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Set Maps, Umbral Calculus, and the Chromatic Polynomial

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Set Maps, Umbral Calculus, and the Chromatic Polynomial Set maps, umbral calculus, and the chromatic polynomial Gus Wiseman 1 Department of Mathematics, University of California, One Shields Ave., Davis, CA 95616 Abstract Some important properties of the chromatic polynomial also hold for any polynomial set map satisfying pS(x + y)= pT (x)pU (y). T ⊎XU=S Using umbral calculus, we give a formula for the expansion of such a set map in terms of any polynomial sequence of binomial type. This leads to some new expansions of the chromatic polynomial. We also describe a set map generalization of Abel polynomials. Key words: chromatic polynomial, umbral calculus, Abel polynomials 2000 MSC: 05C15, 05A40, 05A18 1 Introduction arXiv:math/0507038v2 [math.CO] 5 Jul 2007 In order to motivate our results we will begin by describing two classical expansions of the chromatic polynomial. Let G be a simple graph with finite vertex set V . Let χG(x) be the number of proper colorings of G with x colors (assignments of colors 1, 2,...,x to the vertices of G so that no two adjacent vertices have the same color). If e is an edge of G, let G\e be G with e removed, and let G/e be G with e contracted to a single vertex. Then χG(x) satisfies the deletion-contraction property, χG(x)= χG\e(x) − χG/e(x). Email address: [email protected] (Gus Wiseman). 1 Research supported in part by NSF VIGRE Grant No. DMS-0135345. Preprint submitted to Elsevier 23 July 2021 This property can be used to obtain the subgraph expansion, |E(H)| c(H) χG(x)= (−1) x , (1) E(HX)⊆E(G) where the sum is over all spanning subgraphs of G (which correspond to sub- sets of the edge set E(G)), and c(H) is the number of connected components of H (see [1, Thm. 10.4] for a proof). In particular, χG(x) is a polynomial, called the chromatic polynomial of G. The coefficient of x is ′ |E(H)| χG(0) = (−1) , E(H)⊆E(G) connectedX where the sum is over all connected spanning subgraphs. Choosing a spanning subgraph of G is equivalent to choosing a set partition of the vertices and a connected spanning subgraph of the restriction of G to each block. Hence (1) may be expressed in the form ℓ(σ) ′ χG(x)= x χG|T (0), (2) σX⊢V TY∈σ where the sum is over all set partitions of the vertex set V , ℓ(σ) denotes the number of blocks of σ, and G|T is the subgraph induced by T ⊆ V . The second expansion is simpler. A proper coloring of G induces a set partition σ of the vertices where two vertices are in the same block if and only if they have the same color. This set partition will be stable, meaning G has no edges with both ends in the same block. Corresponding to such a set partition are (x)ℓ(σ) := x(x − 1)(x − 2) ··· (x − ℓ(σ) + 1) proper colorings. Hence χG(x)= (x)ℓ(σ). σ⊢V stableX Since 1 if G has no edges χG(1) = , 0 otherwise we have χG(x)= (x)ℓ(σ) χG|T (1). (3) σX⊢V TY∈σ Notice the structural similarity between (2) and (3). The present work was motivated by the observation that both expansions are implied by the fact that the chromatic polynomial satisfies the identity χG(x + y)= χG|T (x)χG|U (y), (4) T ⊎XU=V 2 where the sum is over all ordered pairs (T, U) of disjoint sets whose union is the vertex set V . The abstraction of this identity leads to an object that we call a polynomial set map of binomial type. Such a structure, which may be thought of as a generating function satisfying an exponential-like property, has some of the same properties as the chromatic polynomial, yet its definition does not suggest any connection to graph theory. The idea of studying the chromatic polynomial through a generating function such as χG|S (x)S SX⊆V appears to have first been considered by Tutte [22, Eq. (13)], who showed essentially that x x χ (x)S = S = χ (1)S , G|S G|S S⊆V S⊆V S⊆V X X X stable where vertex sets are multiplied by the rule ST = S∪T if S ∩ T = ∅, and ST = 0 otherwise. More recently, Lass [10] has applied this approach to enumeration of acyclic orientations. We will show that, coupled with the notion of binomial type, the set map approach provides a useful tool for studying the chromatic polynomial. Our main result is that