Chromatic symmetric function

<h2 id="definition">Definition</h2>
For a finite graph 
 
 
 
 G
 =
 (
 V
 ,
 E
 )
 
 
 {\displaystyle G=(V,E)}
 
 with vertex set 
 
 
 
 V
 =
 {
 
 v
 
 1
 
 
 ,
 
 v
 
 2
 
 
 ,
 …
 ,
 
 v
 
 n
 
 
 }
 
 
 {\displaystyle V=\{v_{1},v_{2},\ldots ,v_{n}\}}
 
, a vertex coloring is a function 
 
 
 
 κ
 :
 V
 →
 C
 
 
 {\displaystyle \kappa :V\to C}
 
 where 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is a set of colors. A vertex coloring is called proper if all adjacent vertices are assigned distinct colors (i.e., 
 
 
 
 {
 i
 ,
 j
 }
 ∈
 E
 
 ⟹
 
 κ
 (
 i
 )
 ≠
 κ
 (
 j
 )
 
 
 {\displaystyle \{i,j\}\in E\implies \kappa (i)\neq \kappa (j)}
 
). The chromatic symmetric function denoted 
 
 
 
 
 X
 
 G
 
 
 (
 
 x
 
 1
 
 
 ,
 
 x
 
 2
 
 
 ,
 …
 )
 
 
 {\displaystyle X_{G}(x_{1},x_{2},\ldots )}
 
 is defined to be the weight generating function of proper vertex colorings of 
 
 
 
 G
 
 
 {\displaystyle G}
 
:<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
 
 
 
 
 X
 
 G
 
 
 (
 
 x
 
 1
 
 
 ,
 
 x
 
 2
 
 
 ,
 …
 )
 :=
 
 ∑
 
 
 
 κ
 :
 V
 →
 
 N
 
 
 proper
 
 
 
 
 x
 
 κ
 (
 
 v
 
 1
 
 
 )
 
 
 
 x
 
 κ
 (
 
 v
 
 2
 
 
 )
 
 
 ⋯
 
 x
 
 κ
 (
 
 v
 
 n
 
 
 )
 
 
 
 
 {\displaystyle X_{G}(x_{1},x_{2},\ldots ):=\sum _{\underset {\text{proper}}{\kappa :V\to \mathbb {N} }}x_{\kappa (v_{1})}x_{\kappa (v_{2})}\cdots x_{\kappa (v_{n})}}

<h2 id="examples">Examples</h2>
For 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 a <a href="/facts/Integer_partition/LtyXWqwH">partition</a>, let 
 
 
 
 
 m
 
 λ
 
 
 
 
 {\displaystyle m_{\lambda }}
 
 be the <a href="/facts/Symmetric_polynomial/GNhguVcn">monomial symmetric polynomial</a> associated to 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
.

<h3>Example 1: complete graphs</h3>
Consider the <a href="/facts/Complete_graph/AYD6hUqI">complete graph</a> 
 
 
 
 
 K
 
 n
 
 
 
 
 {\displaystyle K_{n}}
 
 on 
 
 
 
 n
 
 
 {\displaystyle n}
 
 vertices:

<ul><li>There are 
 
 
 
 n
 !
 
 
 {\displaystyle n!}
 
 ways to color 
 
 
 
 
 K
 
 n
 
 
 
 
 {\displaystyle K_{n}}
 
 with exactly 
 
 
 
 n
 
 
 {\displaystyle n}
 
 colors yielding the term 
 
 
 
 n
 !
 
 x
 
 1
 
 
 ⋯
 
 x
 
 n
 
 
 
 
 {\displaystyle n!x_{1}\cdots x_{n}}
 
</li>
<li>Since every pair of vertices in 
 
 
 
 
 K
 
 n
 
 
 
 
 {\displaystyle K_{n}}
 
 is adjacent, it can be properly colored with no fewer than 
 
 
 
 n
 
 
 {\displaystyle n}
 
 colors.</li></ul>
Thus, 
 
 
 
 
 X
 
 
 K
 
 n
 
 
 
 
 (
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 )
 =
 n
 !
 
 x
 
 1
 
 
 ⋯
 
 x
 
 n
 
 
 =
 n
 !
 
 m
 
 (
 1
 ,
 …
 ,
 1
 )
 
 
 
 
 {\displaystyle X_{K_{n}}(x_{1},\ldots ,x_{n})=n!x_{1}\cdots x_{n}=n!m_{(1,\ldots ,1)}}

<h3>Example 2: a path graph</h3>
Consider the <a href="/facts/Path_graph/4Y2f35eM">path graph</a> 
 
 
 
 
 P
 
 3
 
 
 
 
 {\displaystyle P_{3}}
 
 of length 
 
 
 
 3
 
 
 {\displaystyle 3}
 
:

<ul><li>There are 
 
 
 
 3
 !
 
