scholarly journals Counting Forests by Descents and Leaves

10.37236/1266 ◽  
1995 ◽  
Vol 3 (2) ◽  
Author(s):  
Ira M. Gessel
Keyword(s):  
Functional Equation ◽  
Generating Function ◽  
Rooted Tree ◽  
Rooted Trees ◽  
Exponential Generating Function ◽  
Vertex Set

A descent of a rooted tree with totally ordered vertices is a vertex that is greater than at least one of its children. A leaf is a vertex with no children. We show that the number of forests of rooted trees on a given vertex set with $i+1$ leaves and $j$ descents is equal to the number with $j+1$ leaves and $i$ descents. We do this by finding a functional equation for the corresponding exponential generating function that shows that it is symmetric.

Enumeration of smooth labelled graphs

1982 ◽  
Vol 91 (3-4) ◽  
pp. 205-212 ◽  
Author(s):  
E. M. Wright
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Functional Equation ◽  
Asymptotic Expansion ◽  
Explicit Form ◽  
Generating Function ◽  
Asymptotic Approximation ◽  
Recurrence Formula ◽  
Exponential Generating Function ◽  
Labelled Graphs

SynopsisAn (n, q) graph is a graph on n labelled points and q lines without loops or multiple lines. We write ν(n, q) for the number of smooth (n, q) graphs, i.e. connected graphs without end points, and ν = V(Z, Y) = ∑n,q ν(n,q)ZnYq /n! for the exponential generating function of ν(n,q). We use the Riddell “core and mantle” method to find an explicit form for V (not, as usual with this method, only a functional equation). From this we deduce a partial differential equation satisfied by V. We interpret this equation in purely combinatorial terms. We write Vk = ∑ n ν(n, n + k)Xn/n! and find a recurrence formula for Vk for successive k. We use these and other results to find an asymptotic expansion for ν(n,q) as n→∞ when (q/n) − log n − log log n→ + ∞ and an asymptotic approximation to ν(n,n + k) when 0 < k = o and to log ν(n, n + k) when k < (1−ε).

scholarly journals On the distribution of winning moves in random game trees

1981 ◽  
Vol 24 (2) ◽  
pp. 227-237
Author(s):  
J. A. Flanigan
Keyword(s):  
Asymptotic Distribution ◽  
Rooted Tree ◽  
Random Variable ◽  
Initial Position ◽  
Rooted Trees ◽  
Contour Integral ◽  
Game Trees ◽  
Vertex Set ◽  
Natural Way

In a natural way, associated with each rooted tree there exists a pair of two-person games, each game possessing the root as initial position. When a rooted tree is selected at random from the set of all rooted trees which possess {1, 2, …, n} as vertex set, the number of winning moves available to the first player to move in each of the associated games is a random variable. For fixed n, we determine the distribution of this random variable. As an immediate consequence, we find the probability that the first player to move has no winning move at all. The “saddlepoint method” is applied to a certain contour integral to obtain the asymptotic distribution of the number of winning moves as n → ∞.

scholarly journals IDENTITIES OF SYMMETRY FOR GENERALIZED TWISTED BERNOULLI POLYNOMIALS TWISTED BY RAMIFIED ROOTS OF UNITY

2014 ◽  
Vol 60 (1) ◽  
pp. 19-36
Author(s):  
Dae San Kim
Keyword(s):  
Generating Function ◽  
Bernoulli Polynomials ◽  
Roots Of Unity ◽  
Integral Expression ◽  
Power Sums ◽  
Exponential Generating Function

Abstract We derive eight identities of symmetry in three variables related to generalized twisted Bernoulli polynomials and generalized twisted power sums, both of which are twisted by ramified roots of unity. All of these are new, since there have been results only about identities of symmetry in two variables. The derivations of identities are based on the p-adic integral expression of the generating function for the generalized twisted Bernoulli polynomials and the quotient of p-adic integrals that can be expressed as the exponential generating function for the generalized twisted power sums.

