Chromatic Symmetric MacMahon Function

• The chromatic symmetric MacMahon function is a vertex-weighted graph invariant that uses two sets of indeterminates to record both color assignments and vertex weights in proper graph colorings.
• It recovers detailed generating functions for vertex subsets by tracking cardinality, weight, and edge statistics—especially illuminating for trees and forests.
• The invariant generalizes traditional chromatic symmetric functions and connects with Hopf-algebra and Tutte polynomial methods to offer refined subgraph enumeration.

The chromatic symmetric MacMahon function is a graph invariant for vertex-weighted graphs that takes values in the ring of MacMahon symmetric functions in two alphabets and records information about both cardinalities and weights of vertex sets. In "Chromatic MacMahon symmetric functions of graphs" (Martin et al., 31 Jul 2025), it is defined as an analogue of the chromatic symmetric function for vertex-weighted graphs, and for trees it is shown to determine the generating function for vertex subsets by cardinality, weight, and the numbers of internal and external edges. This generalizes the corresponding unweighted result first conjectured by Crew and proved independently by Aliste-Prieto–Martin–Wagner–Zamora and Liu–Tang.

1. Definition and ambient ring

Let $G=(V,E,w)$ be a finite simple graph whose vertex-weight function

$w:V\to\mathbb{N}_+$

assigns each $v\in V$ a positive integer weight $w(v)$. Introduce two countable alphabets of commuting indeterminates

$$X=\{x_1,x_2,x_3,\ldots\},\qquad Y=\{y_1,y_2,y_3,\ldots\},$$

and consider the ring

$$\mathrm{Mac}^2\subset\mathbb{C}[[x_j,y_j\mid j\ge 1]]$$

of MacMahon symmetric functions in two alphabets, namely those formal power series invariant under the simultaneous, or diagonal, action of $S_\infty$ on the indices $j$ of both $X$ and $Y$.

A proper coloring of $w:V\to\mathbb{N}_+$0 is a map $w:V\to\mathbb{N}_+$1 such that $w:V\to\mathbb{N}_+$2 whenever $w:V\to\mathbb{N}_+$3. The chromatic symmetric MacMahon function of the weighted graph $w:V\to\mathbb{N}_+$4 is

$w:V\to\mathbb{N}_+$5

By construction, $w:V\to\mathbb{N}_+$6 is invariant under any finite permutation of the labels $w:V\to\mathbb{N}_+$7 in both the $w:V\to\mathbb{N}_+$8's and $w:V\to\mathbb{N}_+$9's, and so lies in $v\in V$0. The definition simultaneously records the color-class structure through the $v\in V$1-variables and the vertex-weight contribution through the exponents of the $v\in V$2-variables. This suggests that $v\in V$3 is designed to interpolate between ordinary chromatic symmetric data and weighted refinements (Martin et al., 31 Jul 2025).

2. Bigrading, vector partitions, and the power-sum expansion

The ring $v\in V$4 is bigraded by total $v\in V$5-degree and total $v\in V$6-degree. Its most elementary basis is given by MacMahon power-sums indexed by vector partitions.

A vector partition of $v\in V$7 is a multiset

$v\in V$8

of nonzero vectors in $v\in V$9 whose sum is $w(v)$0. One writes $w(v)$1, and its length is $w(v)$2. The corresponding MacMahon power-sum is

$w(v)$3

where

$w(v)$4

For $w(v)$5, let the bitype of the spanning subgraph $w(v)$6 be the vector partition

$w(v)$7

and let $w(v)$8 be the number of edges in $w(v)$9. The power-sum expansion is

$$X=\{x_1,x_2,x_3,\ldots\},\qquad Y=\{y_1,y_2,y_3,\ldots\},$$0

The proof follows the same inclusion–exclusion pattern as Stanley’s proof for $$X=\{x_1,x_2,x_3,\ldots\},\qquad Y=\{y_1,y_2,y_3,\ldots\},$$1: the contribution of each subgraph $$X=\{x_1,x_2,x_3,\ldots\},\qquad Y=\{y_1,y_2,y_3,\ldots\},$$2 is exactly the sum of monomials of all colorings that are constant on each component of $$X=\{x_1,x_2,x_3,\ldots\},\qquad Y=\{y_1,y_2,y_3,\ldots\},$$3, with sign $$X=\{x_1,x_2,x_3,\ldots\},\qquad Y=\{y_1,y_2,y_3,\ldots\},$$4, and summing over $$X=\{x_1,x_2,x_3,\ldots\},\qquad Y=\{y_1,y_2,y_3,\ldots\},$$5 removes all non-proper colorings.