 
 {\displaystyle 3!}
 
 ways to color 
 
 
 
 
 P
 
 3
 
 
 
 
 {\displaystyle P_{3}}
 
 with exactly 
 
 
 
 3
 
 
 {\displaystyle 3}
 
 colors, yielding the term 
 
 
 
 6
 
 x
 
 1
 
 
 
 x
 
 2
 
 
 
 x
 
 3
 
 
 
 
 {\displaystyle 6x_{1}x_{2}x_{3}}
 
</li>
<li>For each pair of colors, there are 
 
 
 
 2
 
 
 {\displaystyle 2}
 
 ways to color 
 
 
 
 
 P
 
 3
 
 
 
 
 {\displaystyle P_{3}}
 
 yielding the terms 
 
 
 
 
 x
 
 i
 
 
 2
 
 
 
 x
 
 j
 
 
 
 
 {\displaystyle x_{i}^{2}x_{j}}
 
 and 
 
 
 
 
 x
 
 i
 
 
 
 x
 
 j
 
 
 2
 
 
 
 
 {\displaystyle x_{i}x_{j}^{2}}
 
 for 
 
 
 
 i
 ≠
 j
 
 
 {\displaystyle i\neq j}
 
</li></ul>
Altogether, the chromatic symmetric function of 
 
 
 
 
 P
 
 3
 
 
 
 
 {\displaystyle P_{3}}
 
 is then given by:<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
 
 
 
 
 X
 
 
 P
 
 3
 
 
 
 
 (
 
 x
 
 1
 
 
 ,
 
 x
 
 2
 
 
 ,
 
 x
 
 3
 
 
 )
 =
 6
 
 x
 
 1
 
 
 
 x
 
 2
 
 
 
 x
 
 3
 
 
 +
 
 x
 
 1
 
 
 2
 
 
 
 x
 
 2
 
 
 +
 
 x
 
 1
 
 
 
 x
 
 2
 
 
 2
 
 
 +
 
 x
 
 1
 
 
 2
 
 
 
 x
 
 3
 
 
 +
 
 x
 
 1
 
 
 
 x
 
 3
 
 
 2
 
 
 +
 
 x
 
 2
 
 
 2
 
 
 
 x
 
 3
 
 
 +
 
 x
 
 2
 
 
 
 x
 
 3
 
 
 2
 
 
 =
 6
 
 m
 
 (
 1
 ,
 1
 ,
 1
 )
 
 
 +
 
 m
 
 (
 1
 ,
 2
 )
 
 
 
 
 {\displaystyle X_{P_{3}}(x_{1},x_{2},x_{3})=6x_{1}x_{2}x_{3}+x_{1}^{2}x_{2}+x_{1}x_{2}^{2}+x_{1}^{2}x_{3}+x_{1}x_{3}^{2}+x_{2}^{2}x_{3}+x_{2}x_{3}^{2}=6m_{(1,1,1)}+m_{(1,2)}}

<h2 id="properties">Properties</h2>
<ul><li>Let 
 
 
 
 
 χ
 
 G
 
 
 
 
 {\displaystyle \chi _{G}}
 
 be the chromatic polynomial of 
 
 
 
 G
 
 
 {\displaystyle G}
 
, so that 
 
 
 
 
 χ
 
 G
 
 
 (
 k
 )
 
 
 {\displaystyle \chi _{G}(k)}
 
 is equal to the number of proper vertex colorings of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 using at most 
 
 
 
 k
 
 
 {\displaystyle k}
 
 distinct colors. The values of 
 
 
 
 
 χ
 
 G
 
 
 
 
 {\displaystyle \chi _{G}}
 
 can then be computed by <a href="/facts/Ring_of_symmetric_functions/SmENv9o4">specializing</a> the chromatic symmetric function, setting the first 
 
