Automated Counting and Statistical Analysis of Labeled Trees with Degree Restrictions
Shalosh B. Ekhad, Doron Zeilberger
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5 pages. Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and this arxiv. Accompanied by a Maple package, and extensive output files available from https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/etsim.html . This version adds references to previous work of A. Meir and J. W. Moon
Arthur Cayley famously proved that there are n to the power n-2 labeled trees on n vertices. Here we go much further and show how to enumerate, fully automatically, labeled trees such that every vertex has a number of neighbors that belongs to a specified finite set, and also count trees where the number of neighbors is not allowed to be in a given finite set. We also give detailed statistical analysis, and show that in the sample space of labeled trees with n vertices, the random variable "number of vertices with d neighbors" is asymptotically normal, and for any different degrees, are jointly asymptotically normal, but of course, not independently so (except for the pair (1,3), i.e. the number of leaves and the number of degree-3 vertices, where there are asymptotically independent). We also give new proofs to Amram Meir and John Noon's expressions for the limiting expectation and variance for these, and derive an explicit expression for the covariance.
We provide a short combinatorial proof of Cayley's formula by means of a bijective map to an outcome space of an urn-drawing problem. Furthermore we introduce an algebraic structure on the set of labeled trees, which provides a more standard approach to Cayley's formula. Moreover, this algebraic structure sheds light on the problem of counting the unlabeled trees. In particular, it indicates how counting the number of unlabeled trees on $n$ vertices is connected to finding the number of partitions of $n-2$
A threshold graph is any graph which can be constructed from the empty graph by repeatedly adding a new vertex that is either adjacent to every vertex or to no vertices. The Eulerian number $\genfrac{\langle}{\rangle}{0pt}{}{n}{k}$ counts the number of permutations of size $n$ with exactly $k$ ascents. Implicitly Beissinger and Peled proved that the number of labeled threshold graphs on $n\ge 2$ vertices is \[\sum_{k=1}^{n-1}(n-k)\genfrac{\langle}{\rangle}{0pt}{}{n-1}{k-1}2^k.\] Their proof used generating functions. We give a direct combinatorial proof of this result.
In his Ph.D. thesis, Ira Gessel proved a reciprocity formula for noncommutative symmetric functions which enables one to count words and permutations with restrictions on the lengths of their increasing runs. We generalize Gessel's theorem to allow for a much wider variety of restrictions on increasing run lengths, and use it to complete the enumeration of permutations with parity restrictions on peaks and valleys, and to give a systematic method for obtaining generating functions for permutation statistics that are expressible in terms of increasing runs. Our methods can also be used to obtai…
Given $A \subset F_{p}^2$ a sufficiently small set in the plane over a prime residue field, we prove that there are at most $O_ε(|A|^{\frac{99}{41}+ε})$ rectangles with corners in $A$. The exponent $\frac{99}{41} = 2.413\ldots$ improves slightly on the exponent of $\frac{17}{7} = 2.428\ldots$ due to Rudnev and Shkredov. Using this estimate we prove that the extension operator for the three dimensional paraboloid in prime order fields maps $L^2 \rightarrow L^{r}$ for $r >\frac{188}{53}=3.547\ldots$ improving the previous range of $r\geq \frac{32}{9}= 3.\overline{555}$.
Pólya's enumeration theorem is concerned with counting labeled sets up to symmetry. Given a finite group acting on a finite set of labeled elements it states that the number of labeled sets up to symmetry is given by a polynomial in the number of labels. We give a new perspective on this theorem by generalizing it to partially ordered sets and order preserving maps. Further we prove a reciprocity statement in terms of strictly order preserving maps generalizing a classical result by Stanley (1970). We apply our results to counting graph colorings up to symmetry.
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.
We consider labeled $r$-uniform hypertrees having $n \ge r \ge 2$ vertices. The number of hyperedges in such a hypertree is $m = (n - 1)/(r - 1)$. We show that there are exactly $f(n, r) = \frac{(n-1)! n^{m-1}}{(r-1)!^m m!}$ $r$-uniform hypertrees with $n$ vertices labeled with distinct integers. We also give an encoding scheme that encodes such hypertrees using, on an average, at most $1 + \log_2 e$ bits more than $\log_2(f(n, r))$.
Let C(n,k) denote the number of connected graphs with n labeled vertices and n+k-1 edges. For any sequence (k_n), the limit of C(n,k_n) as n tends to infinity is known. It has been observed that, if k_n=o(\sqrt{n}), this limit is asymptotically equal to the $k_n$th moment of the area under the standard Brownian excursion. These moments have been computed in the literature via independent methods. In this article we show why this is true for k_n=o(\sqrt[3]{n}) starting from an observation made by Joel Spencer. The elementary argument uses a result about strong embedding of the Uniform empirical…
The classical 1991 result by Brightwell and Winkler states that the number of linear extensions of a poset is #P-complete. We extend this result to posets with certain restrictions. First, we prove that the number of linear extension for posets of height two is #P-complete. Furthermore, we prove that this holds for incidence posets of graphs. Finally, we prove that the number of linear extensions for posets of dimension two is #P-complete.