When $$X=\{x_1,x_2,x_3,\ldots\},\qquad Y=\{y_1,y_2,y_3,\ldots\},$$6 is a forest on $$X=\{x_1,x_2,x_3,\ldots\},\qquad Y=\{y_1,y_2,y_3,\ldots\},$$7 vertices of total weight $$X=\{x_1,x_2,x_3,\ldots\},\qquad Y=\{y_1,y_2,y_3,\ldots\},$$8, there is no cancellation among the $$X=\{x_1,x_2,x_3,\ldots\},\qquad Y=\{y_1,y_2,y_3,\ldots\},$$9 because every $$\mathrm{Mac}^2\subset\mathbb{C}[[x_j,y_j\mid j\ge 1]]$$0 has exactly $$\mathrm{Mac}^2\subset\mathbb{C}[[x_j,y_j\mid j\ge 1]]$$1 edges. In that case,

$$\mathrm{Mac}^2\subset\mathbb{C}[[x_j,y_j\mid j\ge 1]]$$2

where

$$\mathrm{Mac}^2\subset\mathbb{C}[[x_j,y_j\mid j\ge 1]]$$3

The same source notes that one may also expand $$\mathrm{Mac}^2\subset\mathbb{C}[[x_j,y_j\mid j\ge 1]]$$4 in the MacMahon monomial basis $$\mathrm{Mac}^2\subset\mathbb{C}[[x_j,y_j\mid j\ge 1]]$$5 by grouping colorings $$\mathrm{Mac}^2\subset\mathbb{C}[[x_j,y_j\mid j\ge 1]]$$6 with the same type, namely the same multiset of weight-augmented color-classes, although for the tree-uniqueness theorem the power-sum basis is the most convenient.

3. The extended generalized degree polynomial and the forest theorem

For any vertex-weighted graph $$\mathrm{Mac}^2\subset\mathbb{C}[[x_j,y_j\mid j\ge 1]]$$7 and $$\mathrm{Mac}^2\subset\mathbb{C}[[x_j,y_j\mid j\ge 1]]$$8, define:

• $$\mathrm{Mac}^2\subset\mathbb{C}[[x_j,y_j\mid j\ge 1]]$$9 as the number of edges with both endpoints in $S_\infty$0,
• $S_\infty$1 as the number of edges with exactly one endpoint in $S_\infty$2,
• $S_\infty$3 as the number of vertices in $S_\infty$4,
• $S_\infty$5.

The extended generalized degree polynomial is

$S_\infty$6

The main theorem states that if $S_\infty$7 is any vertex-weighted forest, then $S_\infty$8 uniquely determines $S_\infty$9 (Martin et al., 31 Jul 2025). Equivalently, there is an explicit $j$0-linear operator

$j$1

such that

$j$2

where $j$3 is the number of components of $j$4.

In particular, one recovers all the four-variable statistics

$j$5

This theorem identifies $j$6 as substantially more informative than a coloring enumerator alone. It determines a vertex-subset generating function that tracks four statistics simultaneously: external edges, cardinality, total weight, and internal edges. A plausible implication is that the two-alphabet MacMahon framework is not merely a repackaging of weighted colorings, but a mechanism for recovering refined substructure enumerators on forests.

4. Hopf-theoretic and combinatorial proof frameworks

The proof sketch given for the main theorem is Hopf-theoretic, in the style of Liu–Tang. The ring $j$7 carries a natural commutative, cocommutative Hopf-algebra structure in which the power-sums $j$8 multiply by concatenation and the coproduct is given by splitting the list of parts. The chromatic map

$j$9

is multiplicative under disjoint union and respects that Hopf structure. More precisely,

$X$0

Two linear evaluation maps

$X$1

are defined so that each map sends $X$2 to a simple monomial in $X$3 extracted from $X$4, and the convolution

$X$5

applied to $X$6 reproduces precisely $X$7. Recovering the parameters $X$8 of $X$9 from $Y$0 is done by a third evaluation $Y$1 on $Y$2, which reads off $Y$3 and yields a two-variable monomial that forces a single term in the expansion.

