2.1 Partitions and symmetric functions

In this section, we give a brief overview of the necessary background material. Further details can be found in the textbooks of Stanley [Reference Stanley and Fomin41], Manivel [Reference Manivel26], and Macdonald [Reference Macdonald25].

An $\lambda = (\lambda _1 \geq \lambda _2 \geq \dots \geq \lambda _k)$ is a finite nonincreasing sequence of positive integers. We define $\ell (\lambda )$ to be the length of the sequence $\lambda $ (so above, $\ell (\lambda ) = k$ ). We define $r_i(\lambda )$ to be the number of occurrences of i as a part of $\lambda $ (so, e.g., $r_1(2,1,1,1) = 3$ ). If

$$\begin{align*}\sum_{i=1}^{\ell(\lambda)} \lambda_i = n, \end{align*}$$

we say that $\lambda $ is a partition of n, and we write $\lambda \vdash n$ . The $\lambda $ is a set of squares called , left- and top-justified (i.e., in “English notation”), such that the ith row from the top contains $\lambda _i$ cells. For example, the Young diagram of shape $(2,2,1)$ is . Let $C(\lambda )$ denote the set of cells of the Young diagram of shape $\lambda $ . If $\mathsf {c} \in C(\lambda )$ is a cell of the Young diagram of shape $\lambda $ , we write $\mathsf {c}^\uparrow $ for the cell immediately above $\mathsf {c}$ (assuming it exists), $\mathsf {c}^\rightarrow $ for the cell immediately right of $\mathsf {c}$ , and so on. We write $\lambda ^{\mathsf {T}}$ for the of $\lambda $ , the integer partition whose Young diagram is obtained from that of $\lambda $ by exchanging rows and columns.

Let $S_{\mathbb {N}}$ denote the set of all permutations of the set $\mathbb {N}$ fixing all but finitely-many elements. A is a power series of bounded degree such that for each permutation $\sigma \in S_{\mathbb {N}}$ , we have $f(x_1,x_2,\dots ) = f(x_{\sigma (1)}, x_{\sigma (2)}, \dots )$ . The set of symmetric functions forms a $\mathbb {C}$ -vector space. Furthermore, if $\Lambda ^d$ denotes the set of symmetric functions that are homogeneous of degree d, then each $\mathrm {Sym}^d$ is a vector space, and

$$\begin{align*}\mathrm{Sym} = \bigoplus_{d=0}^{\infty} \mathrm{Sym}^d \end{align*}$$

as graded vector spaces.

The dimension of $\mathrm {Sym}^d$ as a $\mathbb {C}$ -vector space is equal to the number of integer partitions of d, and many bases of symmetric functions are conveniently indexed by integer partitions. Below, we provide some commonly used bases that will be used in this article.

The space of symmetric functions is equipped with a natural inner product $\langle \cdot , \cdot \rangle $ ; it may be defined by

$$\begin{align*}\langle h_{\lambda}, m_{\mu} \rangle = \delta_{\lambda,\mu}, \end{align*}$$

where $\delta _{\bullet , \bullet }$ denotes the Kronecker delta function.

We will also need the basis of Schur functions. A of shape $\lambda $ is a function $T: C(\lambda ) \rightarrow \mathbb {N},$ typically visualized by writing the value $T(\mathsf {c})$ in the cell $\mathsf {c}$ . A Young tableau T of shape $\lambda $ is if for each cell $\mathsf {c} \in C(\lambda )$ , we have ${T(\mathsf {c}) \leq T(\mathsf {c}^\rightarrow )}$ and $T(\mathsf {c}) < T(\mathsf {c}^\downarrow )$ whenever the cells in question exist. We write $\mathrm {SSYT}(\lambda )$ for the set of all semistandard Young tableaux of shape $\lambda $ . The $s_{\lambda }$ is defined by

$$\begin{align*}s_{\lambda} = \sum_{T\in \mathrm{SSYT}(\lambda)} x^T, \quad \text{where} \quad x^T = \prod_{\mathsf{c} \in C(\lambda)} x_{T(\mathsf{c})}.\end{align*}$$

As $\lambda $ ranges over integer partitions, the Schur functions are another basis of $\mathrm {Sym}$ . The inner product on $\mathrm {Sym}$ also satisfies

$$\begin{align*}\langle s_{\lambda}, s_{\mu} \rangle = \delta_{\lambda,\mu}. \end{align*}$$

When $f \in \mathrm {Sym}$ is a symmetric function and $\{b_{\lambda }\}$ is a basis of symmetric functions indexed by integer partitions $\lambda $ , the notation $[b_{\mu }]f$ denotes the coefficient of $b_{\mu }$ when f is expanded in the b-basis. A symmetric function $f \in \mathrm {Sym}$ is said to be if $[b_{\mu }]f$ is nonnegative for every integer partition $\mu $ .

2.2 K-theoretic Schubert calculus

The $\Gamma _k = \mathrm {Gr}_k(\mathbb {C}^\infty )$ is the parameter space of k-dimensional vector subspaces of the space of all eventually-zero sequences of complex numbers. The space $\Gamma _k$ can be given the structure of a projective Ind-variety and has a cell decomposition into cells $\Gamma _\lambda $ indexed by partitions with at most k parts. Each $\Gamma _\lambda $ induces a cohomology class $\sigma _\lambda \in H^\star (\Gamma _k)$ and classically we have $H^\star (\Gamma _k) \cong \mathrm {Sym}_k = \mathrm {Sym} \cap \mathbb {C}[x_1, \dots , x_k]$ with the isomorphism taking the class of the cell $\sigma _\lambda $ to the Schur polynomial $s_\lambda (x_1, \dots , x_k)$ .

Each cell-closure in $\Gamma _k$ also has a structure sheaf, inducing a class in the representable ring $K^0(\Gamma _k)$ . These K-theoretic classes are represented by inhomogeneous symmetric polynomials called Grothendieck polynomials $\overline {s}_\lambda (x_1, \dots , x_k)$ .

