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Register a new userA simple bijection between binary trees and colored ternary trees
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Yidong Sun
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In this short note, we first present a simple bijection between binary trees and colored ternary trees and then derive a new identity related to generalized Catalan numbers.
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A di-sk tree is a rooted binary tree whose nodes are labeled by $oplus$ or $ominus$, and no node has the same label as its right child. The di-sk trees are in natural bijection with separable permutations. We construct a combinatorial bijection on di-sk trees proving the two quintuples $(LMAX,LMIN,DESB,iar,comp)$ and $(LMAX,LMIN,DESB,comp,iar)$ have the same distribution over separable permutations. Here for a permutation $pi$, $LMAX(pi)/LMIN(pi)$ is the set of values of the left-to-right maxima/minima of $pi$ and $DESB(pi)$ is the set of descent bottoms of $pi$, while $comp(pi)$ and $iar(pi)$ are respectively the number of components of $pi$ and the length of initial ascending run of $pi$. Interestingly, our bijection specializes to a bijection on $312$-avoiding permutations, which provides (up to the classical {em Knuth--Richards bijection}) an alternative approach to a result of Rubey (2016) that asserts the two triples $(LMAX,iar,comp)$ and $(LMAX,comp,iar)$ are equidistributed on $321$-avoiding permutations. Rubeys result is a symmetric extension of an equidistribution due to Adin--Bagno--Roichman, which implies the class of $321$-avoiding permutations with a prescribed number of components is Schur positive. Some equidistribution results for various statistics concerning tree traversal are presented in the end.
We give a new algorithm to construct optimal alphabetic ternary trees, where every internal node has at most three children. This algorithm generalizes the classic Hu-Tucker algorithm, though the overall computational complexity has yet to be determined.
K{a}rolyi, Pach, and T{o}th proved that every 2-edge-colored straight-line drawing of the complete graph contains a monochromatic plane spanning tree. It is open if this statement generalizes to other classes of drawings, specifically, to simple drawings of the complete graph. These are drawings where edges are represented by Jordan arcs, any two of which intersect at most once. We present two partial results towards such a generalization. First, we show that the statement holds for cylindrical simple drawings. (In a cylindrical drawing, all vertices are placed on two concentric circles and no edge crosses either circle.) Second, we introduce a relaxation of the problem in which the graph is $k$-edge-colored, and the target structure must be hypochromatic, that is, avoid (at least) one color class. In this setting, we show that every $lceil (n+5)/6rceil$-edge-colored monotone simple drawing of $K_n$ contains a hypochromatic plane spanning tree. (In a monotone drawing, every edge is represented as an $x$-monotone curve.)
These notes are a written version of my talk given at the CARMA workshop in June 2017, with some additional material. I presented a few concepts that have recently been used in the computation of tree-level scattering amplitudes (mostly using pure spinor methods but not restricted to it) in a context that could be of interest to the combinatorics community. In particular, I focused on the appearance of {it planar binary trees} in scattering amplitudes and presented some curious identities obeyed by related objects, some of which are known to be true only via explicit examples.
Two subclasses of Motzkin paths, S-Motzkin and T-Motzkin paths, are introduced. We provide bijections between S-Motzkin paths and ternary trees, S-Motzkin paths and non-crossing trees, and T-Motzkin paths and ordered pairs of ternary trees. Symbolic equations for both paths, and thus generating functions for the paths, are provided. Using these, various parameters involving the two paths are analyzed.
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