scholarly journals The Number of Intervals in the $m$-Tamari Lattices

10.37236/2027 ◽  
2012 ◽  
Vol 18 (2) ◽  
Author(s):  
Mireille Bousquet-Mélou ◽  
Éric Fusy ◽  
Louis-François Préville-Ratelle
Keyword(s):  
Functional Equation ◽  
Generating Function ◽  
Open Problem ◽  
Lattice Structure ◽  
Square Grid ◽  
Tamari Lattice ◽  
Bijective Proof ◽  
Tamari Lattices

An $m$-ballot path of size $n$ is a path on the square grid consisting of north and east steps, starting at $(0,0)$, ending at $(mn,n)$, and never going below the line $\{x=my\}$. The set of these paths can be equipped with a lattice structure, called the $m$-Tamari lattice and denoted by $\mathcal{T}_n^{(m)}$, which generalizes the usual Tamari lattice $\mathcal{T}_n$ obtained when $m=1$. We prove that the number of intervals in this lattice is $$ \frac {m+1}{n(mn+1)} {(m+1)^2 n+m\choose n-1}. $$ This formula was recently conjectured by Bergeron in connection with the study of diagonal coinvariant spaces. The case $m=1$ was proved a few years ago by Chapoton. Our proof is based on a recursive description of intervals, which translates into a functional equation satisfied by the associated generating function. The solution of this equation is an algebraic series, obtained by a guess-and-check approach. Finding a bijective proof remains an open problem.

scholarly journals A Consecutive Lehmer Code for Parabolic Quotients of the Symmetric Group

2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Wenjie Fang ◽  
Henri Mühle ◽  
Jean-Christophe Novelli
Keyword(s):  
Symmetric Group ◽  
Simple Proof ◽  
Open Problem ◽  
Lattice Isomorphism ◽  
Tamari Lattice ◽  
Lehmer Code

In this article we define an encoding for parabolic permutations that distinguishes between parabolic $231$-avoiding permutations. We prove that the componentwise order on these codes realizes the parabolic Tamari lattice, and conclude a direct and simple proof that the parabolic Tamari lattice is isomorphic to a certain $\nu$-Tamari lattice, with an explicit bijection. Furthermore, we prove that this bijection is closely related to the map $\Theta$ used when the lattice isomorphism was first proved in (Ceballos, Fang and Mühle, 2020), settling an open problem therein.

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)).

scholarly journals A probablistic proof of the series representation of the MacDonald function with applications

1998 ◽  
Vol 21 (3) ◽  
pp. 463-466
Author(s):  
M. Aslam Chaudhry ◽  
Munir Ahmad
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Generating Function ◽  
Open Problem ◽  
Series Representation ◽  
Moment Generating Function ◽  
Macdonald Function

A series representation of the Macdonald function is obtained using the properties of a probability density function and its moment generating function. Some applications of the result are discussed and an open problem is posed.

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 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.

scholarly journals A 'Nice' Bijection for a Content Formula for Skew Semistandard Young Tableaux

10.37236/1635 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Martin Rubey
Keyword(s):  
Generating Function ◽  
Young Tableaux ◽  
Jeu De Taquin ◽  
Enumerative Combinatorics ◽  
Bijective Proof

Based on Schützenberger's evacuation and a modification of jeu de taquin, we give a bijective proof of an identity connecting the generating function of reverse semistandard Young tableaux with bounded entries with the generating function of all semistandard Young tableaux. This solves Exercise 7.102 b of Richard Stanley's book 'Enumerative Combinatorics 2'.

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 Noncrossing Arc Diagrams, Tamari Lattices, and Parabolic Quotients of the Symmetric Group

2021 ◽  
Author(s):  
Henri Mühle
Keyword(s):  
Symmetric Group ◽  
Partial Order ◽  
Structural Investigation ◽  
Canonical Representation ◽  
Weak Order ◽  
Noncrossing Partitions ◽  
Tamari Lattice ◽  
Semidistributive Lattice ◽  
Tamari Lattices

AbstractOrdering permutations by containment of inversion sets yields a fascinating partial order on the symmetric group: the weak order. This partial order is, among other things, a semidistributive lattice. As a consequence, every permutation has a canonical representation as a join of other permutations. Combinatorially, these canonical join representations can be modeled in terms of arc diagrams. Moreover, these arc diagrams also serve as a model to understand quotient lattices of the weak order. A particularly well-behaved quotient lattice of the weak order is the well-known Tamari lattice, which appears in many seemingly unrelated areas of mathematics. The arc diagrams representing the members of the Tamari lattices are better known as noncrossing partitions. Recently, the Tamari lattices were generalized to parabolic quotients of the symmetric group. In this article, we undertake a structural investigation of these parabolic Tamari lattices, and explain how modified arc diagrams aid the understanding of these lattices.

scholarly journals Tamari Lattices for Parabolic Quotients of the Symmetric Group

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Henri Mühle ◽  
Nathan Williams
Keyword(s):  
Symmetric Group ◽  
Coxeter Groups ◽  
Weak Order ◽  
Set Partitions ◽  
Tamari Lattice ◽  
Nous Montrons ◽  
International Audience ◽  
Tamari Lattices

International audience We present a generalization of the Tamari lattice to parabolic quotients of the symmetric group. More precisely, we generalize the notions of 231-avoiding permutations, noncrossing set partitions, and nonnesting set partitions to parabolic quotients, and show bijectively that these sets are equinumerous. Furthermore, the restriction of weak order on the parabolic quotient to the parabolic 231-avoiding permutations is a lattice quotient. Lastly, we suggest how to extend these constructions to all Coxeter groups. Nous présentons une généralisation du treillis de Tamari aux quotients paraboliques du groupe symétrique. Plus précisément, nous généralisons les notions de permutations qui évitent le motif 231, les partitions non-croisées, et les partitions non-emboîtées aux quotients paraboliques, et nous montrons de façon bijective que ces ensembles sont équipotents. En restreignant l’ordre faible du quotient parabolique aux permutations paraboliques qui évitent le motif 231, on obtient un quotient de treillis d’ordre faible. Enfin, nous suggérons comment étendre ces constructions à tous les groupes de Coxeter.

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