A of shape $\lambda $ is a filling T of each cell of $C(\lambda )$ with a nonempty set of positive integers. The set-valued tableau T is if for each cell ${\mathsf {c} \in C(\lambda )}$ , we have $\max T(\mathsf {c}) \leq \min T(\mathsf {c}^\rightarrow )$ and $\max T(\mathsf {c}) < \min T(\mathsf {c}^\downarrow )$ whenever the cells in question exist. In other words, T is semistandard if every Young tableau formed by choosing one number from the set of each cell is semistandard. For example,

is a set-valued tableau of shape $\lambda = (3,2)$ . Let $\mathrm {SV}(\lambda )$ denote the set of all semistandard set-valued tableaux of shape $\lambda $ . The $\overline {s}_{\lambda }$ is

$$\begin{align*}\overline{s}_{\lambda} = \sum_{T \in \mathrm{SV}(\lambda)} (-1)^{|T|-\ell(\lambda)}x^T, \end{align*}$$

where $|T| = \sum _{\mathsf {c} \in C(\lambda )} |T(\mathsf {c})|$ and $x^T = \prod _{\mathsf {c} \in C(\lambda )} \prod _{i \in T(\mathsf {c})} x_i$ . Note that $\overline {s}_{\lambda }$ contains terms of degree greater than or equal to $|\lambda |$ , and that the sum of all of its lowest-degree terms is equal to $s_{\lambda }$ . This tableau formula for $\overline {s}_\lambda $ is due to Buch [Reference Buch6]. For further background on K-theoretic Schubert calculus and symmetric Grothendieck functions, see [Reference Monical, Pechenik and Searles28, Reference Pechenik and Yong32].

We will also need the $\underline {s}_\lambda $ defined by

$$\begin{align*}\langle \overline{s}_{\lambda}, \underline{s}_{\mu} \rangle = \delta_{\lambda,\mu}. \end{align*}$$

Dual symmetric Grothendieck functions were first introduced explicitly in [Reference Lam and Pylyavskyy22] in relation to the K-homology of $\Gamma _k$ ; however, they are also implicit in the earlier work [Reference Buch6]. Each $\underline {s}_\lambda $ contains terms of degree less than or equal to $|\lambda |$ ; moreover, the sum of all of its highest-degree terms is equal to $s_{\lambda }$ . Although an attractive tableau formula for $\underline {s}_\lambda $ was given in [Reference Lam and Pylyavskyy22], we do not recall it here, as we will not need it.

2.3 Graphs and coloring

Here, we recall basic notions, terminology, and notations from graph theory. For further details, see the textbooks [Reference Diestel14, Reference West47].

A G consists of a set V of , and a set E of unordered pairs of distinct vertices called . All graphs in this article are simple, so there are no loops and no multi-edges. When $\{v_1,v_2\} \in E(G)$ , we will typically denote this edge by $v_1v_2$ and say $v_1$ and $v_2$ are . Two graphs $G,G'$ are if there is a bijection $\phi : V(G) \to V(G')$ such that, for all vertices $v, w \in V(G)$ , we have $vw \in E(G)$ if and only if $\phi (v) \phi (w) \in E(G')$ . In this article, we consider graphs up to isomorphism.

The $K_d$ with d vertices is the graph such that $V(K_d) = [d]$ , and

$$\begin{align*}E(K_d) = \{vw: v, w \in [d], v \neq w\}.\end{align*}$$

The n-vertex $P_n$ has vertex set $V(P_n) {\kern-1pt}={\kern-1pt} [n]$ and edge set $E(P_n) {\kern-1pt}={\kern-1pt} \{ uv : u,v {\kern-1pt}\in{\kern-1pt} [n], v-u = 1\}$ . The $K_{1,3}$ has vertex set $V(K_{1,3}) = [4]$ and edge set $E(K_{1,3}) = \{\{1,2\},\{1,3\},\{1,4\}\}$ .

An of a graph G is a graph H such that $V(H) \subseteq V(G)$ and

$$\begin{align*}E(H) = \{ vw \in E(G) : v,w \in V(H)\}. \end{align*}$$

We say the graph G is if no induced subgraph of G is isomorphic to H. We will be especially interested in claw-free graphs.

A (or ) of a graph G is a set $S \subseteq V(G)$ of vertices such that for each $v, w \in S$ , $vw \notin E(G)$ . A of a graph G is a set $S \subseteq V(G)$ of vertices such that for each $v \neq w \in S$ , $vw \in E(G)$ .

For $\alpha : V(G) \to \mathbb {N}$ a vertex weight function of the graph G, the $\boldsymbol {\alpha }$ is the graph $C_\alpha (G)$ obtained by blowing up each vertex v into a clique of $\alpha (v)$ vertices. More formally, $C_\alpha (G)$ has vertex set $ V(C_\alpha (G)) = \{(v,i) : v \in V(G), i \in [\alpha (v)] \}. $ In $C_\alpha (G)$ , the vertices $(v,i)$ and $(w,j)$ are adjacent either if $vw \in E(G)$ or if both $v=w$ and $i \neq j$ .

Given a vertex $v \in V(G)$ , its $N(v)$ is defined by $N(v) = \{w: vw \in E(G)\}$ . Given $S \subseteq V(G)$ and $v \in V(G)$ with $v \notin S$ , we let $vS \subseteq E(G)$ denote the set of edges $\{vs: s \in S\}$ . The of a graph G by a pair of distinct vertices $v, w \in V(G)$ , denoted $G/vw$ , is the graph with vertex set

$$\begin{align*}V(G/vw) = \left(V(G) \backslash \{v,w\}\right) \cup \{z_{vw}\}, \end{align*}$$

where $z_{vw}$ is a new vertex, and edge set

$$\begin{align*}E(G/vw) = \left(E(G) \backslash \big( vN(v) \cup wN(w) \big)\right) \cup \big( z_{vw} N(v) \cup z_{vw} N(w) \big). \end{align*}$$

A of a graph G is a function $\kappa : V(G) \rightarrow \mathbb {N}$ . A coloring $\kappa $ of G is if $\kappa (a) \neq \kappa (b)$ whenever $ab \in E(G)$ .

The [Reference Stanley39] of a graph G is the power series

$$\begin{align*}X_{G} = \sum_{\kappa} \prod_{v \in V(G)} x_{\kappa(v)} \end{align*}$$

where the first sum ranges over all proper colorings $\kappa $ of G. Note that, for every graph G, $X_G \in \mathrm {Sym}$ .