A purely combinatorial proof also exists. It generalizes the approach of Aliste-Prieto–Martin–Wagner–Zamora by expressing each coefficient $Y$4 as an explicit signed linear combination of the $Y$5, and hence of the coefficients of $Y$6 in the power-sum basis, using a standard inclusion–exclusion identity. The coexistence of Hopf-theoretic and combinatorial proofs places the invariant at the intersection of algebraic combinatorics and subgraph enumeration.

5. Small examples

Two minimal examples illustrate both the definition and the recovery of the extended generalized degree polynomial.

For the unweighted edge $Y$7, with $Y$8 and total weight $Y$9, the proper colorings are exactly those with $w:V\to\mathbb{N}_+$00. Hence

$w:V\to\mathbb{N}_+$01

Here $w:V\to\mathbb{N}_+$02 and $w:V\to\mathbb{N}_+$03, with signs determined by $w:V\to\mathbb{N}_+$04. Its extended generalized degree polynomial is

$w:V\to\mathbb{N}_+$05

For a $w:V\to\mathbb{N}_+$06-vertex path with weights $w:V\to\mathbb{N}_+$07 and $w:V\to\mathbb{N}_+$08, the total weight is $w:V\to\mathbb{N}_+$09. Proper colorings again satisfy $w:V\to\mathbb{N}_+$10, and each term

$w:V\to\mathbb{N}_+$11

has the form

$w:V\to\mathbb{N}_+$12

Therefore

$w:V\to\mathbb{N}_+$13

The corresponding extended generalized degree polynomial has the form

$w:V\to\mathbb{N}_+$14

as recorded in the source.

Larger examples, including stars and paths of length $w:V\to\mathbb{N}_+$15, proceed by listing all proper colorings or by applying the subgraph formula. These examples make explicit that the $w:V\to\mathbb{N}_+$16-degree tracks the number of vertices while the $w:V\to\mathbb{N}_+$17-degree tracks total vertex weight.

6. Specializations, antecedents, and relation to other invariants

The chromatic symmetric MacMahon function generalizes several existing graph invariants (Martin et al., 31 Jul 2025). If all $w:V\to\mathbb{N}_+$18, then $w:V\to\mathbb{N}_+$19 specializes by setting $w:V\to\mathbb{N}_+$20 for all $w:V\to\mathbb{N}_+$21 and recovers exactly Stanley’s chromatic symmetric function $w:V\to\mathbb{N}_+$22. More generally, the ordinary weighted chromatic symmetric function

$w:V\to\mathbb{N}_+$23

is obtained by setting $w:V\to\mathbb{N}_+$24 in $w:V\to\mathbb{N}_+$25. Conversely, $w:V\to\mathbb{N}_+$26 is recovered by setting $w:V\to\mathbb{N}_+$27 in $w:V\to\mathbb{N}_+$28.

The same source states that $w:V\to\mathbb{N}_+$29 admits a deletion–contraction recurrence inherited from the weighted chromatic symmetric function, which makes it a specialization of the dichromatic, or Tutte, polynomial in two alphabets. It also states that the new extended generalized degree polynomial $w:V\to\mathbb{N}_+$30 refines the classical GDP of a tree, described as Crew’s statistic, by also tracking vertex-weights, and is determined by $w:V\to\mathbb{N}_+$31.

Historically, the forest result is presented as a weighted generalization of the unweighted case first conjectured by Crew and proved independently by Aliste-Prieto–Martin–Wagner–Zamora and Liu–Tang. In that sense, the chromatic symmetric MacMahon function sits in a lineage of results showing that chromatic-type symmetric invariants can recover unexpectedly fine information for trees and forests. The weighted extension demonstrates that the same phenomenon persists when vertex weights are incorporated into the symmetric-function framework.