scholarly journals Pattern Avoiding Permutations with a Unique Longest Increasing Subsequence

10.37236/9506 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Miklós Bóna ◽  
Elijah DeJonge
Keyword(s):  
Growth Rate ◽  
Explicit Formula ◽  
Generating Function ◽  
Longest Increasing Subsequence ◽  
Pattern Avoiding Permutations

We investigate permutations and involutions that avoid a pattern of length three and have a unique longest increasing subsequence (ULIS). We prove an explicit formula for 231-avoiders, we show that the growth rate for 321-avoiding permutations with a ULIS is 4, and prove that their generating function is not rational. We relate the case of 132-avoiders to the existing literature, raising some interesting questions. For involutions, we construct a bijection between 132-avoiding involutions with a ULIS and bidirectional ballot sequences.

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 Growth rate of ample heights and the Dynamical Mordell–Lang conjecture

2018 ◽  
Vol 14 (10) ◽  
pp. 2673-2685
Author(s):  
Kaoru Sano
Keyword(s):  
Growth Rate ◽  
Explicit Formula ◽  
Number Fields ◽  
Rational Points ◽  
Positive Answer ◽  
Projective Varieties ◽  
Smooth Projective Varieties

We provide an explicit formula on the growth rate of ample heights of rational points under iteration of endomorphisms of smooth projective varieties over number fields. As an application, we give a positive answer to a variant of the Dynamical Mordell–Lang conjecture for pairs of étale endomorphisms, which is also a variant of the original one stated by Bell, Ghioca, and Tucker in their monograph.

scholarly journals The Number of Chains of Subgroups in the Lattice of Subgroups of the Dicyclic Group

2012 ◽  
Vol 2012 ◽  
pp. 1-17
Author(s):  
Ju-Mok Oh ◽  
Yunjae Kim ◽  
Kyung-Won Hwang
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Lattice Of Subgroups

We give an explicit formula for the number of chains of subgroups in the lattice of subgroups of the dicyclic groupB4nof order4nby finding its generating function of multivariables.

scholarly journals Explicit formula of a new class of q-Hermite-based Apostol-type polynomials and generalization

2020 ◽  
Vol 26 (4) ◽  
pp. 93-102
Author(s):  
Mouloud Goubi ◽  
Keyword(s):  
Explicit Formula ◽  
Present Article ◽  
Generating Function ◽  
New Class

The present article deals with a recent study of a new class of q-Hermite-based Apostol-type polynomials introduced by Waseem A. Khan and Divesh Srivastava. We give their explicit formula and study a generalized class depending in any q-analog generating function.

PARTITIONS WITH AN ARBITRARY NUMBER OF SPECIFIED DISTANCES

2019 ◽  
Vol 101 (1) ◽  
pp. 35-39 ◽  
Author(s):  
BERNARD L. S. LIN
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Generating Functions ◽  
Arbitrary Number ◽  
Positive Integers

For positive integers $t_{1},\ldots ,t_{k}$, let $\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$ (respectively $p(n,t_{1},t_{2},\ldots ,t_{k})$) be the number of partitions of $n$ such that, if $m$ is the smallest part, then each of $m+t_{1},m+t_{1}+t_{2},\ldots ,m+t_{1}+t_{2}+\cdots +t_{k-1}$ appears as a part and the largest part is at most (respectively equal to) $m+t_{1}+t_{2}+\cdots +t_{k}$. Andrews et al. [‘Partitions with fixed differences between largest and smallest parts’, Proc. Amer. Math. Soc.143 (2015), 4283–4289] found an explicit formula for the generating function of $p(n,t_{1},t_{2},\ldots ,t_{k})$. We establish a $q$-series identity from which the formulae for the generating functions of $\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$ and $p(n,t_{1},t_{2},\ldots ,t_{k})$ can be obtained.

INTERSECTION THEORY ON SYMPLECTIC QUOTIENTS OF PRODUCTS OF SPHERES

2001 ◽  
Vol 12 (01) ◽  
pp. 97-111 ◽  
Author(s):  
TATSURU TAKAKURA
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Cohomology Ring ◽  
Geometric Quantization ◽  
Intersection Theory ◽  
Symplectic Reduction ◽  
Poincare Duality ◽  
Diagonal Action ◽  
Symplectic Quotients ◽  
Products Of Spheres

We present an explicit formula for cohomology intersection pairings on an arbitrary smooth symplectic quotient of products of 2-spheres, by the standard diagonal action of SO3, without using known results on relations in the cohomology ring. By the Poincaré duality, it contains all the information enough to recover the structure of the cohomology ring. Our method is based on the commutativity of geometric quantization and symplectic reduction, originating from a conjecture of Guillemin-Sternberg. In fact, it enables us to derive a formula for the generating function of the intersection pairings.