2.4 Posets and their incomparability graphs

A (partially-ordered set) $(P, \leq )$ is a set P together with a binary relation $\leq $ that is ( $a \leq b$ and $b \leq c$ implies $a \leq c$ ), ( $a \leq a$ ), and ( $a\leq b$ and $b \leq a$ implies $a = b$ ). For $a,b \in P$ , we write $a < b$ if $a \leq b$ and $a \neq b$ . We often write P as shorthand for $(P, \leq )$ and decorate the relation as $\leq _P$ for clarity as needed. For more background on posets than is provided here, see [Reference West47].

When $a,b \in P$ are such that $a \not \leq b$ and $b \not \leq a$ , we say a and b are . We write $\mathbf {n}$ for the unique totally ordered n-element poset and call such a poset a . The $P + Q$ of posets $(P, \leq _P), (Q, \leq _Q)$ is the disjoint union of sets $P \sqcup Q$ with the relation $a \leq _{P+Q} b$ if and only if either $a,b \in P$ with $a \leq _P b$ or $a,b \in Q$ with $a \leq _Q b$ .

We say $(Q, \leq _Q)$ is a of $(P, \leq _P)$ if Q is a subset of P and, for all $a,b \in Q$ , we have $a \leq _Q b$ if and only if $a \leq _P b$ . Two posets $(P, \leq _P), (Q, \leq _Q)$ are if there is a bijection $\phi : P \to Q$ such that, for all $a,b \in P$ , we have $a \leq _P b$ if and only if $\phi (a) \leq _Q \phi (b)$ . If $(P, \leq _P), (Q, \leq _Q)$ are any two posets, we say that $(P, \leq _P)$ is $\boldsymbol {(Q, \leq _Q)}$ if no subposet of P is isomorphic to Q. We will be mostly interested in posets that are $(\mathbf {3} + \mathbf {1})$ -free.

Associated with any poset P is its $I(P)$ . This is the graph whose vertex set is $V(I(P)) = P$ and whose edge set is $E(I(P)) = \{ab: a,b \in P, a \not \leq b, b \not \leq a \}$ . That is to say, edges connect incomparable elements of the poset. It is straightforward to see that the poset P is $(\mathbf {3} + \mathbf {1})$ -free if and only if its incomparability graph is claw-free; however, many claw-free graphs are not incomparability graphs of posets.

3.1 Main definition

A $(G,\alpha )$ consists of a graph G together with a function $\alpha \colon V(G) \rightarrow \mathbb {N}; $ we call $\alpha $ the on the vertices of G. A $\boldsymbol {\alpha }$ of G is a function $ \kappa : V(G) \to 2^{\mathbb {N}} \backslash \{\emptyset \} $ assigning to each $v \in V(G)$ a set of $\alpha (v)$ distinct colors in $\mathbb {N}$ , subject to the constraint that when $uv \in E(G)$ , we have $\kappa (u) \cap \kappa (v) = \emptyset $ . Note that these conditions are equivalent to saying that every choice of a single element from each $\kappa (v)$ yields a proper coloring of G. A of G is a proper $\alpha $ -coloring for some weight function on the vertices of G.

The of the vertex-weighted graph $(G,\alpha )$ is

$$\begin{align*}X_{G}^\alpha = \sum_{\kappa} \prod_{v \in V(G)} \prod_{i \in \kappa(v)} x_i, \end{align*}$$

where the first sum runs over all proper $\alpha $ -colorings of G. Note that up to a scalar factor depending only on $\alpha $ , the set chromatic symmetric function $X_{G}^\alpha $ equals the chromatic symmetric function $X_{C_\alpha (G)}$ of the $\alpha $ -clan graph of G.

Definition 3.1 The of a graph G is the symmetric power series

$$\begin{align*}\overline{X}_G = \sum_{\alpha} X_{G}^\alpha, \end{align*}$$

where $\alpha $ ranges over all weight functions of the vertex set $V(G)$ .

In other words, $\overline {X}_G$ enumerates all colorings of G by nonempty sets of colors, such that adjacent vertices receive disjoint sets of colors. Note that $\overline {X}_G$ is not a homogeneous symmetric function, but rather consists of $X_G$ plus terms of degree higher than $|V(G)|$ .

Although weight functions are used in the definition of $\overline {X}_G$ , the function $\overline {X}_G$ is independent of any particular one. We will find it useful to also consider a vertex-weighted analog of $\overline {X}_G$ . Let $\alpha $ and $\omega $ be independent vertex weight functions on G. Define

$$\begin{align*}X_{(G,\omega)}^\alpha = \sum_{\kappa} \prod_{v \in V(G)} \left(\prod_{i \in \kappa(v)} x_i\right)^{\omega(v)}, \end{align*}$$

where again the first sum runs over all proper $\alpha $ -colorings of G. Finally, we define the of the vertex-weighted graph $(G,\omega )$ to be

$$\begin{align*}\overline{X}_{(G,\omega)} = \sum_{\alpha} X_{(G,\omega)}^\alpha, \end{align*}$$

where the sum is over all weight functions $\alpha $ . In this way, $\overline {X}_{(G,\omega )}$ is a generating function for proper set colorings of G.

3.2 A K-theoretic monomial expansion

For $\lambda $ an integer partition, let $K_\lambda $ denote the vertex-weighted complete graph $(K_{\ell (\lambda )}, \omega )$ , where $\omega (i) = \lambda _i$ for each i. It is straightforward to see that $X_{K_\lambda } = \widetilde {m}_{\lambda }$ , the augmented monomial symmetric function. Thus, by analogy, we define

We call $\overline {\widetilde {m}}_{\lambda }$ the . To justify this definition, we show that the Kromatic symmetric function of every graph (even every vertex-weighted graph) is a positive sum of K-theoretic augmented monomial symmetric functions.

First, we need some additional definitions. We define a C of a graph G to be a collection of (distinct) stable sets of G such that every vertex of $V(G)$ is in at least one element of C. In symbols, this means that

$$\begin{align*}\bigcup_{S \in C} S = V(G); \end{align*}$$

note that this union is not required to be disjoint. We write $\mathsf {SSC}(G)$ for the family of all stable set covers of G. For $C \in \mathsf {SSC}(G)$ , if G is endowed with a vertex weight function $\omega $ , let $\lambda (C)$ be the partition of length $|C|$ whose parts are $\sum _{v \in S} \omega (v)$ for $S \in C$ . Finally, let the of the color i in a proper set coloring $\kappa $ be

$$\begin{align*}\{v \in V(G) : i \in \kappa(v)\}, \end{align*}$$

the set of vertices of G that receive color i (possibly among other colors) under $\kappa $ .