We find the number of compositions over finite abelian groups under two types of restrictions: (i) each part belongs to a given subset and (ii) small runs of consecutive parts must have given properties. Waring's problem over finite fields can be converted to type~(i) compositions, whereas Carlitz and locally Mullen compositions can be formulated as type~(ii) compositions. We use the multisection formula to translate the problem from integers to group elements, the transfer matrix method to do exact counting, and finally the Perron-Frobenius theorem to derive asymptotics. We also exhibit bijec…
Motzkin paths are simple yet important combinatorial objects. In this paper, we consider families of Motzkin paths with restrictions on peak heights, valley heights, upward-run lengths, downward-run lengths, and flat-run lengths. This paper presents two fully automated methods for enumerating the paths of such families. The first method uses numeric dynamic programming. While this method often times works, it can be slow and may not work for larger problems. The second method uses symbolic dynamic programming to solve such problems. These methods are implemented in the maple packages accompany…
In this paper we enumerate $k$-noncrossing tangled-diagrams. A tangled-diagram is a labeled graph whose vertices are $1,...,n$ have degree $\le 2$, and are arranged in increasing order in a horizontal line. Its arcs are drawn in the upper halfplane with a particular notion of crossings and nestings. Our main result is the asymptotic formula for the number of $k$-noncrossing tangled-diagrams $T_{k}(n) \sim c_k n^{-((k-1)^2+(k-1)/2)} (4(k-1)^2+2(k-1)+1)^n$ for some $c_k>0$.
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; for most of these a simple test can be used to quickly show that the form of the asymptotics is the same as that for the class of rooted trees: C ρ^{-n} n^{-3/2} where ρis the radius of convergence of T.
Site swap is a mathematical notation used by jugglers to communicate, create and study complex juggling patterns. Determining the number of possible site swap juggling patterns with respect to certain limiting parameters such as number of balls etc., is a problem that has been much studied and solved by many mathematicians. However, when the patterns have a throw height restriction (ceiling) the problem becomes difficult and is in general still open. In this article we derive some formulae for computing the number of possible juggling patterns with respect to certain ceiling types.
We count the number of occurrences of certain patterns in given words. We choose these words to be the set of all finite approximations of a sequence generated by a morphism with certain restrictions. The patterns in our considerations are either classical patterns 1-2, 2-1, 1-1-...-1, or arbitrary generalized patterns without internal dashes, in which repetitions of letters are allowed. In particular, we find the number of occurrences of the patterns 1-2, 2-1, 12, 21, 123 and 1-1-...-1 in the words obtained by iterations of the morphism 1->123, 2->13, 3->2, which is a classical example of a m…
We consider maps on genus-$g$ surfaces with $n$ (labeled) faces of prescribed even degrees. It is known since work of Norbury that, if one disallows vertices of degree one, the enumeration of such maps is related to the counting of lattice point in the moduli space of genus-$g$ curves with $n$ labeled points and is given by a symmetric polynomial $N_{g,n}(\ell_1,\ldots,\ell_n)$ in the face degrees $2\ell_1, \ldots, 2\ell_n$. We generalize this by restricting to genus-$g$ maps that are essentially $2b$-irreducible for $b\geq 0$, which loosely speaking means that they are not allowed to possess …
Phylogenetic networks generalize phylogenetic trees, and have been introduced in order to describe evolution in the case of transfer of genetic material between coexisting species. There are many classes of phylogenetic networks, which can all be modeled as families of graphs with labeled leaves. In this paper, we focus on rooted and unrooted level-k networks and provide enumeration formulas (exact and asymptotic) for rooted and unrooted level-1 and level-2 phylogenetic networks with a given number of leaves. We also prove that the distribution of some parameters of these networks (such as the…
Delone sets of finite local complexity in Euclidean space are investigated. We show that such a set has patch counting and topological entropy 0 if it has uniform cluster frequencies and is pure point diffractive. We also note that the patch counting entropy is 0 whenever the repetitivity function satisfies a certain growth restriction.