χG(x)= aℓ(σ)(x) AχG|T (x), σX⊢V TY∈σ where an(x) is any polynomial sequence of binomial type (meaning it satisfies n n an(x+y)= k=0 k ak(x)an−k(y)), A is the linear functional (on polynomials) defined by AaP n(x) = δn=1, and the sum is over all set partitions of V . This provides a description of the coefficients in the expansion of the chromatic polynomial in terms of any polynomial sequence of binomial type. This result has several interesting consequences. For example, we will show that the coefficients in the expansion of the chromatic polynomial in terms of rising factorials x(n) := x(x+1)(x+2) ··· (x+n−1) can be given the following combinatorial interpretation, originally due to Brenti [2, Theorem 5.5]. Let (k) χG(x)= ckx . kX≥0 |V |−k Then (−1) ck is the number of pairs (σ, α), where σ is a set partition of V with k blocks and α is an acyclic orientation of all edges of G with both ends in the same block of σ. 3 Though the chromatic polynomial is our primary example of a polynomial set map of binomial type, we describe a less trivial example in Section 7. We call it the Abel polynomial set map because it generalizes the Abel polynomial sequence x(x − an)n−1. The coefficients of Abel polynomials are well-known to enumerate planted forests, and we show that the coefficients of the Abel polynomial set map enumerate a similar, more general structure. We begin in Section 3 by studying the algebra of set maps. In Section 4 we briefly review some definitions and results from umbral calculus. This allows us to prove our main result (Theorem 12) in Section 5. Section 6 is devoted to applications to the chromatic polynomial. Finally, in Section 7 we describe a set map generalization of Abel polynomials. 2 Notation Let K be a field of characteristic zero. A polynomial sequence a(x) is a se- quence of polynomials a0(x), a1(x),... ∈ K[x] such that deg(an(x)) = n. This restriction, that degrees agree with indices, is essential in umbral calculus be- cause it ensures that polynomial sequences are bases for K[x]. Because we will not need to regard polynomial set maps as bases, we do not make a similar restriction in defining polynomial set maps. A set partition is a set of disjoint, nonempty sets. We use the symbols σ, π, γ, and ρ exclusively for set partitions. The elements of a set partition are called blocks. The length ℓ(σ) of a set partition σ is the number of blocks. If the union of the blocks is S, then we say that σ is a set partition of S and write σ ⊢ S. The same notation is used for integer partitions, but the meaning should always be evident from context (we have set partitions of sets and integer partitions of integers). Note that the only set partition of ∅ is ∅. We use the following notations: • [n]= {1, 2,...,n}. x! • (x)n = x(x − 1)(x − 2) ··· (x − n +1) = (x−n)! . (n) (x+n−1)! • x = x(x + 1)(x + 2) ··· (x + n − 1) = (x−1)! . 1 if P is true • If P is a proposition, then δP = . 0 if P is false • T ⊎ U = S means T ∪ U = S and T ∩ U = ∅. 4 3 Set Maps In this section we study the ring of set maps. This ring and numerous appli- cations have been studied by Lass (see [10]). Our approach differs from Lass’s in that we define composition directly by (5), whereas Lass defines it using generating functions. The latter approach has the advantage of trivializing most of the results of this section. However, our approach makes clear the relationship between composition and the combinatorics of set partitions. Let V be a set. If p is a function whose domain is the set of all finite subsets of V and whose range is some set E, we call p an E-set map on V , or simply an E-set map. The image of S ⊆ V will always be denoted by pS. Throughout this section we let R be a fixed commutative, associative ring. Definition 1 Let g and h be R-set maps. Define their sum g + h and product g · h to be the R-set maps (g + h)S = gS + hS and (g · h)S = gT hU . T ⊎XU=S It is clear that this makes the set of R-set maps into a commutative, associative ring with unit δS=∅. This is a generalization of the ring of exponential formal power series in the following sense. We can regard a sequence a as the set map aS = a|S|. If h is a set map and hS depends only on |S|, we can likewise regard h as a sequence.
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