Proposition 3.4 For any vertex-weighted graph $(G,\omega )$ , we have

$$\begin{align*}\overline{X}_{(G,\omega)} = \sum_{C \in \mathsf{SSC}(G)} \overline{\widetilde{m}}_{\lambda(C)}. \end{align*}$$

We define the $\overline {p}_\lambda $ to be the Kromatic symmetric function of a disjoint union of vertices with weights $\lambda _1, \dots , \lambda _{\ell (\lambda )}$ . Observe that the ordinary chromatic symmetric function of this graph yields the classical power sum symmetric function $p_\lambda $ . For some small graphs G, the expansions of $\overline {X}_G$ in the $\overline {\widetilde {m}}_{\lambda }$ -basis and the $\overline {p}$ -basis are collected in Table 1.

Table 1

Kromatic symmetric functions of some small graphs as determined by implementing the deletion–contraction relation of Proposition 3.7 in Python, expressed in the K-theoretic $\overline {\widetilde {m}}$ -basis, as well as in the $\overline {p}$ -basis. Since the latter expansion is infinite, we write explicitly only the $\overline {p}_\lambda $ with $|\lambda | \leq |V(G)|+1$ , suppressing higher order terms (“h.o.t.”).

3.3 A deletion–contraction relation

The Kromatic symmetric function for vertex-weighted graphs also admits a deletion–contraction relation, analogous to that of [Reference Crew and Spirkl10] for the chromatic symmetric function, although somewhat more complicated. We first need to set up some additional notation.

Recall that, given $S \subseteq V(G)$ and $v \in V(G)$ with $v \notin S$ , $vS$ denotes the set of edges $\{vs: s \in S\} \subseteq E(G)$ . Let $(G,\omega )$ be a vertex-weighted graph, and let $v, w$ be distinct vertices such that $e = vw \notin E(G)$ . The graph $G^\star $ has vertex set

$$\begin{align*}V(G^\star) = V(G) \cup \{ z^\star\}, \end{align*}$$

where $z^\star $ is a new vertex, and edge set

$$\begin{align*}E(G^\star) = E(G) \cup \{vw, vz^\star, wz^\star\} \cup z^\star N(v) \cup z^\star N(w). \end{align*}$$

If G has a vertex weight function $\omega $ , we define an induced vertex weight function $\omega ^\star $ on $G^\star $ by

$$\begin{align*}\omega^\star(u) = \begin{cases} \omega(v)+\omega(w), \quad &\text{if}\ u = z^\star;\\ \omega(u), \quad &\text{if}\ u \in V(G). \end{cases} \end{align*}$$

We also define graphs $G^1, G^2$ with vertex sets

$$\begin{align*}V(G^i) = V(G) \end{align*}$$

and edge sets

$$\begin{align*}E(G^1) = E(G) \cup e \cup vN(w) \quad \text{and} \quad E(G^2) = E(G) \cup e \cup wN(v). \end{align*}$$

When G has a vertex weight function $\omega $ , there are induced vertex weight functions $\omega ^i$ on $G^i$ given by

$$\begin{align*}\omega^1(u) = \begin{cases} \omega(v)+\omega(w), \quad &\text{if}\ u = v; \\ \omega(u), \quad &\text{otherwise}; \end{cases} \end{align*}$$

and

$$\begin{align*}\omega^2(u) = \begin{cases} \omega(v)+\omega(w), \quad &\text{if}\ u = w; \\ \omega(u), \quad &\text{otherwise}. \end{cases} \end{align*}$$

In the contracted graph $G/e$ , we give it the weight function $\omega /e$ defined by

$$\begin{align*}(\omega/e)(u) = \begin{cases} \omega(v)+\omega(w), \quad &\text{if}\ u = z_{vw}; \\ \omega(u), \quad &\text{otherwise}. \end{cases} \end{align*}$$

Finally, let $G \cup e$ be the graph $(V(G),E(G)\cup \{e\}$ ).

Proposition 3.7 Let $(G,\omega )$ be a vertex-weighted graph, and let v and w be distinct vertices such that $e = vw \notin E(G)$ . Then

(3.1) $$ \begin{align} \overline{X}_{(G,\omega)} = \overline{X}_{(G / e, \omega / e)}+ \overline{X}_{(G \cup e, \omega)}+ \overline{X}_{(G^1, \omega^1)}+\overline{X}_{(G^2, \omega^2)}+\overline{X}_{(G^\star, \omega^\star)}. \end{align} $$

The deletion–contraction relation of Proposition 3.7 can be used to yield algorithmically the $\overline {\widetilde {m}}_\lambda $ -expansion of a Kromatic symmetric function $\overline {X}_{(G,\omega )}$ in an alternative fashion to Proposition 3.4. Define the of a graph G to be $\operatorname {\mathrm {\mathsf {ts}}}(G) = |\operatorname {\mathrm {\mathsf {SS}}}(G)|-|V(G)|$ , where $\operatorname {\mathrm {\mathsf {SS}}}(G)$ denotes the collection of all stable sets of G. Since any single vertex of a graph is a stable set, we may view the total stability as the number of nontrivial stable sets. Thus, note that $\operatorname {\mathrm {\mathsf {ts}}}(G) \geq 0$ and that equality holds if and only if G is a complete graph.

Proof We proceed by induction on the total stability $\operatorname {\mathrm {\mathsf {ts}}}(G)$ . If $\operatorname {\mathrm {\mathsf {ts}}}(G) = 0$ , then G is a complete graph and the result is trivial. Otherwise, it is sufficient to show that after applying Proposition 3.7 to $(G,\omega )$ , each of the resulting five graphs

$$\begin{align*}G/e, G \cup e, G^1, G^2, G^\star \end{align*}$$

has strictly smaller total stability than G does. We consider each of these five graphs in turn.