 
 
 k
 
 
 {\displaystyle k}
 
 variables 
 
 
 
 
 x
 
 i
 
 
 
 
 {\displaystyle x_{i}}
 
 equal to 
 
 
 
 1
 
 
 {\displaystyle 1}
 
 and the remaining variables equal to 
 
 
 
 0
 
 
 {\displaystyle 0}
 
:<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> 
 
 
 
 
 X
 
 G
 
 
 (
 
 1
 
 k
 
 
 )
 =
 
 X
 
 G
 
 
 (
 1
 ,
 …
 ,
 1
 ,
 0
 ,
 0
 ,
 …
 )
 =
 
 χ
 
 G
 
 
 (
 k
 )
 
 
 {\displaystyle X_{G}(1^{k})=X_{G}(1,\ldots ,1,0,0,\ldots )=\chi _{G}(k)}
 
</li>
<li>If 
 
 
 
 G
 ⨿
 H
 
 
 {\displaystyle G\amalg H}
 
 is the disjoint union of two graphs, then the chromatic symmetric function for 
 
 
 
 G
 ⨿
 H
 
 
 {\displaystyle G\amalg H}
 
 can be written as a product of the corresponding functions for 
 
 
 
 G
 
 
 {\displaystyle G}
 
 and 
 
 
 
 H
 
 
 {\displaystyle H}
 
:<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>
 
 
 
 
 X
 
 G
 ⨿
 H
 
 
 =
 
 X
 
 G
 
 
 ⋅
 
 X
 
 H
 
 
 
 
 {\displaystyle X_{G\amalg H}=X_{G}\cdot X_{H}}
 
</li>
<li>A stable partition 
 
 
 
 π
 
 
 {\displaystyle \pi }
 
 of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is defined to be a <a href="/facts/Set_partition/VeP9g8CA">set partition</a> of vertices 
 
 
 
 V
 
 
 {\displaystyle V}
 
 such that each block of 
 
 
 
 π
 
 
 {\displaystyle \pi }
 
 is an <a href="/facts/Independent_set_(graph_theory)/ksu46kdz">independent set</a> in 
 
 
 
 G
 
 
 {\displaystyle G}
 
. The type of a stable partition 
 
 
 
 
 type
 
 (
 π
 )
 
 
 {\displaystyle {\text{type}}(\pi )}
 
 is the partition consisting of parts equal to the sizes of the <a href="/facts/Connected_component_(graph_theory)/5owVSXtY">connected components</a> of the vertex induced subgraphs. For a partition 
 
 
 
 λ
 ⊢
 n
 
 
 {\displaystyle \lambda \vdash n}
 
, let 
 
 
 
 
 z
 
 λ
 
 
 
 
 {\displaystyle z_{\lambda }}
 
 be the number of stable partitions of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 with 
 
 
 
 
 type
 
 (
 π
 )
 =
 λ
 =
 ⟨
 
 1
 
 
 r
 
 1
 
 
 
 
 
 2
 
 r
 2
 
 
 …
 ⟩
 
 
 {\displaystyle {\text{type}}(\pi )=\lambda =\langle 1^{r_{1}}2^{r2}\ldots \rangle }
 
. Then, 
 
 
 
 
 X
 
 G
 
 
 
 
 {\displaystyle X_{G}}
 
 expands into the augmented monomial symmetric functions, 
 
 
 
 
 
 
 
 m
 ~
 
 
 
 
 λ
 
 
 :=
 
 r
 
 1
 
 
 !
 
 r
 
 2
 
 
 !
 ⋯
 
 m
 
 λ
 
 
 
 
 {\displaystyle {\tilde {m}}_{\lambda }:=r_{1}!r_{2}!\cdots m_{\lambda }}
 
 with coefficients given by the number of stable partitions of 
 
 
 
 G
 
 
 {\displaystyle G}
 
:<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>
 
 
 
 
 X
 
 G
 
 
 =
 
 ∑
 
 λ
 ⊢
 n
 
 
 
 z
 
 λ
 
 
 
 
 
 
 m
 ~
 
 
 
 
 λ
 
 
 
 
 {\displaystyle X_{G}=\sum _{\lambda \vdash n}z_{\lambda }{\tilde {m}}_{\lambda }}
 
</li>
<li>Let 
 
 
 
 
 p
 
 λ
 
 
 
