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A chromatic partition polynomial
Authors:Einar Steingrímsson  
Institution:

aMatematiska Institutionen, Chalmers Tekniska Högskola, Göteborgs Universitet, 412 96 Göteborg, Sweden

Abstract:A polynomial in two variables is defined by Cn(x,t)=Σπset membership, variantΠnx(Gπ,x)t|π|, where Πn is the lattice of partitions of the set {1, 2, …, n}, Gπ is a certain interval graph defined in terms of the partition gp, χ(Gπ, x) is the chromatic polynomial of Gπ and |π| is the number of blocks in π. It is shown that View the MathML source, where S(n, i) is the Stirling number of the second kind and (x)i = x(x − 1) ··· (xi + 1). As a special case, Cn(−1, −t) = An(t), where An(t) is the nth Eulerian polynomial. Moreover, An(t)=Σπset membership, variantΠnaπt|π| where aπ is the number of acyclic orientations of Gπ.
Keywords:
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