• • ( $\underline{G/e}$ ): We have $|V(G/e)| = |V(G)|-1$ . On the other hand, the stable sets of $G/e$ not containing $z_{vw}$ are in obvious bijection with the stable sets of G containing neither v nor w. Moreover, there is a bijection between the stable sets of $G/e$ containing $z_{vw}$ and $\{ S \in \operatorname {\mathrm {\mathsf {SS}}}(G) : v, w\in S\}$ . Together, this gives a bijection between stable sets of $G/e$ and those stable sets of G which contain either both or neither of v and w. Since $\{v\}$ and $\{w\}$ are stable sets of G, we have $|\operatorname {\mathrm {\mathsf {SS}}}(G/e)| \leq |\operatorname {\mathrm {\mathsf {SS}}}(G)| - 2$ , and so

  $$\begin{align*}\operatorname{\mathrm{\mathsf{ts}}}(G/e) = |\operatorname{\mathrm{\mathsf{SS}}}(G/e)| - |V(G/e)| \leq \big( |\operatorname{\mathrm{\mathsf{SS}}}(G)| - 2 \big) - \big( |V(G)|-1 \big) = \operatorname{\mathrm{\mathsf{ts}}}(G) -1, \end{align*}$$
  as needed.

• • ( $\underline{G \cup e}$ ): We have $|V(G \cup e)| = |V(G)|$ . Clearly, $\operatorname {\mathrm {\mathsf {SS}}}(G \cup e) \subseteq \operatorname {\mathrm {\mathsf {SS}}}(G)$ . However, this inclusion is strict since $\{v,w\} \in \operatorname {\mathrm {\mathsf {SS}}}(G) \backslash \operatorname {\mathrm {\mathsf {SS}}}(G \cup e)$ . Thus, $\operatorname {\mathrm {\mathsf {ts}}}(G \cup e) < \operatorname {\mathrm {\mathsf {ts}}}(G)$ .

• • ( $\underline{G^1}$ ): We have $|V(G^1)| = |V(G)|$ . Again, it is clear that $\operatorname {\mathrm {\mathsf {SS}}}(G^1) \subseteq \operatorname {\mathrm {\mathsf {SS}}}(G)$ and the inclusion is strict since $\{v,w\} \in \operatorname {\mathrm {\mathsf {SS}}}(G) \backslash \operatorname {\mathrm {\mathsf {SS}}}(G^1)$ .

• • ( $\underline{G^2}$ ): The analysis is the same as for $G^1$ .

• • ( $\underline{G^\star} $ ): We have $|V(G^\star )| = |V(G)|+1$ . Let $X = \{S \in \operatorname {\mathrm {\mathsf {SS}}}(G) : v, w \in S \}$ and let $Y = \operatorname {\mathrm {\mathsf {SS}}}(G) \backslash X$ . There is an obvious injection of $\{ S : \operatorname {\mathrm {\mathsf {SS}}}(G^\star ) : z^\star \notin S \}$ into Y, since $vw \in E(G^\star )$ . We may also biject $\{ S : \operatorname {\mathrm {\mathsf {SS}}}(G^\star ) : z^\star \in S \}$ with X by mapping $S \mapsto (S \backslash \{z^\star \}) \cup \{v,w\}$ . This latter map is well-defined since such an S does not include $v, w$ , or any vertex in $N(v)$ or $N(w)$ . Combining these bijections yields a bijection of $\operatorname {\mathrm {\mathsf {SS}}}(G^\star )$ with $\operatorname {\mathrm {\mathsf {SS}}}(G)$ , so $|\operatorname {\mathrm {\mathsf {SS}}}(G^\star )| = |\operatorname {\mathrm {\mathsf {SS}}}(G)|$ . We conclude that

  $$\begin{align*}\operatorname{\mathrm{\mathsf{ts}}}(G^\star) = |\operatorname{\mathrm{\mathsf{SS}}}(G^\star)| - |V(G^\star)| \leq |\operatorname{\mathrm{\mathsf{SS}}}(G)| - \left( |V(G)|+1 \right) < |\operatorname{\mathrm{\mathsf{SS}}}(G)| - |V(G)| = \operatorname{\mathrm{\mathsf{ts}}}(G), \end{align*}$$
  as needed.

Therefore, the corollary follows by induction on $\operatorname {\mathrm {\mathsf {ts}}}$ .

3.4 Grothendieck positivity

In 1996, Gasharov [Reference Gasharov15] proved that $X_G$ is Schur-positive for G a claw-free incomparability graph of a poset P by showing that $[s_{\lambda }]X_G$ enumerates objects he called P-tableaux. We now define a generalization of these objects that is enumerated by $[\overline {s}_{\lambda }]\overline {X}_G$ . Informally, a Grothendieck P-tableau of shape $\lambda $ consists of a P-tableau (as defined in [Reference Gasharov15]) of shape $\mu $ for some $\mu \subseteq \lambda $ layered with a semistandard Young tableau of shape $\lambda /\mu $ that also satisfies “flagging” restrictions on each row. (Similar flagging conditions appear in [Reference Lenart24] with relation to Schur-basis expansions of Grothendieck symmetric functions; we do not know a direct relation between our Grothendieck P-tableaux and [Reference Lenart24], nor with the flagged tableaux of [Reference Wachs46].)

Example 3.10 Let $\lambda = (3,3,2,1)$ . Consider the poset $P=\mathbf {3}+ \mathbf {1}$ with four elements $a,b,c,d$ satisfying $a < b < c$ and no other inequalities. Then

are Grothendieck P-tableaux of shape $\lambda $ .

Proof The basic structure of our proof is as follows. We use a generalized Jacobi–Trudi formula [Reference Lascoux and Naruse23, Equation (4)] to write a dual symmetric Grothendieck function as a sum of products of complete homogeneous symmetric functions. Then, for any graph G, the inner product of this expression with $\overline {X}_G$ yields a formula for the coefficient of $\overline {s}_\lambda $ in the Grothendieck expansion of $\overline {X}_G$ in terms of its monomial expansion. In the case that G is a claw-free incomparability graph, we then extend Gasharov’s [Reference Gasharov15] theory of P-arrays to collect terms in this expansion and extend his sign-reversing involution to cancel all terms except those corresponding to Grothendieck P-tableaux.

Now, we give the details of this argument. Let G be a claw-free incomparability graph and let P be a poset such that $G = I(P)$ . The graph G being claw-free is equivalent to the poset P being $(\mathbf {3} + \mathbf {1})$ -free.

Fix a positive integer $n \in \mathbb {N}$ and a partition $\lambda \vdash n$ . Let $k = \ell (\lambda )$ . Let $N \geq 2n$ be fixed, and let $S_N$ denote the symmetric group of permutations of $[N]$ , with identity element $\mathrm {id}_{S_N}$ . For $\pi \in S_N$ , the of $\pi $ is

$$\begin{align*}\operatorname{\mathrm{sgn}} \pi = \begin{cases} +1, \quad \text{if}\ \pi\ \text{is in the}\ \textit{alternating group}\ A_N \subset S_N; \\ -1, \quad \text{otherwise.} \end{cases} \end{align*}$$

We will write

as a shorthand for the number of k-element multisets with elements of n types.

