scholarly journals Phylogenetic Trees, Augmented Perfect Matchings, and a Thron-type Continued Fraction (T-fraction) for the Ward Polynomials

10.37236/9571 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Andrew Elvey Price ◽  
Alan D. Sokal
Keyword(s):  
Generating Function ◽  
Phylogenetic Trees ◽  
Continued Fraction ◽  
Infinite Family ◽  
Perfect Matchings ◽  
Dyck Paths ◽  
Schröder Paths

We find a Thron-type continued fraction (T-fraction) for the ordinary generating function of the Ward polynomials, as well as for some generalizations employing a large (indeed infinite) family of independent indeterminates. Our proof is based on a bijection between super-augmented perfect matchings and labeled Schröder paths, which generalizes Flajolet's bijection between perfect matchings and labeled Dyck paths.

scholarly journals ON GENERALIZED THUE–MORSE FUNCTIONS AND THEIR VALUES

2019 ◽  
Vol 108 (2) ◽  
pp. 177-201
Author(s):  
DZMITRY BADZIAHIN ◽  
EVGENIY ZORIN
Keyword(s):  
Generating Function ◽  
Continued Fraction ◽  
Laurent Series ◽  
Regular Structure ◽  
Continued Fraction Expansion ◽  
Infinite Products ◽  
Morse Functions ◽  
Morse Number ◽  
Badly Approximable

In this paper we extend and generalize, up to a natural bound of the method, our previous work Badziahin and Zorin [‘Thue–Morse constant is not badly approximable’, Int. Math. Res. Not. IMRN 19 (2015), 9618–9637] where we proved, among other things, that the Thue–Morse constant is not badly approximable. Here we consider Laurent series defined with infinite products $f_{d}(x)=\prod _{n=0}^{\infty }(1-x^{-d^{n}})$, $d\in \mathbb{N}$, $d\geq 2$, which generalize the generating function $f_{2}(x)$ of the Thue–Morse number, and study their continued fraction expansion. In particular, we show that the convergents of $x^{-d+1}f_{d}(x)$ have a regular structure. We also address the question of whether the corresponding Mahler numbers $f_{d}(a)\in \mathbb{R}$, $a,d\in \mathbb{N}$, $a,d\geq 2$, are badly approximable.

Positivity and continued fractions from the binomial transformation

2019 ◽  
Vol 149 (03) ◽  
pp. 831-847 ◽  
Author(s):  
Bao-Xuan Zhu
Keyword(s):  
Generating Function ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Hankel Matrix ◽  
Eulerian Polynomials ◽  
Binomial Convolution ◽  
Image Position ◽  
Moment Property

AbstractGiven a sequence of polynomials$\{x_k(q)\}_{k \ges 0}$, define the transformation$$y_n(q) = a^n\sum\limits_{i = 0}^n {\left( \matrix{n \cr i} \right)} b^{n-i}x_i(q)$$for$n\ges 0$. In this paper, we obtain the relation between the Jacobi continued fraction of the ordinary generating function ofyn(q) and that ofxn(q). We also prove that the transformation preservesq-TPr+1(q-TP) property of the Hankel matrix$[x_{i+j}(q)]_{i,j \ges 0}$, in particular forr= 2,3, implying ther-q-log-convexity of the sequence$\{y_n(q)\}_{n\ges 0}$. As applications, we can give the continued fraction expressions of Eulerian polynomials of typesAandB, derangement polynomials typesAandB, general Eulerian polynomials, Dowling polynomials and Tanny-geometric polynomials. In addition, we also prove the strongq-log-convexity of derangement polynomials typeB, Dowling polynomials and Tanny-geometric polynomials and 3-q-log-convexity of general Eulerian polynomials, Dowling polynomials and Tanny-geometric polynomials. We also present a new proof of the result of Pólya and Szegö about the binomial convolution preserving the Stieltjes moment property and a new proof of the result of Zhu and Sun on the binomial transformation preserving strongq-log-convexity.

scholarly journals Inversions in 312–permutations

2014 ◽  
Vol 23 (2) ◽  
pp. 147-152
Author(s):  
BEATA BENYI ◽  
Keyword(s):  
Perfect Matchings ◽  
Unified Approach ◽  
Dyck Paths ◽  
Ordered Graphs

We introduce an inversion diagram and use it to describe a simple bijection between 312–avoiding permutations and Dyck paths that sends inversion statistic to area statistic. We define Tamari and Dyck lattice using inversion tables. Our unified approach allows to understand the connections between these lattices better. We present a simple bijection between pairs of 312–avoiding permutations and perfect matchings of ordered graphs that avoid the pattern abccab.

scholarly journals Distribution of Crossings, Nestings and Alignments of Two Edges in Matchings and Partitions

10.37236/1059 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Anisse Kasraoui ◽  
Jiang Zeng
Keyword(s):  
Recent Result ◽  
Generating Functions ◽  
Continued Fraction ◽  
Perfect Matchings ◽  
Symmetric Distribution ◽  
Set Partitions ◽  
Continued Fraction Expansions

We construct an involution on set partitions which keeps track of the numbers of crossings, nestings and alignments of two edges. We derive then the symmetric distribution of the numbers of crossings and nestings in partitions, which generalizes a recent result of Klazar and Noy in perfect matchings. By factorizing our involution through bijections between set partitions and some path diagrams we obtain the continued fraction expansions of the corresponding ordinary generating functions.

