arXiv · math/0407235
This note gives a short proof on characterizations of a forest to be equitably k-colorable.
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Gerard J. Chang. 2004-07-14. A note on equitable colorings of forests. https://arxiv.org/abs/math/0407235
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We give a precise definition of incidence theorems in plane projective geometry and introduce the notion of absolute incidence theorems, which hold over any ring. Fomin and Pylyavskyy describe how to obtain incidence theorems from tilings of an orientable surface; they call this result the master theorem. Instances of the master theorem are always absolute incidence theorems. As most classically known incidence theorems are instances of the master theorem, they are absolute incidence theorems. We give an explicit example of an incidence theorem involving 13 points that is not an absolute incidence theorem, and therefore is not an instance of the master theorem.
We prove cyclic sieving phenomena satisfied by corner-rooted plane trees (alias ordered trees). The sets of rooted plane trees that we consider are: (1) all trees with $n$ nodes; (2) all trees with $n$ nodes and $k$ leaves; (3) all trees with a given degree distribution of the nodes. Moreover, we consider four different cyclic group actions: (1) the root is moved to the next corner along a tour of the tree; (2) only trees in which the root is at a leaf are considered, and the action moves the root to the next leaf; (3) only trees in which the root is at a non-leaf are considered, and the action moves the root to the next non-leaf corner; (4) only trees in which the root is at a node of degree $δ$ are considered, for a fixed $δ$, and the action moves the root to the next corner of this type. We prove a cyclic sieving phenomenon for each meaningful combination of these sets and actions. As a bonus, we also establish corresponding cyclic sieving phenomena for tree-rooted planar maps.
We prove injectively that the pattern-avoiding permutation classes $\mathrm{Av}(1324, 231)$ and $\mathrm{Av}(1324, 2314, 3214, 4213)$ are inversion monotone: for every $k$, the number of length-$n$ permutations with $k$ inversions in either class is weakly increasing in $n$. These are the first known nontrivial examples of the phenomenon. The latter is a special case of a general construction. We also characterize the finitely based permutation classes in which the number of length-$n$ members with a fixed number of inversions stabilizes as $n$ grows, and determine the limiting values for all classes $\mathrm{Av}(1324, p)$ with $p \in S_4$. Connections to several nice families of integer partitions are revealed. Lastly, we show that for asymptotically almost every pattern $p \in \mathrm{Av}(1324)$, the class $\mathrm{Av}(1324, p)$ is inversion monotone in the region $n \geq \frac{k+7}{2}$, expanding on the work of Linusson and Verkama. This work is motivated by the conjectures of Claesson, Jelínek and Steingrímsson that $\mathrm{Av}(1324)$, and more generally $\mathrm{Av}(p)$ for every nonidentity pattern $p$, is inversion monotone.