scholarly journals Enumerating Parking Completions Using Join and Split

10.37236/9194 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Ayomikun Adeniran ◽  
Steve Butler ◽  
Galen Dorpalen-Barry ◽  
Pamela E. Harris ◽  
Cyrus Hettle ◽  
...  
Keyword(s):  
Lattice Paths ◽  
Parking Functions ◽  
Volume Formula

Given a strictly increasing sequence $\mathbf{t}$ with entries from $[n]:=\{1,\ldots,n\}$, a parking completion is a sequence $\mathbf{c}$ with $|\mathbf{t}|+|\mathbf{c}|=n$ and $|\{t\in \mathbf{t}\mid t\leqslant i\}|+|\{c\in \mathbf{c}\mid c\leqslant i\}|\geqslant i$ for all $i$ in $[n]$.  We can think of $\mathbf{t}$ as a list of spots already taken in a street with $n$ parking spots and $\mathbf{c}$ as a list of parking preferences where the $i$-th car attempts to park in the $c_i$-th spot and if not available then proceeds up the street to find the next available spot, if any.  A parking completion corresponds to a set of preferences $\mathbf{c}$ where all cars park. We relate parking completions to enumerating restricted lattice paths and give formulas for both the ordered and unordered variations of the problem by use of a pair of operations termed Join and Split.  Our results give a new volume formula for most Pitman-Stanley polytopes, and enumerate the \emph{signature parking functions} of Ceballos and González D'León.

scholarly journals Corrigendum to “Two-boundary lattice paths and parking functions” [Adv. in Appl. Math. 39 (4) (2007) 515–524]

2009 ◽  
Vol 43 (2) ◽  
pp. 197-198
Author(s):  
Joseph P.S. Kung ◽  
Xinyu Sun ◽  
Catherine Yan
Keyword(s):  
Lattice Paths ◽  
Parking Functions ◽  
Appl Math

scholarly journals Type C parking functions and a zeta map

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Robin Sulzgruber ◽  
Marko Thiel
Keyword(s):  
Lattice Paths ◽  
Parking Functions ◽  
Natural Analogue ◽  
International Audience ◽  
Type C ◽  
Shi Arrangement

International audience We introduce type $C$ parking functions, encoded as vertically labelled lattice paths and endowed with a statistic dinv'. We define a bijection from type $C$ parking functions to regions of the Shi arrangement of type $C$, encoded as diagonally labelled ballot paths and endowed with a natural statistic area'. This bijection is a natural analogue of the zeta map of Haglund and Loehr and maps dinv' to area'. We give three different descriptions of it. Nous introduisons les fonctions de stationnement de type $C$, encodées par des chemins étiquetés verticalement et munies d’une statistique dinv'. Nous définissons une bijection entre les fonctions de stationnement de type $C$ et les régions de l’arrangement de Shi de type $C$, encodées par des chemins étiquetés diagonalement et munies d’une statistique naturelle area'. Cette bijection est un analogue naturel à la fonction zeta de Haglund et Loehr, et envoie dinv' sur area'. Nous donnons trois différentes descriptions de celle-ci.

scholarly journals On Parking Functions and the Zeta Map in Types B, C and D

10.37236/6714 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Robin Sulzgruber ◽  
Marko Thiel
Keyword(s):  
Hilbert Series ◽  
Root Systems ◽  
The Other ◽  
Lattice Paths ◽  
Square Lattice ◽  
Parking Functions ◽  
Combinatorial Objects ◽  
Coxeter Number ◽  
Dyck Paths ◽  
Diagonal Harmonics

Let $\Phi$ be an irreducible crystallographic root system with Weyl group $W$, coroot lattice $\check{Q}$ and Coxeter number $h$. Recently the second named author defined a uniform $W$-isomorphism $\zeta$ between the finite torus $\check{Q}/(mh+1)\check{Q}$ and the set of non-nesting parking functions $\operatorname{Park}^{(m)}(\Phi)$. If $\Phi$ is of type $A_{n-1}$ and $m=1$ this map is equivalent to a map defined on labelled Dyck paths that arises in the study of the Hilbert series of the space of diagonal harmonics. In this paper we investigate the case $m=1$ for the other infinite families of root systems ($B_n$, $C_n$ and $D_n$). In each type we define models for the finite torus and for the set of non-nesting parking functions in terms of labelled lattice paths. The map $\zeta$ can then be viewed as a map between these combinatorial objects. Our work entails new bijections between (square) lattice paths and ballot paths.

scholarly journals Conjectured Combinatorial Models for the Hilbert Series of Generalized Diagonal Harmonics Modules

