scholarly journals Enumerations of bargraphs with respect to corner statistics

2020 ◽  
Vol 14 (1) ◽  
pp. 221-238
Author(s):  
Toufik Mansour ◽  
Gökhan Yıldırım
Keyword(s):  
Generating Functions ◽  
Explicit Formulas ◽  
Set Partitions ◽  
Number Of Cells

We study the enumeration of bargraphs with respect to some corner statistics. We find generating functions for the number of bargraphs that track the corner statistics of interest, the number of cells, and the number of columns. We also consider bargraph representation of set partitions and obtain some explicit formulas for the number of specific types of corners in such representations.

scholarly journals Counting water cells in bargraphs of compositions and set partitions

2018 ◽  
Vol 12 (2) ◽  
pp. 413-438 ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Horizontal Line ◽  
Explicit Formulas ◽  
Set Partitions ◽  
Unit Square ◽  
Joint Distributions ◽  
Water Cell ◽  
First Moment

In this paper, we consider statistics on compositions and set partitions represented geometrically as bargraphs. By a water cell, we mean a unit square exterior to a bargraph that lies along a horizontal line between any two squares contained within the area subtended by the bargraph. That is, if a large amount of a liquid were poured onto the bargraph from above and allowed to drain freely, then the water cells are precisely those cells where the liquid would collect. In this paper, we count both compositions and set partitions according to the number of descents and water cells in their bargraph representations and determine generating function formulas for the joint distributions on the respective structures. Comparable generating functions that count non-crossing and non-nesting partitions are also found. Finally, we determine explicit formulas for the sign balance and for the first moment of the water cell statistic on set partitions, providing both algebraic and combinatorial proofs.

scholarly journals Enumerating set partitions by the number of positions between adjacent occurrences of a letter

2010 ◽  
Vol 4 (2) ◽  
pp. 284-308 ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck ◽  
Stephan Wagner
Keyword(s):  
Generating Functions ◽  
Stirling Numbers ◽  
Asymptotic Estimates ◽  
Explicit Formulas ◽  
Set Partitions ◽  
Minimal Elements

A partition ? of the set [n] = {1, 2,...,n} is a collection {B1,...,Bk} of nonempty disjoint subsets of [n] (called blocks) whose union equals [n]. Suppose that the subsets Bi are listed in increasing order of their minimal elements and ? = ?1, ?2...?n denotes the canonical sequential form of a partition of [n] in which iEB?i for each i. In this paper, we study the generating functions corresponding to statistics on the set of partitions of [n] with k blocks which record the total number of positions of ? between adjacent occurrences of a letter. Among our results are explicit formulas for the total value of the statistics over all the partitions in question, for which we provide both algebraic and combinatorial proofs. In addition, we supply asymptotic estimates of these formulas, the proofs of which entail approximating the size of certain sums involving the Stirling numbers. Finally, we obtain comparable results for statistics on partitions which record the total number of positions of ? of the same letter lying between two letters which are strictly larger.

New family of Whitney numbers

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 309-320 ◽  
Author(s):  
B.S. El-Desouky ◽  
Nenad Cakic ◽  
F.A. Shiha
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Explicit Formulas ◽  
Basic Properties ◽  
New Family ◽  
Whitney Numbers ◽  
Special Cases

In this paper we give a new family of numbers, called ??-Whitney numbers, which gives generalization of many types of Whitney numbers and Stirling numbers. Some basic properties of these numbers such as recurrence relations, explicit formulas and generating functions are given. Finally many interesting special cases are derived.

scholarly journals Liouville quantum gravity — holography, JT and matrices

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Thomas G. Mertens ◽  
Gustavo J. Turiaci
Keyword(s):  
String Theory ◽  
Generating Functions ◽  
Preliminary Evidence ◽  
Quantum Deformation ◽  
Dilaton Gravity ◽  
Continuum Approach ◽  
Genus Zero ◽  
Explicit Formulas ◽  
Liouville Gravity ◽  
The Continuum

Abstract We study two-dimensional Liouville gravity and minimal string theory on spaces with fixed length boundaries. We find explicit formulas describing the gravitational dressing of bulk and boundary correlators in the disk. Their structure has a striking resemblance with observables in 2d BF (plus a boundary term), associated to a quantum deformation of SL(2, ℝ), a connection we develop in some detail. For the case of the (2, p) minimal string theory, we compare and match the results from the continuum approach with a matrix model calculation, and verify that in the large p limit the correlators match with Jackiw-Teitelboim gravity. We consider multi-boundary amplitudes that we write in terms of gluing bulk one-point functions using a quantum deformation of the Weil-Petersson volumes and gluing measures. Generating functions for genus zero Weil-Petersson volumes are derived, taking the large p limit. Finally, we present preliminary evidence that the bulk theory can be interpreted as a 2d dilaton gravity model with a sinh Φ dilaton potential.

