scholarly journals Asymptotics of 3-Stack-Sortable Permutations

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Colin Defant ◽  
Andrew Elvey Price ◽  
Anthony Guttmann
Keyword(s):  
Functional Equation ◽  
Lower Bound ◽  
Generating Function ◽  
Growth Constant

We derive a simple functional equation with two catalytic variables characterising the generating function of 3-stack-sortable permutations. Using this functional equation, we extend the 174-term series to 1000 terms. From this series, we conjecture that the generating function behaves as  $$W(t) \sim C_0(1-\mu_3 t)^\alpha \cdot \log^\beta(1-\mu_3 t), $$ so that $$[t^n]W(t)=w_n \sim \frac{c_0\mu_3^n}{  n^{(\alpha+1)}\cdot \log^\lambda{n}} ,$$ where $\mu_3 = 9.69963634535(30),$ $\alpha = 2.0 \pm 0.25.$ If $\alpha = 2$ exactly, then $\lambda = -\beta+1$, and we estimate $\beta \approx -2,$ but with a wide uncertainty of $\pm 1.$  If $\alpha$ is not an integer, then $\lambda=-\beta$, but we cannot give a useful estimate of $\beta$. The growth constant estimate (just) contradicts a conjecture of the first author that $$9.702 < \mu_3 \le 9.704.$$ We also prove a new rigorous lower bound of $\mu_3\geq 9.4854$, allowing us to disprove a conjecture of Bóna.  We then further extend the series using differential-approximants to obtain approximate coefficients $O(t^{2000}),$ expected to be accurate to $20$ significant digits, and use the approximate coefficients to provide additional evidence supporting the results obtained from the exact coefficients.

Polynomial bounds for probability generating functions

1975 ◽  
Vol 12 (3) ◽  
pp. 507-514 ◽  
Author(s):  
Henry Braun
Keyword(s):  
Lower Bound ◽  
Generating Function ◽  
Generating Functions ◽  
Branching Process ◽  
Probability Generating Function ◽  
Extinction Time ◽  
Probability Generating Functions ◽  
Extinction Probabilities ◽  
Polynomial Bounds

The problem of approximating an arbitrary probability generating function (p.g.f.) by a polynomial is considered. It is shown that if the coefficients rj are chosen so that LN(·) agrees with g(·) to k derivatives at s = 1 and to (N – k) derivatives at s = 0, then LN is in fact an upper or lower bound to g; the nature of the bound depends only on k and not on N. Application of the results to the problems of finding bounds for extinction probabilities, extinction time distributions and moments of branching process distributions are examined.

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)).

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 Weakly directed self-avoiding walks

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Axel Bacher ◽  
Mireille Bousquet-Mélou
Keyword(s):  
Generating Function ◽  
Square Lattice ◽  
Growth Constant ◽  
Complex Singularity ◽  
Singularity Structure ◽  
New Family ◽  
End Distance ◽  
International Audience ◽  
Directed Walks ◽  
Self Avoiding Walks

International audience We define a new family of self-avoiding walks (SAW) on the square lattice, called $\textit{weakly directed walks}$. These walks have a simple characterization in terms of the irreducible bridges that compose them. We determine their generating function. This series has a complex singularity structure and in particular, is not D-finite. The growth constant is approximately 2.54 and is thus larger than that of all natural families of SAW enumerated so far (but smaller than that of general SAW, which is about 2.64). We also prove that the end-to-end distance of weakly directed walks grows linearly. Finally, we study a diagonal variant of this model. Nous définissons une nouvelle famille de chemins auto-évitants (CAE) sur le réseau carré, appelés $\textit{chemins faiblement dirigés}$. Ces chemins ont une caractérisation simple en termes des ponts irréductibles qui les composent. Nous déterminons leur série génératrice. Cette série a une structure de singularité complexe et n'est en particulier pas D-finie. La constante de croissance est environ 2,54, ce qui est supérieur à toutes les familles naturelles de SAW étudiées jusqu'à présent, mais inférieur aux CAE généraux (dont la constante est environ 2,64). Nous prouvons également que la distance moyenne entre les extrémités du chemin croît linéairement. Enfin, nous étudions une variante diagonale du modèle.

