Sviatlana Kniahnitskaya, Michele Ortiz, Olivia Ramirez, Katharine Shultis, Hays Whitlatch
Motivated by the question of computing the probability of successful power domination by placing k monitors uniformly at random, in this paper we give a recursive formula to count the number of power domination sets of size k in a labeled complete m-ary tree. As a corollary we show that the desired probability can be computed in exponential with linear exponent time.
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Using the Lagrange inversion formula, $t$-ary trees are enumerated with respect to edge type (left, middle, right for ternary trees).
The sequence A120986 in the Encyclopedia of Integer Sequences counts ternary trees according to the number of nodes and the number of middle edges. Using a certain substition, the underlying cubic equation can be factored. This leads to an extension of the concept of $(3/2)$-ary trees, introduced by Knuth in his christmas lecture from 2014.
Counting largest mutual-visibility and general position sets of glued $t$-ary trees
研究粘合 $t$-ary 树的最大互视集和一般位置集的计数问题，确定相关不变量并扩展至更一般的情况。
We count the number of vertices in plane trees and $k$-ary trees with given outdegree, and prove that the total number of vertices of outdegree $i$ over all plane trees with $n$ edges is ${2n-i-1 \choose n-1}$, and the total number of vertices of outdegree $i$ over all $k$-ary trees with $n$ edges is ${k\choose i}{kn\choose n-i}$. For both results we give bijective proofs as well as generating function proofs.
Using double counting, we prove Delsarte inequalities for $q$-ary codes and their improvements. Applying the same technique to $q$-ary constant-weight codes, we obtain new inequalities for $q$-ary constant-weight codes.
We present some exact expressions for the number of paths of a given length in a perfect $m$-ary tree. We first count the paths in perfect rooted $m$-ary trees and then use the results to determine the number of paths in perfect unrooted $m$-ary trees, extending a known result for binary trees.
A rooted tree is called a $k$-ary tree, if all non-leaf vertices have exactly $k$ children, except possibly one non-leaf vertex has at most $k-1$ children. Denote by $h(k)$ the minimum integer such that every tournament of order at least $h(k)$ contains a $k$-ary spanning tree. It is well-known that every tournament contains a Hamiltonian path, which implies that $h(1)=1$. Lu et al. [J. Graph Theory {\bf 30}(1999) 167--176] proved the existence of $h(k)$, and showed that $h(2)=4$ and $h(3)=8$. The exact values of $h(k)$ remain unknown for $k\geq 4$. A result of Erdős on the domination number o…
Counting k-ary words by number of adjacency differences of a prescribed size
本文研究了 k-ary words 上指定大小相邻差异数的分布问题，计算了相应统计量的常生成函数显式公式，并扩展至有限集合分块和绝对差情形，同时提供了相关递推式的组合证明。
We calculate the exact number of contours of size $n$ containing a fixed vertex in $d$-ary trees and provide sharp estimates for this number for more general trees. We also obtain a characterization of the locally finite trees with infinitely many contours of the same size containing a fixed vertex.
Random tensor models are generalizations of matrix models which also support a 1/N expansion. The dominant observables are in correspondence with some trees, namely rooted trees with vertices of degree at most $D$ and lines colored by a number $i$ from 1 to $D$ such that no two lines connecting a vertex to its descendants have the same color. In this Letter we study by independent methods a generating function for these observables. We prove that the number of such trees with exactly $p_i$ lines of color $i$ is $\frac{1}{\sum_{i=1}^D p_i +1} \binom{\sum_{i=1}^D p_i+1}{p_1} ... \binom{\sum_{i=1…
Counting words without strictly increasing subwords of fixed length
本文推导了计数不含长度为k的严格递增子词的n元单词数量的生成函数精确公式，并给出其应用。
Counting dominating sets in a graph $G$ is closely related to the neighborhood complex of $G$. We exploit this relation to prove that the number of dominating sets $d(G)$ of a graph is determined by the number of complete bipartite subgraphs of its complement. More precisely, we state the following. Let $G$ be a simple graph of order $n$ such that its complement has exactly $a(G)$ subgraphs isomorphic to $K_{2p,2q}$ and exactly $b(G)$ subgraphs isomorphic to $K_{2p+1,2q+1}$. Then $d(G) = 2^n -1 + 2[a(G)-b(G)]$. We also show some new relations between the domination polynomial and the neighborh…
A $k$-plane tree is a plane tree whose vertices are assigned labels between $1$ and $k$ in such a way that the sum of the labels along any edge is no greater than $k+1$. These trees are known to be related to $(k+1)$-ary trees, and they are counted by a generalised version of the Catalan numbers. We prove a surprisingly simple refined counting formula, where we count trees with a prescribed number of labels of each kind. Several corollaries are derived from this formula, and an analogous theorem is proven for $k$-noncrossing trees, a similarly defined family of labelled noncrossing trees that …
We associate with a Bienayme-Galton-Watson branching process a family tree rooted at the ancestor. For a positive integer N, define a complete N-ary tree to be the family tree of a deterministic branching process with offspring generating function s^N. We study the random variables V(N,n) and V(N) counting the number of disjoint complete N-ary subtrees, rooted at the ancestor, and having height n and infinity, respectively. Dekking (1991) and Pakes and Dekking (1991) find recursive relations for Pr(V(N,n)>0) and Pr(V(N)>0) involving the offspring probability generation function (pgf) and its d…
Our goal is to prove new results in graph theory and combinatorics thanks to the speed of computers, used with smart algorithms. We tackle four problems. The four-colour theorem states that any map whose countries are connected can be coloured with 4 colours such that neighbouring countries have differnt colours. It was the first result proved using computers, in 1989. We wished to automatise further its proof. We explain the proof and provide a program which replays the proof. It also makes it possible to obtain other results with the same method. We give ideas to automatise the search for di…
We introduce new combinatorial objects, the interval- posets, that encode intervals of the Tamari lattice. We then find a combinatorial interpretation of the bilinear operator that appears in the functional equation of Tamari intervals described by Chapoton. Thus, we retrieve this functional equation and prove that the polynomial recursively computed from the bilinear operator on each tree T counts the number of trees smaller than T in the Tamari order. Then we show that a similar m + 1-linear operator is also used in the functionnal equation of m-Tamari intervals. We explain how the m-Tamari …
Counting Weight-$k$ Italian Dominating Sets on Trees and Cycles
本文研究路径、树和环上意大利支配函数的计数问题，提出动态规划算法计算所有权重系数，并给出环上系数的闭式表达式，验证了已知的意大利支配数。
Laplacian Spectrum and Domination in Trees
本文研究树中最小支配集大小与拉普拉斯矩阵在[0,1)内特征值个数之比，证明该比值小于4/3并构造无穷树族说明界的最优性，同时对非叶顶点度数受限的树给出更精细上界。
Algorithmic counting of nonequivalent compact Huffman codes
本文证明了可通过一次幂级数除法计算所有 $n<N$ 的不等价紧致Huffman码序列 $f_t(n)$，仅需 $N^{2+\varepsilon}$ 次位运算，改进了先前 $O(N^4)$ 的界限。
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