相关论文: A Combinatorial proof of a result of Hetyei and Re…
In this paper we enumerate and give bijections for the following four sets of vertices among rooted ordered trees of a fixed size: (i) first-children of degree $k$ at level $\ell$, (ii) non-first-children of degree $k$ at level $\ell-1$,…
For any integer s >= 0, we derive a combinatorial interpretation for the family of sequences generated by the recursion (parameterized by s) h_s(n) = h_s(n - s - h_s(n - 1)) + h_s(n - 2 - s - h_s(n - 3)), n > s + 3, with the initial…
The classes of tree permutations and forest permutations were defined by Acan and Hitczenko (2016). We study random permutations of a given length from these classes, and in particular the number of occurrences of a fixed pattern in one of…
Given a permutation $\sigma$, its corresponding binary search tree is obtained by recursively inserting the values $\sigma(1),\ldots,\sigma(n)$ into a binary tree so that the label of each node is larger than the labels of its left subtree…
A subset of vertices is a {\it maximum independent set} if no two of the vertices are adjacent and the subset has maximum cardinality. A subset of vertices is called a {\it maximum dissociation set} if it induces a subgraph with vertex…
We state and solve a discrete version of the classical Riemann-Hilbert problem. In particular, we associate a Riemann-Hilbert problem to every dessin d'enfants. We show how to compute the solution for a dessin that is a tree. This amounts…
In the combinatorial study of the coefficients of a bivariate polynomial that generalizes both the length and the reflection length generating functions for finite Coxeter groups, Petersen introduced a new Mahonian statistic $sor$, called…
A phylogenetic tree shows the evolutionary relationships among species. Internal nodes of the tree represent speciation events and leaf nodes correspond to species. A goal of phylogenetics is to combine such trees into larger trees, called…
The Horton-Strahler (HS) number, a classical measure of branching complexity arising in hydrology and register allocation, is studied for butterfly trees, a recursive family of binary trees generated by block-merging operations. These trees…
A permutation is (1-23-4)-avoiding if it contains no four entries, increasing left to right, with the middle two adjacent in the permutation. Here we give a 2-variable recurrence for the number of such permutations, improving on the…
Splay trees are a simple and efficient dynamic data structure, invented by Sleator and Tarjan. The basic primitive for transforming a binary tree in this scheme is a rotation. Sleator, Tarjan, and Thurston proved that the maximum rotation…
Blanc showed in his thesis that a compact minimal foliated space with a residual subset of 2-ended leaves can contain only 1 or 2 ended leaves. In this paper we give examples of compact minimal foliated spaces where a topologically generic…
The Gy\'arf\'as tree packing conjecture states that any set of $n-1$ trees $T_{1},T_{2},..., T_{n-1}$ such that $T_i$ has $n-i+1$ vertices pack into $K_n$. We show that $t=1/10n^{1/4}$ trees $T_1,T_2,..., T_t$ such that $T_i$ has $n-i+1$…
We prove that, if $m$ is sufficiently large, every graph on $m+1$ vertices that has a universal vertex and minimum degree at least $\lfloor \frac{2m}{3} \rfloor$ contains each tree $T$ with $m$ edges as a subgraph. Our result confirms, for…
In the critical beta-splitting model of a random $n$-leaf rooted tree, clades are recursively split into sub-clades, and a clade of $m$ leaves is split into sub-clades containing $i$ and $m-i$ leaves with probabilities $\propto 1/(i(m-i))$.…
A branch vertex in a tree is a vertex of degree at least three. We prove that, for all $s\geq 1$, every connected graph on $n$ vertices with minimum degree at least $(\frac{1}{s+3}+o(1))n$ contains a spanning tree having at most $s$ branch…
We construct natural representations of the Connes-Kreimer Lie algebras on rooted trees/Feynman graphs arising from Hecke correspondences in the categories $\LRF, \LFG$ constructed by K. Kremnizer and the author. We thus obtain the…
Recently, Andrews and El Bachraoui considered the number of integer partitions whose smallest part is repeated exactly $k$ times and the remaining parts are not repeated. They presented several interesting results and posed questions…
Motivated by a question of Defant and Propp (2020) regarding the connection between the degrees of noninvertibility of functions and those of their iterates, we address the combinatorial optimization problem of minimizing the sum of squares…
Combinatorial classes T that are recursively defined using combinations of the standard multiset, sequence, directed cycle and cycle constructions, and their restrictions, have generating series T(z) with a positive radius of convergence;…