10.37236/1821 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Nicholas A. Loehr ◽  
Jeffrey B. Remmel
Keyword(s):  
Generating Function ◽  
Hilbert Series ◽  
Lattice Paths ◽  
Weighted Sums ◽  
Parking Functions ◽  
Dyck Paths ◽  
Diagonal Harmonics ◽  
Combinatorial Formulas ◽  
New Statistics ◽  
Univariate Distribution

Haglund and Loehr previously conjectured two equivalent combinatorial formulas for the Hilbert series of the Garsia-Haiman diagonal harmonics modules. These formulas involve weighted sums of labelled Dyck paths (or parking functions) relative to suitable statistics. This article introduces a third combinatorial formula that is shown to be equivalent to the first two. We show that the four statistics on labelled Dyck paths appearing in these formulas all have the same univariate distribution, which settles an earlier question of Haglund and Loehr. We then introduce analogous statistics on other collections of labelled lattice paths contained in trapezoids. We obtain a fermionic formula for the generating function for these statistics. We give bijective proofs of the equivalence of several forms of this generating function. These bijections imply that all the new statistics have the same univariate distribution. Using these new statistics, we conjecture combinatorial formulas for the Hilbert series of certain generalizations of the diagonal harmonics modules.

scholarly journals Two-boundary lattice paths and parking functions

2007 ◽  
Vol 39 (4) ◽  
pp. 515-524 ◽  
Author(s):  
Joseph P.S. Kung ◽  
Xinyu Sun ◽  
Catherine Yan
Keyword(s):  
Lattice Paths ◽  
Parking Functions

Rancang Bangun Sistem Monitoring Pengukuran Volume Air Berbasis IoT Menggunakan Arduino Wemos

2020 ◽  
Vol 6 (2) ◽  
pp. 147-153
Author(s):  
Muhamad Yusup ◽  
Po. Abas Sunarya ◽  
Krisandi Aprilyanto
Keyword(s):  
Waste Water ◽  
Waste Water Treatment ◽  
Water Reservoir ◽  
Ultrasonic Sensor ◽  
Automated System ◽  
Volume Formula ◽  
Air Volume ◽  
Database Server ◽  
A Company ◽  
Height Data

System The process of counting and storing in a manual water reservoir analysis has a high percentage of error rate compared to an automated system. In a company industry, especially in the WWT (Waste Water Treatment) section, it has several reservoir tanks as stock which are still counted manually. The ultrasonic sensor is placed at the top of the WWT tank in a hanging position. Basically, to measure the volume in a tank only variable height is always changing. So by utilizing the function of the ultrasonic sensor and also the tube volume formula, the stored AIR volume can be monitored in real time based on IoT using the Blynk application. From the sensor, height data is obtained which then the formula is processed by Arduino Wemos and then information is sent to the MySQL database server via the WIFI network.

scholarly journals The Volume of the Quiver Vortex Moduli Space

2021 ◽  
Author(s):  
Kazutoshi Ohta ◽  
Norisuke Sakai
Keyword(s):  
Gauge Theory ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Gauge Theories ◽  
Target Space ◽  
Contour Integral ◽  
Quiver Gauge Theory ◽  
Volume Formula ◽  
Volume Operator ◽  
Space Volume

Abstract We study the moduli space volume of BPS vortices in quiver gauge theories on compact Riemann surfaces. The existence of BPS vortices imposes constraints on the quiver gauge theories. We show that the moduli space volume is given by a vev of a suitable cohomological operator (volume operator) in a supersymmetric quiver gauge theory, where BPS equations of the vortices are embedded. In the supersymmetric gauge theory, the moduli space volume is exactly evaluated as a contour integral by using the localization. Graph theory is useful to construct the supersymmetric quiver gauge theory and to derive the volume formula. The contour integral formula of the volume (generalization of the Jeffrey-Kirwan residue formula) leads to the Bradlow bounds (upper bounds on the vorticity by the area of the Riemann surface divided by the intrinsic size of the vortex). We give some examples of various quiver gauge theories and discuss properties of the moduli space volume in these theories. Our formula are applied to the volume of the vortex moduli space in the gauged non-linear sigma model with CPN target space, which is obtained by a strong coupling limit of a parent quiver gauge theory. We also discuss a non-Abelian generalization of the quiver gauge theory and “Abelianization” of the volume formula.

scholarly journals Combinatorics on lattice paths in strips

2021 ◽  
Vol 94 ◽  
pp. 103310
Author(s):  
Nancy S.S. Gu ◽  
Helmut Prodinger
Keyword(s):  
Lattice Paths

scholarly journals Enumerations of plane trees with multiple edges and Raney lattice paths

2014 ◽  
Vol 337 ◽  
pp. 9-24 ◽  
Author(s):  
M. Dziemiańczuk
Keyword(s):  
Lattice Paths ◽  
Plane Trees ◽  
Multiple Edges
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