scholarly journals The proof of the Perepechko’s conjecture concerning near-perfect matchings on Cm x Pn cylinders of odd order

2019 ◽  
Vol 13 (2) ◽  
pp. 361-377
Author(s):  
Rade Doroslovacki ◽  
Jelena Djokic ◽  
Bojana Pantic ◽  
Olga Bodroza-Pantic
Keyword(s):  
Recurrence Relation ◽  
Generating Function ◽  
Perfect Matchings ◽  
Odd Order

For all odd values of m, we prove that the sequence of the numbers of near-perfect matchings on Cm x P2n+1 cylinder with a vacancy on the boundary obeys the same recurrence relation as the sequence of the numbers of perfect matchings on Cm x P2n. Further more, we prove that for all odd values of m denominator of the generating function for the total number of the near-perfect matchings on Cm x P2n+1 graph is always the square of denominator of generating function for the sequence of the numbers of perfect matchings on Cm x P2n graph, as recently conjectured by Perepechko.

scholarly journals SIF Permutations and Chord-Connected Permutations

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Natasha Blitvić
Keyword(s):  
Probability Theory ◽  
Generating Function ◽  
Probability Distributions ◽  
Free Probability ◽  
Perfect Matchings ◽  
Free Probability Theory ◽  
Diagrammatic Representation ◽  
International Audience ◽  
Binary Strings

International audience A <i>stabilized-interval-free </i> (SIF) permutation on [n], introduced by Callan, is a permutation that does not stabilize any proper interval of [n]. Such permutations are known to be the irreducibles in the decomposition of permutations along non-crossing partitions. That is, if $s_n$ denotes the number of SIF permutations on [n], $S(z)=1+\sum_{n\geq1} s_n z^n$, and $F(z)=1+\sum_{n\geq1} n! z^n$, then $F(z)= S(zF(z))$. This article presents, in turn, a decomposition of SIF permutations along non-crossing partitions. Specifically, by working with a convenient diagrammatic representation, given in terms of perfect matchings on alternating binary strings, we arrive at the \emphchord-connected permutations on [n], counted by $\{c_n\}_{n\geq1}$, whose generating function satisfies $S(z)= C(zS(z))$. The expressions at hand have immediate probabilistic interpretations, via the celebrated <i>moment-cumulant formula </i>of Speicher, in the context of the <i>free probability theory </i>of Voiculescu. The probability distributions that appear are the exponential and the complex Gaussian.

scholarly journals Permutation Patterns and Continued Fractions

10.37236/1470 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Aaron Robertson ◽  
Herbert S. Wilf ◽  
Doron Zeilberger
Keyword(s):  
Generating Function ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Permutation Patterns

We find, in the form of a continued fraction, the generating function for the number of $(132)$-avoiding permutations that have a given number of $(123)$ patterns, and show how to extend this to permutations that have exactly one $(132)$ pattern. We also find some properties of the continued fraction, which is similar to, though more general than, those that were studied by Ramanujan.

The generalised state-dependent queue: the busy period Erlangian

1974 ◽  
Vol 11 (03) ◽  
pp. 618-623
Author(s):  
B. W. Conolly
Keyword(s):  
Laplace Transform ◽  
Generating Function ◽  
Busy Period ◽  
Continued Fraction ◽  
Queueing System ◽  
Joint Probability ◽  
Single Server ◽  
System State ◽  
State Dependent ◽  
The Laplace Transform

A continued fraction representation is presented of the Laplace transform of the generating function of the fundamental joint probability and density of busy period length measured in customers served and duration in time. The setting is the single server Erlang queueing system where the parameters of negative exponentially distributed arrival and service times have a general dependence on instantaneous system state.

scholarly journals Short note on the number of 1-ascents in dispersed Dyck paths

2017 ◽  
Vol 09 (06) ◽  
pp. 1750077
Author(s):  
Kairi Kangro ◽  
Mozhgan Pourmoradnasseri ◽  
Dirk Oliver Theis
Keyword(s):  
Generating Function ◽  
Short Note ◽  
Lattice Path ◽  
Dyck Path ◽  
Closed Formula ◽  
Dyck Paths ◽  
Maximal Sequence

A dispersed Dyck path (DDP) of length [Formula: see text] is a lattice path on [Formula: see text] from [Formula: see text] to [Formula: see text] in which the following steps are allowed: “up” [Formula: see text]; “down” [Formula: see text]; and “right” [Formula: see text]. An ascent in a DDP is an inclusion-wise maximal sequence of consecutive up steps. A 1-ascent is an ascent consisting of exactly 1 up step. We give a closed formula for the total number of 1-ascents in all dispersed Dyck paths of length [Formula: see text], #A191386 in Sloane’s OEIS. Previously, only implicit generating function relations and asymptotics were known.

scholarly journals An infinite family of graphs with a facile count of perfect matchings

2014 ◽  
Vol 166 ◽  
pp. 210-214 ◽  
Author(s):  
Vladimir R. Rosenfeld ◽  
Douglas J. Klein
Keyword(s):  
Infinite Family ◽  
Perfect Matchings
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