A generalized class of restricted Stirling and Lah numbers

2018 ◽  
Vol 68 (4) ◽  
pp. 727-740 ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Generating Function ◽  
Stirling Numbers ◽  
Combinatorial Properties ◽  
Exponential Generating Function ◽  
Cauchy Numbers ◽  
Restricted Stirling Numbers

Abstract In this paper, we consider a polynomial generalization, denoted by $\begin{array}{} u_m^{a,b} \end{array}$ (n, k), of the restricted Stirling numbers of the first and second kind, which reduces to these numbers when a = 1 and b = 0 or when a = 0 and b = 1, respectively. If a = b = 1, then $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) gives the cardinality of the set of Lah distributions on n distinct objects in which no block has cardinality exceeding m with k blocks altogether. We derive several combinatorial properties satisfied by $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) and some additional properties in the case when a = b = 1. Our results not only generalize previous formulas found for the restricted Stirling numbers of both kinds but also yield apparently new formulas for these numbers in several cases. Finally, an exponential generating function formula is derived for $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) as well as for the associated Cauchy numbers.

A note on the time-dependent and the stationary behaviour of a semi-infinite reservoir subject to a combination of Markovian inflows

1971 ◽  
Vol 8 (04) ◽  
pp. 708-715 ◽  
Author(s):  
Emlyn H. Lloyd
Keyword(s):  
Functional Equation ◽  
Explicit Formula ◽  
Generating Function ◽  
Present Theory ◽  
The Other ◽  
Time Dependent ◽  
Independent Sequence ◽  
Time Dependent Solution ◽  
Infinite Reservoir ◽  
Stationary Behaviour

The present theory of finite reservoirs is not rich in general theorems even when of the simple Moran type, with unit draft and stationary discrete independent-sequence inflows. For the corresponding systems with unbounded capacity however there are two classes of results which have been known for some time. One of these classes is concerned with the time-dependent solution, where the theory provides a functional equation for the generating function of the time to first emptiness (Kendall (1957)), and the other with the asymptotic stationary distribution of reservoir contents, for which an explicit formula for the generating function is available (Moran (1959)).

Random Graphs from a Minor-Closed Class

2009 ◽  
Vol 18 (4) ◽  
pp. 583-599 ◽  
Author(s):  
COLIN McDIARMID
Keyword(s):  
Poisson Distribution ◽  
Random Graph ◽  
Generating Function ◽  
Random Graphs ◽  
Giant Component ◽  
Closed Class ◽  
Exponential Generating Function ◽  
Excluded Minor ◽  
A Minor ◽  
Labelled Graphs

A minor-closed class of graphs is addable if each excluded minor is 2-connected. We see that such a classof labelled graphs has smooth growth; and, for the random graphRnsampled uniformly from then-vertex graphs in, the fragment not in the giant component asymptotically has a simple ‘Boltzmann Poisson distribution’. In particular, asn→ ∞ the probability thatRnis connected tends to 1/A(ρ), whereA(x) is the exponential generating function forand ρ is its radius of convergence.

A note on a functional equation arising in Galton-Watson branching processes

1971 ◽  
Vol 8 (3) ◽  
pp. 589-598 ◽  
Author(s):  
Krishna B. Athreya
Keyword(s):  
Functional Equation ◽  
Continuous Function ◽  
Generating Function ◽  
Branching Process ◽  
Branching Processes ◽  
Probability Generating Function ◽  
Age Dependent ◽  
Watson Process ◽  
Natural Way

The functional equation ϕ(mu) = h(ϕ(u)) where is a probability generating function with 1 < m = h'(1 –) < ∞ and where F(t) is a non-decreasing right continuous function with F(0 –) = 0, F(0 +) < 1 and F(+ ∞) = 1 arises in a Galton-Watson process in a natural way. We prove here that for any if and only if This unifies several results in the literature on the supercritical Galton-Watson process. We generalize this to an age dependent branching process case as well.