Coordination sequences and layer-by-layer growth of periodic structures

2019 ◽  
Vol 234 (5) ◽  
pp. 291-299
Author(s):  
Anton Shutov ◽  
Andrey Maleev
Keyword(s):  
Generating Functions ◽  
Diophantine Equations ◽  
Periodic Structures ◽  
Layer Growth ◽  
Layer By Layer ◽  
Explicit Formulas ◽  
New Approach ◽  
Coordination Numbers ◽  
Linear Diophantine Equations ◽  
Recurrent Relations

Abstract A new approach to the problem of coordination sequences of periodic structures is proposed. It is based on the concept of layer-by-layer growth and on the study of geodesics in periodic graphs. We represent coordination numbers as sums of so called sector coordination numbers arising from the growth polygon of the graph. In each sector we obtain a canonical form of the geodesic chains and reduce the calculation of the sector coordination numbers to solution of the linear Diophantine equations. The approach is illustrated by the example of the 2-homogeneous kra graph. We obtain three alternative descriptions of the coordination sequences: explicit formulas, generating functions and recurrent relations.

scholarly journals Counting visible levels in bargraphs and set partitions

Filomat ◽  
2019 ◽  
Vol 33 (19) ◽  
pp. 6229-6237 ◽  
Author(s):  
Nenad Cakic ◽  
Toufik Mansour ◽  
Rebecca Smith
Keyword(s):  
Generating Functions ◽  
Set Partitions

In this paper, we study the generating functions for the number of visible levels in compositions of n and set partitions of [n].

Explicit formulas for the characteristics of the M/M/2/2 queue with repeated attempts

1987 ◽  
Vol 24 (2) ◽  
pp. 486-494 ◽  
Author(s):  
Thomas Hanschke
Keyword(s):  
Steady State ◽  
Generating Functions ◽  
Explicit Formulas ◽  
Steady State Probabilities

In this paper we study the M/M/2/2 queue with repeated attempts. It is shown that the part generating functions of the steady state probabilities can be expressed in of generalized hypergeometric unctions.

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 Pattern Avoidance in Matchings and Partitions

10.37236/2976 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Jonathan Bloom ◽  
Sergi Elizalde
Keyword(s):  
Fixed Points ◽  
Generating Functions ◽  
Direct Proof ◽  
Pattern Avoidance ◽  
Equivalence Classes ◽  
Set Partitions ◽  
Bijective Proof ◽  
Rook Placements ◽  
Wilf Equivalence ◽  
Diagram Representation

Extending the notion of pattern avoidance in permutations, we study matchings and set partitions whose arc diagram representation avoids a given configuration of three arcs. These configurations, which generalize $3$-crossings and $3$-nestings, have an interpretation, in the case of matchings, in terms of patterns in full rook placements on Ferrers boards.We enumerate $312$-avoiding matchings and partitions, obtaining algebraic generating functions, in contrast with the known D-finite generating functions for the $321$-avoiding (i.e., $3$-noncrossing) case. Our approach provides a more direct proof of a formula of Bóna for the number of $1342$-avoiding permutations. We also give a bijective proof of the shape-Wilf-equivalence of the patterns $321$ and $213$ which greatly simplifies existing proofs by Backelin-West-Xin and Jelínek, and provides an extension of work of Gouyou-Beauchamps for matchings with fixed points. Finally, we classify pairs of patterns of length 3 according to shape-Wilf-equivalence, and enumerate matchings and partitions avoiding a pair in most of the resulting equivalence classes.

scholarly journals Some Properties of the Hermite Polynomials and Their Squares and Generating Functions

2016 ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Hermite Polynomials ◽  
Higher Order ◽  
Explicit Formulas ◽  
Derivatives Of

In the paper, the authors consider the generating functions of the Hermite polynomials and their squares, present explicit formulas for higher order derivatives of the generating functions of the Hermite polynomials and their squares, which can be viewed as ordinary differential equations or derivative polynomials, find differential equations that the generating functions of the Hermite polynomials and their squares satisfy, and derive explicit formulas and recurrence relations for the Hermite polynomials and their squares.

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