 
 {\displaystyle p_{\lambda }}
 
 be the <a href="/facts/Symmetric_polynomial/GNhguVcn">power-sum symmetric function</a> associated to a partition 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
. For 
 
 
 
 S
 ⊆
 E
 
 
 {\displaystyle S\subseteq E}
 
, let 
 
 
 
 λ
 (
 S
 )
 
 
 {\displaystyle \lambda (S)}
 
 be the partition whose parts are the vertex sizes of the connected components of the edge-induced subgraph of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 specified by 
 
 
 
 S
 
 
 {\displaystyle S}
 
. The chromatic symmetric function can be expanded in the power-sum symmetric functions via the following formula:<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>
 
 
 
 
 X
 
 G
 
 
 =
 
 ∑
 
 S
 ⊆
 E
 
 
 (
 −
 1
 
 )
 
 
 |
 
 S
 
 |
 
 
 
 
 p
 
 λ
 (
 S
 )
 
 
 
 
 {\displaystyle X_{G}=\sum _{S\subseteq E}(-1)^{|S|}p_{\lambda (S)}}
 
</li>
<li>Let 
 
 
 
 
 X
 
 G
 
 
 =
 
 ∑
 
 λ
 ⊢
 n
 
 
 
 c
 
 λ
 
 
 
 e
 
 λ
 
 
 
 
 {\textstyle X_{G}=\sum _{\lambda \vdash n}c_{\lambda }e_{\lambda }}
 
 be the expansion of 
 
 
 
 
 X
 
 G
 
 
 
 
 {\displaystyle X_{G}}
 
 in the basis of <a href="/facts/Elementary_symmetric_polynomial/9RFXe9b6">elementary symmetric functions</a> 
 
 
 
 
 e
 
 λ
 
 
 
 
 {\displaystyle e_{\lambda }}
 
. Let 
 
 
 
 
 sink
 
 (
 G
 ,
 s
 )
 
 
 {\displaystyle {\text{sink}}(G,s)}
 
 be the number of <a href="/facts/Acyclic_orientation/cdAUkyFH">acyclic orientations</a> on the graph 
 
 
 
 G
 
 
 {\displaystyle G}
 
 which contain exactly 
 
 
 
 s
 
 
 {\displaystyle s}
 
 sinks. Then we have the following formula for the number of sinks:<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>
 
 
 
 
 sink
 
 (
 G
 ,
 s
 )
 =
 
 ∑
 
 
 
 λ
 ⊢
 n
 
 
 l
 (
 λ
 )
 =
 s
 
 
 
 
 
 c
 
 λ
 
 
 
 
 {\displaystyle {\text{sink}}(G,s)=\sum _{\underset {l(\lambda )=s}{\lambda \vdash n}}c_{\lambda }}
 
</li></ul>
<h2 id="open-problems">Open problems</h2>
There are a number of outstanding questions regarding the chromatic symmetric function which have received substantial attention in the literature surrounding them.

<h3>(3+1)-free conjecture</h3>
For a partition 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
, let 
 
 
 
 
 e
 
 λ
 
 
 
 
 {\displaystyle e_{\lambda }}
 
 be the elementary symmetric function associated to 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
.
A <a href="/facts/Partially_ordered_set/qb4a3FAX">partially ordered set</a> 
 
 
 
 P
 
 
 {\displaystyle P}
 
 is called 
 
 
 
 (
 3
 +
 1
 )
 
 
 {\displaystyle (3+1)}
 
-free if it does not contain a subposet isomorphic to the direct sum of the 
 
 
 
 3
 
 
 {\displaystyle 3}
 
 element <a href="/facts/Chain_(poset)/xnTtmuYJ">chain</a> and the 
 
 
 
 1
 
 
 {\displaystyle 1}
 
 element chain. The <a href="/facts/Comparability_graph/dLht35EJ">incomparability graph</a> 
 
 
 
 
 inc
 
 (
 P
 )
 
 
 {\displaystyle {\text{inc}}(P)}
 
 of a poset 
 
 
 
 P
 
 
 {\displaystyle P}
 
 is the graph with vertices given by the elements of 
 
 
 
 P
 
 
 {\displaystyle P}
 
 which includes an edge between two vertices if and only if their corresponding elements in 
 
 
 
 P
 
 
 {\displaystyle P}
 
 are <a href="/facts/Comparability/PvLeJXol">incomparable</a>.
Conjecture (Stanley–Stembridge) Let 
 