A note on a functional equation arising in Galton-Watson branching processes

1971 ◽  
Vol 8 (03) ◽  
pp. 589-598 ◽  
Author(s):  
Krishna B. Athreya
Keyword(s):  
Functional Equation ◽  
Continuous Function ◽  
Generating Function ◽  
Branching Process ◽  
Branching Processes ◽  
Probability Generating Function ◽  
Age Dependent ◽  
Watson Process ◽  
Natural Way

The functional equation ϕ(mu) = h(ϕ(u)) where is a probability generating function with 1 &lt; m = h'(1 –) &lt; ∞ and where F(t) is a non-decreasing right continuous function with F(0 –) = 0, F(0 +) &lt; 1 and F(+ ∞) = 1 arises in a Galton-Watson process in a natural way. We prove here that for any if and only if This unifies several results in the literature on the supercritical Galton-Watson process. We generalize this to an age dependent branching process case as well.

scholarly journals On Sackin's original proposal: The variance of the leaves' depths as a phylogenetic balance index

2019 ◽  
Author(s):  
Tomás Martínez Coronado ◽  
Arnau Mir ◽  
Francesc Rossello ◽  
Lucía Rotger
Keyword(s):  
Phylogenetic Tree ◽  
Phylogenetic Trees ◽  
Rooted Tree ◽  
Rooted Trees ◽  
Closed Formula ◽  
Original Proposal ◽  
Yule Model ◽  
Uniform Model ◽  
Rooted Phylogenetic Tree ◽  
Balance Index

Abstract Background: The Sackin index S of a rooted phylogenetic tree, defined as the sum of its leaves' depths, is one of the most popular balance indices in phylogenetics, and Sackin's 1972 paper is usually cited as the source for this index. However, what Sackin actually proposed in his paper as a measure of the imbalance of a rooted tree was not the sum of its leaves' depths, but their "variation". This proposal was later implemented as the variance of the leaves' depths by Kirkpatrick and Slatkin, where moreover they posed the problem of finding a closed formula for its expected value under the Yule model. Nowadays, Sackin's original proposal seems to have passed into oblivion in the phylogenetics literature, replaced by the index bearing his name, which, in fact, was introduced a decade later by Sokal.Results: In this paper we study the properties of the variance of the leaves' depths, V, as a balance index. Firstly, we prove that the rooted trees with n leaves and maximum V value are exactly the combs with n leaves. But although V achieves its minimum value on every space BT_n of bifurcating rooted phylogenetic trees with n< 184 leaves at the so-called "maximally balanced trees" with n leaves, this property fails for almost every n>= 184. We provide then an algorithm that finds in O(n) time the trees in BT_n with minimum V value. Secondly, we obtain closed formulas for the expected V value of a bifurcating rooted tree with any number n of leaves under the Yule and the uniform models and, as a by-product of the computations leading to these formulas, we also obtain closed formulas for the variance of the Sackin index and the total cophenetic indexof a bifurcating rooted tree, as well as of their covariance, under the uniform model, thus filling this gap in the literature.Conclusions: The phylogenetics crowd has been wise in preferring as a balance index the sum S(T) of the leaves’ depths of a phylogenetic tree T over their variance V (T), because the latter does not seem to capture correctly the notion of balance of large bifurcating rooted trees. But for bifurcating trees up to 183 leaves, V is a valid and useful balance index.

scholarly journals On a New Family of Generalized Stirling and Bell Numbers

10.37236/564 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Toufik Mansour ◽  
Matthias Schork ◽  
Mark Shattuck
Keyword(s):  
Closed Form ◽  
Generating Function ◽  
Recursion Relation ◽  
Stirling Numbers ◽  
Combinatorial Interpretation ◽  
Bell Numbers ◽  
New Family ◽  
Exponential Generating Function ◽  
Rook Numbers ◽  
Generalized Stirling Numbers

A new family of generalized Stirling and Bell numbers is introduced by considering powers $(VU)^n$ of the noncommuting variables $U,V$ satisfying $UV=VU+hV^s$. The case $s=0$ (and $h=1$) corresponds to the conventional Stirling numbers of second kind and Bell numbers. For these generalized Stirling numbers, the recursion relation is given and explicit expressions are derived. Furthermore, they are shown to be connection coefficients and a combinatorial interpretation in terms of statistics is given. It is also shown that these Stirling numbers can be interpreted as $s$-rook numbers introduced by Goldman and Haglund. For the associated generalized Bell numbers, the recursion relation as well as a closed form for the exponential generating function is derived. Furthermore, an analogue of Dobinski's formula is given for these Bell numbers.

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