We will use the notation that for f a symmetric function, is a restriction to finitely many variables, and is a restriction to $N+r$ variables with r of them set equal to $1$ . Recall the dual symmetric Grothendieck function $\underline {s}_\lambda $ from Section 2.2. Lascoux and Naruse [Reference Lascoux and Naruse23, Equation (4)] show that $\underline {s}_{\lambda }$ may be expanded as

$$ \begin{align*} \underline{s}_{\lambda}[x_N] &= \det(s_{\lambda_i-i+j}[x_N+i-1]) \\ &= \sum_{\pi \in S_N} \operatorname{\mathrm{sgn}}(\pi) \prod_i s_{\lambda_i-i+\pi(i)}[x_N+i-1], \end{align*} $$

a K-theoretic analog of the Jacobi–Trudi formula.

We apply here the simplification from just before [Reference Lascoux and Naruse23, Equation (4)],

to simplify the above to

(3.2)

From here, we can pass to general symmetric functions for convenience since equality on a sufficiently large finite number of variables implies equality on infinitely many variables. Thus, we find that

(3.3)

where the last equality follows by replacing $\pi $ by $\pi ^{-1}$ . We also note that an essentially equivalent formula for $\underline {s}_\lambda $ appears in the work of Iwao [Reference Iwao21].

Taking the inner product of both sides of Equation (3.3) with $\overline {X}_G$ , we find that

(3.4)

where $\lambda (\pi ,l_1,\dots ,l_{l(\lambda )})$ is the partition whose multiset of parts is $\{\lambda _{\pi (i)}-\pi (i)+i-l_{\pi (i)}\}_{i=1}^{l(\lambda )}$ , and $[m_{\lambda (\pi ,l_1,\dots ,l_{l(\lambda )})}]\overline {X}_G$ denotes the coefficient of the corresponding monomial symmetric function in the expansion of $\overline {X}_G$ .

We now proceed to extend the proof of Gasharov [Reference Gasharov15] to the Kromatic symmetric function. Let P be a poset. Taking an isomorphic copy of P if necessary, we can assume that no positive integers are elements of P and that $\emptyset $ is not an element of P. A $\boldsymbol {\lambda }$ is a pair $(\pi , A)$ , where $\pi \in S_N$ and A is a map $A : \mathbb {N} \times \mathbb {N} \to P \sqcup \mathbb {N} \sqcup \{ \emptyset \}$ satisfying the following properties (we write $a_{ij}$ as shorthand for the element $A((i,j))$ ):

• • $a_{ij}$ must equal $\emptyset $ unless $i \leq \ell (\lambda )$ and $j \leq \lambda _{\pi (i)}-\pi (i)+i$ ;

• • for each $p \in P$ , there is some $(i,j) \in \mathbb {N} \times \mathbb {N}$ such that $a_{ij} = p$ ;

• • if $a_{ij} \in P$ for some $(i,j) \in \mathbb {N} \times \mathbb {N}$ with $j> 1$ , then $a_{i(j-1)} \in P$ and $a_{i(j-1)} <_P a_{ij}$ ;

• • if $a_{ij} \in \mathbb {N}$ , then $a_{ij} \leq \pi (i)-1$ ;

• • if $a_{ij} \in \mathbb {N}$ for some $(i,j) \in \mathbb {N} \times \mathbb {N}$ with $j> 1$ , then either $a_{i(j-1)} \in P$ , or $a_{i(j-1)} \in \mathbb {N}$ and $a_{i(j-1)} \leq a_{ij}$ .

We generally think of A as a partial filling of an infinite matrix by elements of P, where coordinates $(i,j)$ with $a_{ij} = \emptyset $ are thought of as unfilled. Under this interpretation, the bullet points state that in addition to the restriction on which integers can appear in which row, the entries in each row of A are left-justified and consist of an increasing chain in P followed by a weakly increasing string of positive integers.

From the definitions, it is now straightforward to verify that the sum in Equation (3.4) is equal to

(3.5) $$ \begin{align} [\overline{s}_{\lambda}]\overline{X}_G = \sum_{(\pi, A)} \operatorname{\mathrm{sgn}}(\pi), \end{align} $$

where the sum ranges over all Grothendieck P-arrays of type $\lambda $ . The choice of $\pi $ determines the shape of the array, the choice of $l_i$ gives the number of cells in row i that contain positive integers, the m-coefficient covers all choices of poset elements filling the appropriate shape (note that each stable set in G corresponds uniquely to a chain in P), and the product of multiset coefficients covers all possible choices of weakly increasing sequences of positive integers for the rows in the remaining cells.

We claim that the sum in Equation (3.5) evaluates to the number of Grothendieck P-tableaux of shape $\lambda $ . This will follow by exhibiting a sign-reversing involution $\Psi $ of the set of all pairs $(\pi , A)$ of Grothendieck P-arrays that are not Grothendieck P-tableaux.

First, note that if $\pi $ is a non-identity permutation, then for i such that $\pi (i)> \pi (i+1)$ we have $\lambda _{\pi (i)} - \pi (i) + i < \lambda _{\pi (i+1)} - \pi (i+1) + i + 1$ , so any Grothendieck P-array with $\pi $ not equal to the identity permutation is not a Grothendieck P-tableau, as it does not have partition shape.

The sign-reversing involution $\Psi $ that we need is a mild extension of that given by Gasharov in his original proof [Reference Gasharov15, Proof of Theorem 3]. We call a position $(i,j)$ of a Grothendieck P-array with $i \geq 2$ a

• • if $a_{ij} \in P$ , and either $a_{(i-1)j} \notin P$ or $a_{ij} <_P a_{(i-1)j}$ ; or

• • if $a_{ij} \in \mathbb {Z}$ , and either $a_{(i-1)j} = \emptyset $ or $a_{(i-1)j} \in \mathbb {Z}$ with $a_{ij} \leq a_{(i-1)j}$ .

That is, $(i,j)$ is a flaw if it and the cell above it violate the conditions for the array to be a Grothendieck P-tableau. In particular, a Grothendieck P-array is a Grothendieck P-tableau if and only if it has no flaws.

