)| equals the number of weakly compatible pairs (c, σ) consisting of a n-coloring c and an acyclic orientation σ of Γ. In particular, |χ Γ (−1)| is the number of acyclic orientations of Γ.
Similarly as in Theorem 2.8 there is a twin reciprocity in the case of graph colorings. We say that a n-coloring c of Γ is even if for all g ∈ Stab(c) we have sgn g = 1 and define Col n,+ (Γ) as the set of all even proper n-colorings of Γ. Then the action of G on Col n (Γ) restricts to an action on Col n,+ (Γ). We further define χ + Γ,G (n) = |Col n,+ (Γ)/G| as the function counting the number of orbits of even proper n-colorings for n ≥ 1. By similar arguments as in Theorem 3.1 and Theorem 3.3 we then have the following: Proposition 3.6. Let Γ be a graph and G a group acting on Γ by automorphisms. Then χ + Γ,G (n) agrees with a polynomial of degree |Γ| for n ≥ 1 and we have χ + Γ,G (−n) = (−1) |Γ| |Σ n (Γ)/G| .