Graph coloring and monotone functions on posets

Proof. We denote V -[p] = { 1 , . . . , p}, A(G) is the set of acyclic orientations of G and a(G) = IA(G)I is their number. An n-coloring of G, c: V---> [n] induces an acyclic orientation DceA(G) as follows: If [x,y]eE is an edge, where c(x) > c(y) then in Dc this edge is oriented from x to y. Every acyclic orientation D ~ A(G) defines a partial order on V, which we denote by i>o. If D e A(G), then we think of D as both an acyclic orientation and as a partial order on V. Note that for an n-coloring c: V--->[n] the function c is a strongly orderpreserving map from the poset (V, ~>n) into [n]. It is easily verified that the correspondence between n-colorings and strong order preserving maps from acyclic orientations into [n], is bijective. For a poset (P, I>) we let #p be its strong order polynomial, namely, # ( n ) = #e(n) is the number of strongly monotone