 
 
 G
 
 
 {\displaystyle G}
 
 be the incomparability graph of a 
 
 
 
 (
 3
 +
 1
 )
 
 
 {\textstyle (3+1)}
 
-free poset, then 
 
 
 
 
 X
 
 G
 
 
 
 
 {\textstyle X_{G}}
 
 is 
 
 
 
 e
 
 
 {\displaystyle e}
 
-positive.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>
A weaker positivity result is known for the case of expansions into the basis of <a href="/facts/Schur_polynomial/mBzF2Mdr">Schur functions</a>.
Theorem (Gasharov) Let 
 
 
 
 G
 
 
 {\displaystyle G}
 
 be the incomparability graph of a 
 
 
 
 (
 3
 +
 1
 )
 
 
 {\textstyle (3+1)}
 
-free poset, then 
 
 
 
 
 X
 
 G
 
 
 
 
 {\textstyle X_{G}}
 
 is 
 
 
 
 s
 
 
 {\displaystyle s}
 
-positive.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>
In the proof of the theorem above, there is a combinatorial formula for the coefficients of the Schur expansion given in terms of 
 
 
 
 P
 
 
 {\displaystyle P}
 
-tableaux which are a generalization of <a href="/facts/Semistandard_Young_tableaux/e5lI4bck">semistandard Young tableaux</a> instead labelled with the elements of 
 
 
 
 P
 
 
 {\displaystyle P}
 
.

<h2 id="generalizations">Generalizations</h2>
There are a number of generalizations of the chromatic symmetric function:

<ul><li>There is a <a href="/facts/Categorification/1HN6SSB1">categorification</a> of the invariant into a <a href="/facts/Homology_theory/VFJmryEW">homology theory</a> which is called chromatic symmetric homology.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> This homology theory is known to be a stronger invariant than the chromatic symmetric function alone.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> The chromatic symmetric function can also be defined for vertex-weighted graphs,<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> where it satisfies a <a href="/facts/Deletion%25E2%2580%2593contraction_formula/H5aXmbrh">deletion-contraction</a> property analogous to that of the chromatic polynomial. If the theory of chromatic symmetric homology is generalized to vertex-weighted graphs as well, this deletion-contraction property lifts to a long exact sequence of the corresponding homology theory.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a></li>
<li>There is also a <a href="/facts/Quasisymmetric_function/VSh6oTCV">quasisymmetric</a> refinement of the chromatic symmetric function which can be used to refine the formulae expressing 
 
 
 
 
 X
 
 G
 
 
 
 
 {\displaystyle X_{G}}
 
 in terms of Gessel's basis of fundamental quasisymmetric functions and the expansion in the basis of Schur functions.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> Fixing an order for the set of vertices, the ascent set of a proper coloring 
 
 
 
 κ
 
 
 {\displaystyle \kappa }
 
 is defined to be 
 
 
 
 
 asc
 
 (
 κ
 )
 =
 {
 {
 i
 ,
 j
 }
 ∈
 E
 :
 i
 <
 j
 
  and 
 
 κ
 (
 i
 )
 <
 κ
 (
 j
 )
 }
 
 
 {\displaystyle {\text{asc}}(\kappa )=\{\{i,j\}\in E:i<j{\text{ and }}\kappa (i)<\kappa (j)\}}
 
. The chromatic quasisymmetric function of a graph 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is then defined to be:<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>
 
 
 
 
 X
 
 G
 
 
 (
 
 x
 
 1
 
 
 ,
 
 x
 
 2
 
 
 ,
 …
 ;
 t
 )
 :=
 
 ∑
 
 
 
 κ
 :
 V
 →
 
 N
 
 
 proper
 
 
 
 
 t
 
 
 |
 
 a
 s
 c
 (
 κ
 )
 
 |
 
 
 
 
 x
 
 κ
 (
 
 v
 
 1
 
 
 )
 
 
 ⋯
 
 x
 
 κ
 (
 
 v
 
 n
 
 
 )
 
 
 
 
 {\displaystyle X_{G}(x_{1},x_{2},\ldots ;t):=\sum _{\underset {\text{proper}}{\kappa :V\to \mathbb {N} }}t^{|asc(\kappa )|}x_{\kappa (v_{1})}\cdots x_{\kappa (v_{n})}}
 