The involution $\Psi $ is as follows. Given $(\pi , A)$ a non-tableau Grothendieck P-array, let c be the leftmost column in which a flaw occurs. Let r be the bottom-most row in which column c has a flaw. We define $\Psi (\pi , A) = (\pi ', A')$ , where

• • $\pi '$ is formed by applying the transposition $(r-1 \,\,\, r)$ to $\pi $ ;

• • $A'$ is formed by swapping each $a_{(r-1)j}$ with $a_{r(j+1)}$ for every $j \geq c$ (i.e., swapping the elements of row $r-1$ that are weakly right of column c with those elements of row r that are strictly right of column c).

We write $a^{\prime }_{ij}$ for the entry in position $(i,j)$ of the array $A'$ .

Clearly, the map $\Psi $ is sign-reversing, so it suffices to show that $\Psi $ takes non-tableau Grothendieck P-arrays to non-tableau Grothendieck P-arrays and that $\Psi $ is an involution. To establish these properties, it is enough to show that $\Psi $ is well-defined and preserves the flaw used (since it is straightforward to observe that no flaw is created to the left of or below the used flaw).

We split into cases based on whether or not $a_{rc} \in P$ . If $a_{rc} \in P$ and $c> 1$ , note that $a_{r(c-1)} \in P$ as well, so necessarily $a_{(r-1)(c-1)} \in P$ , as the opposite would contradict our choice of $(r,c)$ as a leftmost flaw.

Given this observation, it is simple to verify that $\Psi $ is a flaw-preserving involution when $a_{rc} \in P$ , by using the same argument as in [Reference Gasharov15] (the additional cases where some cells contain positive integers are straightforward). The crux of this part of Gasharov’s argument is that, if in the newly formed array $A'$ , we have that $a^{\prime }_{(r-1)(c-1)}$ and $a^{\prime }_{(r-1)c}$ are both in P, then they must satisfy $a^{\prime }_{(r-1)(c-1)} <_P a^{\prime }_{(r-1)c}$ . This follows from the fact that $a_{(r-1)(c-1)} <_P a_{r(c+1)}$ in A whenever both are in P, which in turn follows from P being a ( $\mathbf { 3} + \mathbf {1}$ )-free poset (otherwise consider $a_{(r-1)(c-1)}$ and $a_{r(c-1)} <_P a_{rc} <_P a_{r(c+1)}$ ). Additionally, the integers in rows r and $r+1$ of $A'$ satisfy the required upper bounds, since these bounds changed correspondingly as $\pi $ changed to $\pi '$ .

Suppose instead that $a_{rc} \in \mathbb {N}$ . A key point is that if $a_{(r-1)(c-1)}$ exists, then we cannot have $a_{(r-1)(c-1)} = \emptyset $ or $a_{(r-1)(c-1)} \in \mathbb {N}$ with $a_{(r-1)(c-1)} \geq a_{rc}$ , as otherwise it is straightforward to see that $(r,c-1)$ is a flaw strictly further left than $(r,c)$ . Thus, if it exists, either $a_{(r-1)(c-1)} \in P$ or $a_{(r-1)(c-1)} \in \mathbb {N}$ with $a_{(r-1)(c-1)} < a_{rc}$ .

Either way, we may verify that $\Psi $ produces rows consisting of a chain in P followed by a sequence of weakly increasing integers, since before the swap, if $a_{(r-1)(c-1)} \in \mathbb {N}$ , then we have $a_{(r-1)(c-1)} < a_{rc} \leq a_{r(c+1)}$ . The only extra detail to check is that all integers that remain in their original rows $r-1$ and r are less than or equal to $\pi '(r-1)-1$ and $\pi '(r)-1$ , respectively. For row $r-1$ , this is immediate, since $a_{(r-1)(c-1)} < a_{rc}$ in A. It is also clear for row r, provided that $a_{(r-1)c} \in \mathbb {N}$ , since then $a_{rc} \leq a_{(r-1)c}$ in A.

Thus, we need only consider row r in $A'$ in the case that $a_{(r-1)c} = \emptyset $ . In this case, the length of row r of A is strictly larger than the length of row $r-1$ , so we have

$$\begin{align*}\lambda_{r}-\pi(r)+r> \lambda_{r-1}-\pi(r-1)+r-1, \end{align*}$$

which implies that

$$\begin{align*}\pi(r-1)> \pi(r)+(\lambda_{r-1}-\lambda_r)-1 \geq \pi(r)-1. \end{align*}$$

Therefore, since $\pi (r-1)> \pi (r) - 1$ , clearly $\pi '(r) = \pi (r-1) \geq \pi (r)$ , so $\pi '(r)-1 \geq \pi (r)-1$ , and thus the integers that remain in row r after applying $\Psi $ satisfy the appropriate row bound.

In conclusion, $\Psi $ is a sign-reversing involution on non-tableau Grothendieck P-arrays, so all such terms cancel in Equation (3.5), yielding that the coefficient $[\overline {s}_{\lambda }]\overline {X}_G$ equals the number of Grothendieck P-tableaux of shape $\lambda $ , as desired.

It is highly suggestive that Theorem 3.11 (and Gasharov’s Schur-analog) should have an interpretation and proof via the topology of Grassmannians. We would be very interested in a solution to the following.

4.1 Analogs of the Stanley–Stembridge conjecture

Section 3.4 shows that Schur-positivity of $X_G$ when G is a claw-free incomparability graph lifts to an analog for $\overline {X}_G$ . It is natural to ask if it is similarly possible to lift the – claiming that such $X_G$ are e-positive – to the context of the Kromatic symmetric function. However, it appears that the answer is “no.”

We propose two definitions for a lift of the e-basis to the K-theoretic setting. On the one hand, e-basis elements in usual symmetric function theory may be defined in terms of fillings of single-column Young diagrams, so we may lift this formula.

Definition 4.1 The $\overline {e}_\lambda $ is given by

$$\begin{align*}\overline{e}_n = \overline{s}_{1^n} \quad \text{and} \quad \overline{e}_\lambda = \overline{e}_{\lambda_1} \dots \overline{e}_{\lambda_{\ell(\lambda)}}. \end{align*}$$

On the other hand, we may also define $e_n = \frac {1}{n!}X_{K_n}$ , and lift this characterization.