We focus on counting the number of labeled graphs on $n$ vertices and treewidth at most $k$ (or equivalently, the number of labeled partial $k$-trees), which we denote by $T_{n,k}$. So far, only the particular cases $T_{n,1}$ and $T_{n,2}$ had been studied. We show that $$ \left(c \cdot \frac{k\cdot 2^k \cdot n}{\log k} \right)^n \cdot 2^{-\frac{k(k+3)}{2}} \cdot k^{-2k-2}\ \leq\ T_{n,k}\ \leq\ \left(k \cdot 2^k \cdot n\right)^n \cdot 2^{-\frac{k(k+1)}{2}} \cdot k^{-k}, $$ for $k > 1$ and some explicit absolute constant $c > 0$. The upper bound is an immediate consequence of the well-known num…
F-nomial coefficients encompass among others well-known binomial coefficients or Gaussian coefficients that count subsets of finite set and subspaces of finite vector space respectively. Here, the so called F-cobweb tiling sequences N(a) are considered. For such specific sequences a new interpretation with respect to Kwasniewski general combinatorial interpretation of F-nomial coefficients is unearhed. Namely, for tiling sequences F = N(a)$ the F-nomial coefficients are equal to the number of labeled special bipartite multigraphs denoted here as a-multigraphs G(a,n,k). An explicit relation bet…
Directed acyclic graphs (DAGs) can be characterised as directed graphs whose strongly connected components are isolated vertices. Using this restriction on the strong components, we discover that when $m = cn$, where $m$ is the number of directed edges, $n$ is the number of vertices, and $c < 1$, the asymptotic probability that a random digraph is acyclic is an explicit function $p(c)$, such that $p(0) = 1$ and $p(1) = 0$. When $m = n(1 + μn^{-1/3})$, the asymptotic behaviour changes, and the probability that a digraph is acyclic becomes $n^{-1/3} C(μ)$, where $C(μ)$ is an explicit function of…
We give a short proof of Cayley's tree formula for counting the number of different labeled trees on $n$ vertices. The following nonlinear recursive relation for the number of labeled trees on $n$ vertices is deduced from a combinatorial argument, $$ T_n = \frac{n}{2} \sum_{k=0}^{n-2} \left ( \begin {array} {c} n-2 \\ k \end {array} \right ) T_{k+1} T_{n-k-1}; \ \ for \ n > 1 \ and \ T_1 = 1, $$ and then it is proved that $T_n = n^{n-2}$, which gives yet another proof of the celebrated Cayley's tree formula.
Given a combinatorial decomposition for a counting problem, we resort to the simple scheme of approximating large numbers by floating-point representations in order to obtain efficient Fully Polynomial Time Approximation Schemes (FPTASes) for it. The number of bits employed for the exponent and the mantissa will depend on the error parameter $0 < \varepsilon \leq 1$ and on the characteristics of the problem. Accordingly, we propose the first FPTASes with $1 \pm \varepsilon$ relative error for counting and generating uniformly at random a labeled DAG with a given number of vertices. This is acc…
It is a classical result that the parity-balance of the number of weak excedances of all permutations (derangements, respectively) of length $n$ is the Euler number $E_n$, alternating in sign, if $n$ is odd (even, respectively). Josuat-Vergès obtained a $q$-analog of the results respecting the number of crossings of a permutation. One of the goals in this paper is to extend the results to the permutations (derangements, respectively) of types B and D, on the basis of the joint distribution in statistics excedances, crossings and the number of negative entries obtained by Corteel, Josuat-Vergès…
We investigate the number of variables in two special subclasses of lambda-terms that are restricted by a bound of the number of abstractions between a variable and its binding lambda, the so-called De-Bruijn index, or by a bound of the nesting levels of abstractions, \textit{i.e.}, the number of De Bruijn levels, respectively. These restrictions are on the one hand very natural from a practical point of view, and on the other hand they simplify the counting problem compared to that of unrestricted lambda-terms in such a way that the common methods of analytic combinatorics are applicable. We …
With the current burst of network theory (especially in connection with social and biological networks) there is a renewed interest on realizations of given degree sequences. In this paper we propose an essentially new degree sequence problem: we want to find graphical realizations of a given degree sequence on labeled vertices, where certain would-be edges are {\em forbidden}. Then we want to sample uniformly and efficiently all these possible realizations. (This problem can be considered as a special case of Tutte's $f$-factor problem, however it has a favorable sampling speed.) We solve thi…
Set partitions and permutations with restrictions on the size of the blocks and cycles are important combinatorial sequences. Counting these objects lead to the sequences generalizing the classical Stirling and Bell numbers. The main focus of the present article is the analysis of combinatorial and arithmetical properties of them. The results include several combinatorial identities and recurrences as well as some properties of their $p$-adic valuations.
We describe a general purpose algorithm for counting simple cycles and simple paths of any length $\ell$ on a (weighted di)graph on $N$ vertices and $M$ edges, achieving a time complexity of $O\left(N+M+\big(\ell^ω+\ellΔ\big) |S_\ell|\right)$. In this expression, $|S_\ell|$ is the number of (weakly) connected induced subgraphs of $G$ on at most $\ell$ vertices, $Δ$ is the maximum degree of any vertex and $ω$ is the exponent of matrix multiplication. We compare the algorithm complexity both theoretically and experimentally with most of the existing algorithms for the same task. These comparison…
