Möbius coinvariants and bipartite edge-rooted forests

Abstract The Mobius coinvariant μ ⊥ ( G ) of a graph G is defined to be the Mobius invariant of the dual of the cycle matroid of G . This invariant is known to equal the rank of the reduced homology of the cycle matroid complex of G . For a complete graph K m + 1 , W. Kook gave an interpretation of μ ⊥ ( K m + 1 ) as the number of edge-rooted forests in K m . In this paper, we obtain a new combinatorial interpretation of μ ⊥ ( K m + 1 , n + 1 ) as the number of B-edge-rooted forests in K m , n , which is a bipartite analogue of the previous result. Based on these interpretations, we will give new bijective proofs of the formulas for μ ⊥ ( K m + 1 ) and μ ⊥ ( K m + 1 , n + 1 ) given by I. Novik, A. Postnikov, and B. Sturmfels in terms of the Hermite polynomials. In addition, we will construct a homology basis for the cycle matroid complex of K m + 1 , n + 1 indexed by the B-edge-rooted forests. Also we will discuss the Mobius coinvariant of bi-coned graphs which generalize complete bipartite graphs.

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