Definition 4.2 The is given by

$$\begin{align*}\overline{e}^{\prime}_n = \frac{1}{n!}\overline{X}_{K_n} \quad \text{and} \quad \overline{e}^{\prime}_{\lambda} = \overline{e}^{\prime}_{\lambda_1} \dots \overline{e}^{\prime}_{\lambda_{\ell(\lambda)}}. \end{align*}$$

It is reasonable to hope (for extending the Stanley–Stembridge conjecture) that $\overline {X}_G$ is positive in one of these K-theoretic e-bases, whenever G is a claw-free incomparability graph, or even just when G is a unit interval graph. However, one can compute that $\overline {X}_{P_3}$ is not positive in either K-theoretic e-basis $\{\overline {e}_\lambda \}$ or $\{\overline {e}^{\prime }_\lambda \}$ , dashing any such hopes. (See Table 1 for the $\overline {\widetilde {m}}$ -basis expansion of $\overline {X}_{P_3}$ .)

The terms of $\overline {X}_{P_3}$ that are homogeneous of degree $3$ must come from tableau or graph K-elementary symmetric functions of degree $3$ , and have coefficients corresponding to e-expansion of $X_{P_3}$ . Since $X_{P_3} = 3e_3+e_{21}$ , one sees that the terms of $\overline {X}_{P_3}$ for $|\lambda |=3$ in the $\overline {e}$ -basis are $3\overline {e}_3+\overline {e}_{21}$ , and in the $\overline {e}'$ -basis are $3\overline {e}^{\prime }_3+\overline {e}^{\prime }_{21}$ . However, we now encounter problems with the $|\lambda | = 4$ terms. In particular, both $\overline {e}_{21}$ and $\overline {e}^{\prime }_{21}$ are supported on the monomial $x_1^2x_2^2$ , with two distinct variables each of degree $2$ . However, it is easy to check that there is no proper set coloring of $P_3$ using exactly $1$ twice and $2$ twice; thus, these monomials must be cancelled by $\overline {e}_\mu $ or $\overline {e}^{\prime }_\mu $ terms with strictly negative coefficients.

This breakdown is so fundamental suggests that it may not be possible to reasonably generalize e-positivity to the Kromatic symmetric function, in stark contrast with the generalization of Schur-positivity given in Theorem 3.11. This suggests that the Stanley–Stembridge is not amenable to a topological interpretation along the lines of Problem 3.12.

4.2 Distinguishing graphs by $\overline {X}_G$

It is widely believed that the chromatic symmetric function is a complete invariant for (i.e., connected graphs without cycles). We propose the following weakening of this statement as a stepping stone that may be easier to prove.

As evidence for Conjecture 4.3 being potentially easier than the corresponding statement for $X_T$ , we observe that the Kromatic symmetric function distinguishes some graphs with equal chromatic symmetric function. Indeed, we are not currently aware of any graphs $G \not \cong G'$ with $\overline {X}_G = \overline {X}_{G'}$ . (In a follow-up to this article, Laura Pierson [Reference Pierson33] conjectures that the Kromatic symmetric function is a complete invariant for all graphs and provides some further evidence toward this conjecture.) [Note added in proof: This conjecture has now been disproved by Pierson and Soham Samanta in [Reference Pierson and Samanta35].]

Example 4.4 cf. [Reference Stanley39, p. 170]

Let

It is straightforward to compute that $X_G = X_H$ . By Proposition 3.4, $\overline {X}_H$ has a nonzero coefficient of $\overline {\widetilde {m}}_{(2^3)}$ , as one can easily find a covering of H by three stable sets, each of size $2$ . On the other hand, G has a vertex that is connected to every other vertex, so any stable set containing this vertex must have size $1$ . Hence, by Proposition 3.4, the $\overline {\widetilde {m}}$ -expansion of $\overline {X}_G$ only involves $\overline {\widetilde {m}}_\lambda $ such that $\lambda $ contains a part of size $1$ . In particular, $\overline {X}_G \neq \overline {X}_{H}$ .

Example 4.5 cf. [Reference Orellana and Scott31, Figure 9]

Let

Then $X_G = X_H$ . Note that G has a vertex v adjacent to all but one other vertex. Hence, no stable set containing v can have size greater than $2$ . Therefore, by Proposition 3.4, the $\overline {\widetilde {m}}$ -expansion of $\overline {X}_G$ only involves $\overline {\widetilde {m}}_\lambda $ such that $\lambda $ contains a part of size at most $2$ . On the other hand, H can be covered in a unique fashion by three stable sets of size $3$ . Hence, by Proposition 3.4, $\overline {\widetilde {m}}_{(3^3)}$ appears with coefficient $1$ in the $\overline {\widetilde {m}}$ -expansion of $\overline {X}_H$ . Hence, $\overline {X}_G \neq \overline {X}_{H}$ .

Example 4.6 cf. [Reference Aliste-Prieto, Crew, Spirkl and Zamora4, Figure 5]

Let

Then $X_G = X_H$ , and in fact the stronger condition holds that these graphs have equal Tutte symmetric function [Reference Aliste-Prieto, Crew, Spirkl and Zamora4], or equivalently equal U-polynomial [Reference Noble and Welsh30]. Note that there is a vertex v of H with only two non-neighbors, and that these non-neighbors are adjacent. Hence, no stable set containing v can have size greater than $2$ . Therefore, by Proposition 3.4, the $\overline {\widetilde {m}}$ -expansion of $\overline {X}_H$ only involves $\overline {\widetilde {m}}_\lambda $ such that $\lambda $ contains a part of size at most $2$ . On the other hand, G can be covered by stable sets of size $3$ . Hence, by Proposition 3.4, some $\overline {\widetilde {m}}_{\lambda }$ where all parts of $\lambda $ are at least $3$ appears with positive coefficient in the $\overline {\widetilde {m}}$ -expansion of $\overline {X}_G$ . Hence, $\overline {X}_G \neq \overline {X}_{H}$ .

In each of these examples, it is easy to distinguish the graphs’ Kromatic symmetric functions because the graphs disagree on $\min _v \max |I_v|$ , where the min ranges across all vertices v, and $I_v$ is a stable set containing v. It would be interesting to investigate more generally the extent to which the multiset of numbers $\{\max |I_v|: v \text { a vertex}\}$ distinguishes Kromatic symmetric functions.