</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Chromatic_polynomial/dvr4Hyz9">Chromatic polynomial</a></li>
<li><a href="/facts/Symmetric_function/IB8N4T3m">Symmetric function</a></li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li>Blasiak, Jonah; Eriksson, Holden; Pylyavskyy, Pavlo; Siegl, Isaiah (2022). "Noncommutative Schur functions for posets". <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/2211.03967">2211.03967</a> [<a href="https://arxiv.org/archive/math.CO">math.CO</a>].</li>
<li>Chow, Timothy Y. (1999). "Descents, Quasi-Symmetric Functions, Robinson-Schensted for Posets, and the Chromatic Symmetric Function". Journal of Algebraic Combinatorics. 10 (3): 227–240. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1023%2FA%3A1018719315718">10.1023/A:1018719315718</a>.</li>
<li>Harada, Megumi; Precup, Martha E. (2019). "The cohomology of abelian Hessenberg varieties and the Stanley–Stembridge conjecture". Algebraic Combinatorics. 2 (6): 1059–1108. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1709.06736">1709.06736</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.5802%2Falco.76">10.5802/alco.76</a>.</li>
<li>Hwang, Byung-Hak (2024). "Chromatic quasisymmetric functions and noncommutative 
 
 
 
 P
 
 
 {\displaystyle P}
 
-symmetric functions". Transactions of the American Mathematical Society. 377 (4): 2855–2896. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/2208.09857">2208.09857</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2Ftran%2F9096">10.1090/tran/9096</a>.</li>
<li>Shareshian, John; Wachs, Michelle L. (2012). "Chromatic quasisymmetric functions and Hessenberg varieties". Configuration Spaces. pp. 433–460. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1106.4287">1106.4287</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-88-7642-431-1_20">10.1007/978-88-7642-431-1_20</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-88-7642-430-4.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Stanley, R.P. (1995). "A Symmetric Function Generalization of the Chromatic Polynomial of a Graph". Advances in Mathematics. 111 (1): 166–194. doi:10.1006/aima.1995.1020. ISSN 0001-8708. <a href="/wiki/Richard_P._Stanley" target="_blank">/wiki/Richard_P._Stanley</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Stanley, R.P. (1995). "A Symmetric Function Generalization of the Chromatic Polynomial of a Graph". Advances in Mathematics. 111 (1): 166–194. doi:10.1006/aima.1995.1020. ISSN 0001-8708. <a href="/wiki/Richard_P._Stanley" target="_blank">/wiki/Richard_P._Stanley</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Saliola, Franco (October 15, 2022). "Lectures on Symmetric Functions with a view towards Hessenberg varieties — Draft" (PDF). Archived (PDF) from the original on October 18, 2022. Retrieved April 27, 2024. <a href="https://www.birs.ca/workshops/2022/22w5143/files/Franco%20Saliola/Saliola-Lectures-on-Chromatic-Symmetric-Functions.pdf" target="_blank">https://www.birs.ca/workshops/2022/22w5143/files/Franco%20Saliola/Saliola-Lectures-on-Chromatic-Symmetric-Functions.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Saliola, Franco (October 15, 2022). "Lectures on Symmetric Functions with a view towards Hessenberg varieties — Draft" (PDF). Archived (PDF) from the original on October 18, 2022. Retrieved April 27, 2024. <a href="https://www.birs.ca/workshops/2022/22w5143/files/Franco%20Saliola/Saliola-Lectures-on-Chromatic-Symmetric-Functions.pdf" target="_blank">https://www.birs.ca/workshops/2022/22w5143/files/Franco%20Saliola/Saliola-Lectures-on-Chromatic-Symmetric-Functions.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Stanley, R.P. (1995). "A Symmetric Function Generalization of the Chromatic Polynomial of a Graph". Advances in Mathematics. 111 (1): 166–194. doi:10.1006/aima.1995.1020. ISSN 0001-8708. <a href="/wiki/Richard_P._Stanley" target="_blank">/wiki/Richard_P._Stanley</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Stanley, R.P. (1995). "A Symmetric Function Generalization of the Chromatic Polynomial of a Graph". Advances in Mathematics. 