Generalized k-chromatic polynomials of graphs are expressed via dimensions of grade spaces in the associated free partially commutative Lie algebra using heaps of pieces.
Chromatic Polynomial and Heaps of Pieces
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Stanley in his paper [Stanley, Richard P.: Acyclic orientations of graphs In: Discrete Mathematics 5 (1973), Nr. 2, S. 171-178.] provided interpretations of the chromatic polynomial when it is substituted with negative integers. Greene and Zaslavsky interpreted the coefficients of the chromatic polynomial in [Greene, Curtis ; Zaslavsky, Thomas: On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs. In: Transactions of the American Mathematical Society 280 (1983), jan, Nr. 1, S. 97-97.]. We shall develop an involution on factorisations of heaps of pieces and using this involution, we shall provide bijective proofs to results from both the papers.
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Generalized chromatic polynomials of graphs from Heaps of pieces
Generalized k-chromatic polynomials of graphs are expressed via dimensions of grade spaces in the associated free partially commutative Lie algebra using heaps of pieces.