A chromatic partition polynomial

Abstract

A polynomial in two variables is defined by C n ( x , t ) = Σ π ∈ Π n x ( G π , x ) t | π | , where Π n is the lattice of partitions of the set {1, 2, …, n}, G π is a certain interval graph defined in terms of the partition gp, χ( G π, x ) is the chromatic polynomial of G π and |π| is the number of blocks in π. It is shown that C n ( x , t ) = Σ k = 0 n t k Σ i = 0 k ( n − k n − 1 ) S ( n , i ) ( x ) i , where S( n, i) is the Stirling number of the second kind and ( x) i = x( x − 1) ··· ( x − i + 1). As a special case, C n (−1, − t) = A n ( t), where A n ( t) is the nth Eulerian polynomial. Moreover, A n ( t ) = Σ π ∈ Π n a π t | π | where a π is the number of acyclic orientations of G π .