111 (1): 166–194. doi:10.1006/aima.1995.1020. ISSN 0001-8708. <a href="/wiki/Richard_P._Stanley" target="_blank">/wiki/Richard_P._Stanley</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Stanley, R.P. (1995). "A Symmetric Function Generalization of the Chromatic Polynomial of a Graph". Advances in Mathematics. 111 (1): 166–194. doi:10.1006/aima.1995.1020. ISSN 0001-8708. <a href="/wiki/Richard_P._Stanley" target="_blank">/wiki/Richard_P._Stanley</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Stanley, R.P. (1995). "A Symmetric Function Generalization of the Chromatic Polynomial of a Graph". Advances in Mathematics. 111 (1): 166–194. doi:10.1006/aima.1995.1020. ISSN 0001-8708. <a href="/wiki/Richard_P._Stanley" target="_blank">/wiki/Richard_P._Stanley</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Stanley, R.P. (1995). "A Symmetric Function Generalization of the Chromatic Polynomial of a Graph". Advances in Mathematics. 111 (1): 166–194. doi:10.1006/aima.1995.1020. ISSN 0001-8708. <a href="/wiki/Richard_P._Stanley" target="_blank">/wiki/Richard_P._Stanley</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Stanley, R.P. (1995). "A Symmetric Function Generalization of the Chromatic Polynomial of a Graph". Advances in Mathematics. 111 (1): 166–194. doi:10.1006/aima.1995.1020. ISSN 0001-8708. <a href="/wiki/Richard_P._Stanley" target="_blank">/wiki/Richard_P._Stanley</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Gasharov, Vesselin (1996). "Incomparability graphs of (3+1)-free posets are s-positive" (PDF). Discrete Mathematics. 157 (1–3): 193–197. doi:10.1016/S0012-365X(96)83014-7. <a href="https://core.ac.uk/download/pdf/82067232.pdf" target="_blank">https://core.ac.uk/download/pdf/82067232.pdf</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Sazdanovic, Radmila; Yip, Martha (2018-02-01). "A categorification of the chromatic symmetric function". Journal of Combinatorial Theory. Series A. 154: 218–246. arXiv:1506.03133. doi:10.1016/j.jcta.2017.08.014. ISSN 0097-3165. <a href="https://doi.org/10.1016%2Fj.jcta.2017.08.014" target="_blank">https://doi.org/10.1016%2Fj.jcta.2017.08.014</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Chandler, Alex; Sazdanovic, Radmila; Stella, Salvatore; Yip, Martha (2023-09-01). "On the strength of chromatic symmetric homology for graphs". Advances in Applied Mathematics. 150: 102559. arXiv:1911.13297. doi:10.1016/j.aam.2023.102559. ISSN 0196-8858. <a href="https://www.sciencedirect.com/science/article/pii/S0196885823000775" target="_blank">https://www.sciencedirect.com/science/article/pii/S0196885823000775</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Crew, Logan; Spirkl, Sophie (2020). "A Deletion-Contraction Relation for the Chromatic Symmetric Function". European Journal of Combinatorics. 89: 103143. arXiv:1910.11859. doi:10.1016/j.ejc.2020.103143. <a href="/wiki/European_Journal_of_Combinatorics" target="_blank">/wiki/European_Journal_of_Combinatorics</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Ciliberti, Azzurra (2024-01-01). "A deletion–contraction long exact sequence for chromatic symmetric homology". European Journal of Combinatorics. 115: 103788. arXiv:2211.00699. doi:10.1016/j.ejc.2023.103788. ISSN 0195-6698. <a href="https://www.sciencedirect.com/science/article/pii/S0195669823001051" target="_blank">https://www.sciencedirect.com/science/article/pii/S0195669823001051</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Shareshian, John; Wachs, Michelle L. (June 4, 2016). "Chromatic quasisymmetric functions". Advances in Mathematics. 295: 497–551. arXiv:1405.4629. doi:10.1016/j.aim.2015.12.018. ISSN 0001-8708. <a href="/wiki/Michelle_L._Wachs" target="_blank">/wiki/Michelle_L._Wachs</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Shareshian, John; Wachs, Michelle L. (June 4, 2016). "Chromatic quasisymmetric functions". Advances in Mathematics. 295: 497–551. arXiv:1405.4629. doi:10.1016/j.aim.2015.12.018. ISSN 0001-8708. <a href="/wiki/Michelle_L._Wachs" target="_blank">/wiki/Michelle_L._Wachs</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
</ol>

Chromatic symmetric function open-in-new

Chromatic symmetric function