Björner and Wachs defined a major index for labeled plane forests and showed that it has the same distribution as the number of inversions. We define and study the distributions of a few other natural statistics on labeled forests. Specifically, we introduce the notions of bottom-totop maxima, cyclic bottom-to-top maxima, sorting index, and cycle minima. Then we show that the pairs (inv, Bt-max), (sor, Cyc), and (maj, Cbt-max) are equidistributed. Our results extend the result of Björner and Wachs and generalize results for permutations. We also introduce analogous statistics for signed labeled forests and show equidistribution results which generalize results for signed permutations.

A set S of vertices in a graph G = (V, E) is called cycle independent if the induced subgraph S is acyclic, and called oddcycle indepdendet if S is bipartite. A set S is cycle dominating (resp. odd-cycle dominating) if for every vertex u ∈ V \ S there exists a vertex v ∈ S such that u and v are contained in a (resp. odd cycle) cycle in S \{u} . A set S is cycle irredundant (resp. odd-cycle irredundant) if for every vertex v ∈ S there exists a vertex u ∈ V \ S such that u and v are in a (resp. odd cycle) cycle of S \ {u} , but u is not in a cycle of S ∪ {u} \ {v} . In this paper we present these new concepts, which relate in a natural way to independence, domination and irredundance in graphs. In particular, we construct analogs to the domination inequality chain for these new concepts.

For a poset whose Hasse diagram is a rooted plane forest F , we consider the corresponding tree descent polynomial A F (q), which is a generating function of the number of descents of the labelings of F . When the forest is a path, A F (q) specializes to the classical Eulerian polynomial. We prove that the coefficient sequence of A F (q) is unimodal and that if {T n } is a sequence of trees with |T n | = n and maximal down degree D n = O(n 0.5−ǫ ) then the number of descents in a labeling of T n is asymptotically normal.

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