scholarly journals Euler–Catalan’s Number Triangle and Its Application

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 600 ◽  
Author(s):  
Yuriy Shablya ◽  
Dmitry Kruchinin
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Binary Trees ◽  
Combinatorial Objects ◽  
Left Branch ◽  
Bivariate Generating Function

In this paper, we study such combinatorial objects as labeled binary trees of size n with m ascents on the left branch and labeled Dyck n-paths with m ascents on return steps. For these combinatorial objects, we present the relation of the generated number triangle to Catalan’s and Euler’s triangles. On the basis of properties of Catalan’s and Euler’s triangles, we obtain an explicit formula that counts the total number of such combinatorial objects and a bivariate generating function. Combining the properties of these two number triangles allows us to obtain different combinatorial objects that may have a symmetry, for example, in their form or in their formulas.

scholarly journals Counting Stirling permutations by number of pushes

2020 ◽  
Vol 29 (1) ◽  
pp. 17-27
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Joint Distribution ◽  
Inverse Matrix ◽  
Infinite Matrix ◽  
Do So

AbstractLet 𝒯(k)n denote the set of k-Stirling permutations having n distinct letters. Here, we consider the number of steps required (i.e., pushes) to rearrange the letters of a member of 𝒯(k)n so that they occur in non-decreasing order. We find recurrences for the joint distribution on 𝒯(k)n for the statistics recording the number of levels (i.e., occurrences of equal adjacent letters) and pushes. When k = 2, an explicit formula for the ordinary generating function of this distribution is also found. In order to do so, we determine the LU-decomposition of a certain infinite matrix having polynomial entries which enables one to compute explicitly the inverse matrix.

scholarly journals Unlabeled $(2+2)$-free posets, ascent sequences and pattern avoiding permutations

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Mireille Bousquet-Mélou ◽  
Anders Claesson ◽  
Mark Dukes ◽  
Sergey Kitaev
Keyword(s):  
Generating Function ◽  
Integer Sequences ◽  
Combinatorial Objects ◽  
New Class ◽  
Chord Diagrams ◽  
The Third ◽  
Finite Series ◽  
International Audience ◽  
Ascent Sequences ◽  
Pattern Avoiding Permutations

International audience We present statistic-preserving bijections between four classes of combinatorial objects. Two of them, the class of unlabeled $(\textrm{2+2})$-free posets and a certain class of chord diagrams (or involutions), already appeared in the literature, but were apparently not known to be equinumerous. The third one is a new class of pattern avoiding permutations, and the fourth one consists of certain integer sequences called $\textit{ascent sequences}$. We also determine the generating function of these classes of objects, thus recovering a non-D-finite series obtained by Zagier for chord diagrams. Finally, we characterize the ascent sequences that correspond to permutations avoiding the barred pattern $3\bar{1}52\bar{4}$, and enumerate those permutations, thus settling a conjecture of Pudwell. Nous présentons des bijections, transportant de nombreuses statistiques, entre quatre classes d'objets. Deux d'entre elles, la classe des EPO (ensembles partiellement ordonnés) sans motif $(\textrm{2+2})$ et une certaine classe d'involutions, sont déjà apparues dans la littérature. La troisième est une classe de permutations à motifs exclus, et la quatrième une classe de suites que nous appelons $\textit{suites à montées}$. Nous déterminons ensuite la série génératrice de ces classes, retrouvant ainsi un résultat prouvé par Zagier pour les involutions sus-mentionnées. La série obtenue n'est pas D-finie. Apparemment, le fait qu'elle compte aussi les EPO sans motif $(\textrm{2+2})$ est nouveau. Finalement, nous caractérisons les suites à montées qui correspondent aux permutations évitant le motif barré $3\bar{1}52\bar{4}$ et énumérons ces permutations, ce qui démontre une conjecture de Pudwell.

scholarly journals Continued Fractions and Catalan Problems

10.37236/1523 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Mahendra Jani ◽  
Robert G. Rieper
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Ordered Trees ◽  
Pattern Avoiding Permutations ◽  
Ordered Tree ◽  
Different Levels

We find a generating function expressed as a continued fraction that enumerates ordered trees by the number of vertices at different levels. Several Catalan problems are mapped to an ordered-tree problem and their generating functions also expressed as a continued fraction. Among these problems is the enumeration of $(132)$-pattern avoiding permutations that have a given number of increasing patterns of length $k$. This extends and illuminates a result of Robertson, Wilf and Zeilberger for the case $k=3$.

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