A Chromatic Symmetric Function Conjecture Richard P. Stanley M.I.T.

A Chromatic Symmetric Function Conjecture – p.

Basic notation G: simple graph with d vertices V : vertex set of G E: edge set of G Coloring of G: any κ : V → P = {1, 2, . . . } Proper coloring: uv ∈ E ⇒ κ(u) 6= κ(v)

A Chromatic Symmetric Function Conjecture – p.

The chromatic symmetric function XG = XG (x1 , x2 , . . . ) =

X

xκ ,

proper κ : V →P

the chromatic symmetric function of G, where Y #κ−1 (1) #κ−1 (2) κ xκ(v) = x1 x2 ··· . x = v∈V

A Chromatic Symmetric Function Conjecture – p.

The chromatic symmetric function XG = XG (x1 , x2 , . . . ) =

X

xκ ,

proper κ : V →P

the chromatic symmetric function of G, where Y #κ−1 (1) #κ−1 (2) κ xκ(v) = x1 x2 ··· . x = v∈V

XG (1n ) := XG (1, 1, . . . , 1) = χG (n), | {z } n 1′ s

the chromatic polynomial of G.

A Chromatic Symmetric Function Conjecture – p.

Example of a monomial 3 5 1

3 1

2 xκ = x21 x2 x23 x5 A Chromatic Symmetric Function Conjecture – p.

Simple examples Xpoint = x1 + x2 + x3 + · · · = e1 . More generally, let ek =

X

xi1 · · · xik ,

1≤i1