Twisted Hasse-Weil L-Functions and the Rank of Mordell-Weil Groups

1997 ◽  
Vol 49 (4) ◽  
pp. 749-771 ◽  
Author(s):  
Lawrence Howe
Keyword(s):  
Functional Equation ◽  
Lower Bound ◽  
Irreducible Representation ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Root Number

AbstractFollowing a method outlined by Greenberg, root number computations give a conjectural lower bound for the ranks of certain Mordell–Weil groups of elliptic curves. More specifically, for PQn a PGL2(Z/pnZ)–extension of Q and E an elliptic curve over Q, define the motive E ⊗ ρ, where ρ is any irreducible representation of Gal(PQn /Q). Under some restrictions, the root number in the conjectural functional equation for the L-function of E ⊗ ρ is easily computable, and a ‘–1’ implies, by the Birch and Swinnerton–Dyer conjecture, that ρ is found in E(PQn) ⊗ C. Summing the dimensions of such ρ gives a conjectural lower bound ofp2n–p2n–1–p–1for the rank of E(PQn).

scholarly journals Noncontiguous Pattern Containment in Binary Trees

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Lara Pudwell ◽  
Connor Scholten ◽  
Tyler Schrock ◽  
Alexa Serrato
Keyword(s):  
Functional Equation ◽  
Generating Function ◽  
Binary Tree ◽  
Binary Trees ◽  
Tree Patterns ◽  
Pattern Containment

We consider the enumeration of binary trees containing noncontiguous binary tree patterns. First, we show that any two ℓ-leaf binary trees are contained in the set of all n-leaf trees the same number of times. We give a functional equation for the multivariate generating function for number of n-leaf trees containing a specified number of copies of any path tree, and we analyze tree patterns with at most 4 leaves. The paper concludes with implications for pattern containment in permutations.

scholarly journals Pattern avoidance in inversion sequences

2015 ◽  
Vol 25 (2) ◽  
pp. 157-176 ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Functional Equation ◽  
Generating Function ◽  
Generating Functions ◽  
Kernel Method ◽  
Fibonacci Numbers ◽  
Pattern Avoidance ◽  
Explicit Formulas ◽  
Schröder Numbers ◽  
Single Pattern ◽  
Combinatorial Methods

Abstract A permutation of length n may be represented, equivalently, by a sequence a1a2 • • • an satisfying 0 < ai < i for all z, which is called an inversion sequence. In analogy to the usual case for permutations, the pattern avoidance question is addressed for inversion sequences. In particular, explicit formulas and/or generating functions are derived which count the inversion sequences of a given length that avoid a single pattern of length three. Among the sequences encountered are the Fibonacci numbers, the Schröder numbers, and entry A200753 in OEIS. We make use of both algebraic and combinatorial methods to establish our results. An explicit Injection is given between two of the avoidance classes, and in three cases, the kernel method is used to solve a functional equation satisfied by the generating function enumerating the class in question.

scholarly journals Weakly prudent self-avoiding bridges

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Axel Bacher ◽  
Nicholas Beaton
Keyword(s):  
Generating Function ◽  
Square Lattice ◽  
Growth Constant ◽  
Arbitrary Size ◽  
New Class ◽  
Advantages And Disadvantages ◽  
International Audience ◽  
Self Avoiding Walks

International audience We define and enumerate a new class of self-avoiding walks on the square lattice, which we call <i>weakly prudent bridges</i>. Their definition is inspired by two previously-considered classes of self-avoiding walks, and can be viewed as a combination of those two models. We consider several methods for recursively generating these objects, each with its own advantages and disadvantages, and use these methods to solve the generating function, obtain very long series, and randomly generate walks of arbitrary size. We find that the growth constant of these walks is approximately 2.58, which is larger than that of any previously-solved class of self-avoiding walks.

scholarly journals Multi-Statistic Enumeration of Two-Stack Sortable Permutations

10.37236/1359 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Mireille Bousquet-Mélou
Keyword(s):  
Functional Equation ◽  
Generating Function

Using Zeilberger's factorization of two-stack-sortable permutations, we write a functional equation — of a strange sort — that defines their generating function according to five statistics: length, number of descents, number of right-to-left and left-to-right maxima, and a fifth statistic that is closely linked to the factorization. Then, we show how one can translate this functional equation into a polynomial one. We thus prove that our five-variable generating function for two-stack-sortable permutations is algebraic of degree 20.

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