Graph searching and a generalized parking function

Parking functions have been a focus of mathematical research since the mid-1970s. Various generalizations have been introduced since the mid-1990s and deep relationships between these and other areas of mathematics have been discovered. Here, we introduced a new generalization, the G-multiparking function, where G is a simple graph on a totally ordered vertex set {1, 2, . . . , n}. We give an algorithm that converts a G-multiparking function into a rooted spanning forest of G by using a graph searching technique to build a sequence F1, F2, . . . , Fn, where each Fi is a subforest of G and Fn is a spanning forest of G. We also give another algorithm that converts a rooted spanning forest of G to a G-multiparking function and prove that the resulting functions (between the sets of G-multiparking functions and rooted spanning forests of G) are bijections and inverses of each other. Each of these two algorithms relies on a choice function , which is a function from the set of pairs (F,W) (where F is a subforest of G and W is a set of some of the leaves of F) into W. There are many possible choice functions for a given graph, each giving formality to the concept of choosing how a graph searching algorithm should procede at each step (i.e. if the algorithm has reached Fi, then Fi+1 is some forest on the vertex set V (Fi)∪{(Fi,W)} for some W). We also define F-redundant edges, which are edges whose removal from G does not affect the execution of either algorithm mentioned above. This concept leads to a categorization of the edges of G, which in turn provides a new formula for the Tutte polynomial of G and other enumerative results

A q-analog of certain symmetric functions and one of its specializations

Let be the symmetric functions defined for the pair of integers $\left( n,r\right)$ $n\geq r\geq 1$ by $p_{n}^{\left( r\right) }=\sum m_{\lambda }$ where $m_{\lambda }$ are the monomial symmetric functions, the sum being over the partitions $\lambda$ of the integer $n$ of length $r$. In this article we introduce a $q$-analog of $p_{n}^{\left( r\right) }$, through generating functions and give some of its properties which are $q$-analogs of its classical correspondent in particular when $r=1$. It is proved that this $q$-analog of $p_{n}^{\left( r\right) }$ can be expressed in terms of the classical $p_{n}^{\left( j\right) }$, through the $q$-Stirling numbers of the second kind. We also begin, with the same procedure, the study of a $p,q$-analog of $p_{n}^{\left( r\right) }$. In the rest of the article we specialize in the series $\sum\nolimits_{n=0}^{\infty }q^{\binom{n}{2}}t^{n}/n!$ . We show that $p_{n}^{\left( r\right) }$ is then related to the $q^{r}$-analog of $p_{n-r}$. The existence of a double sequence of polynomials with integer coefficients, denoted $J_{n,r}\left( q\right)$, is deduced. We identify these polynomials with the inversion enumerators introduced for specific rooted forests. These polynomials verify a ''positive'' linear recurrence which allows to build row by row the table of $J_{n,r}$ from the initial conditions $J_{r,r}=1$. The form of the linear recurrence is given for the reciprocal polynomials of \ $J_{n,r}$, which are the sum enumerators of parking functions. The linear recurrence permits to obtain an explicit calculation formula for $J_{n,r}$ . This formula leads us to introduce new statistics on rooted trees and forests for $\ J_{n,r}$ or its reciprocal.Comment: 17 pages, 1 figure. All the results are unchanged. Minor changes to improve English and writing of the text. One reference added and modification of the numbering of references. Equation (2.1) corrected. Section 5: Open problem replaced by Remark and shorcut. Lemma 6.1 replaced by Proposition 6.1, content unchanged. An additional paragraph in the conclusion. Change of legend of Figure

Enumeration of minimal acyclic automata via generalized parking functions

We give an exact enumerative formula for the minimal acyclic deterministic finite automata. This formula is obtained from a bijection between a family of generalized parking functions and the transitions functions of acyclic automata

q-power symmetric functions and q-exponential formula

Let $\lambda =\left( \lambda_{1},\lambda_{2},...,\lambda_{r}\right)$ be an integer partition, and $\left[p_{\lambda }\right]$ the $q$-analog of the symmetric power function $$. This $q$-analogue has been defined as a special case, in the author's previous article: "A $q$-analog of certain symmetric functions and one of its specializations". Here, we prove that a large part of the classical relations between $p_{\lambda }$, on one hand, and the elementary and complete symmetric functions $e_{n}$ and $h_{n}$, on the other hand, have $q$-analogues with $\left[ p_{\lambda }\right]$. In particular, the generating functions $E\left( t\right) =\sum\nolimits_{n\geq 0}e_{n}t^{n}$ and $H\left( t\right) =\sum\nolimits_{n\geq 0}h_{n}t^{n}$ are expressed in terms of $\left[ p_{n}\right]$, using Gessel's $q$-exponential formula and a variant of it. A factorization of these generating functions into infinite $q$-products, which has no classical counterpart, is established. By specializing these results, we show that the $q$-binomial theorem is a special case of these infinite $q$-products. We also obtain new formulas for the tree inversions enumerators and for certain $q$-orthogonal polynomials, detailing the case of dicrete $q$-Hermite polynomials.Comment: 23 page

Parking Functions And Generalized Catalan Numbers

Since their introduction by Konheim and Weiss, parking functions have evolved into objects of surprising combinatorial complexity for their simple definitions. First, we introduce these structures, give a brief history of their development and give a few basic theorems about their structure. Then we examine the internal structures of parking functions, focusing on the distribution of descents and inversions in parking functions. We develop a generalization to the Catalan numbers in order to count subsets of the parking functions. Later, we introduce a generalization to parking functions in the form of k-blocked parking functions, and examine their internal structure. Finally, we expand on the extension to the Catalan numbers, exhibiting examples to explore its internal structure. These results continue the exploration of the deep structures of parking functions and their relationship to other combinatorial objects

Enumeration of minimal acyclic automata via generalized parking functions

We give an exact enumerative formula for the minimal acyclic deterministic finite automata. This formula is obtained from a bijection between a family of generalized parking functions and the transitions functions of acyclic automata

Space programs summary number 37-29, volume iv for the period august 1, 1964 to september 30, 1964. supporting research and advanced development

Systems, guidance and control, engineering mechanics and facilities, propulsion, space sciences, and telecommunications researc

Goncarov Polynomials, Partition Lattices and Parking Sequences

Classical Goncarov polynomials arose in numerical analysis as a basis for the solutions of the Goncarov interpolation problem. These polynomials provide a natural algebraic tool in the enumerative theory of parking functions. By replacing the differentiation operator with a delta operator and using the theory of finite operator calculus, Lorentz, Tringali and Yan introduced the sequence of generalized Goncarov polynomials associated to a pair $(\Delta, Z)$ of a delta operator $\Delta$ and an interpolation grid $Z$. Generalized \gon polynomials share many nice algebraic properties and have a connection with the theories of binomial enumeration and order statistics. Parking functions are combinatorial objects which were introduced in 1966 by Konheim and Weiss. They have been well-studied in the literature due to their numerous connections and have several generalizations and extensions. Ehrenborg and Happ recently introduced a generalization of parking functions called parking sequences in which the $n$ cars have different sizes, and each takes up a number of adjacent parking spaces after a trailer $T$ parked on the first $z-1$ spots. Consequently, this dissertation is divided into two major parts. In the first part, we give a complete combinatorial interpretation for any sequence of generalized Goncarov polynomials. First, we show that they can be realized as weight enumerators in partition lattices. Then, we give a more concrete realization in exponential families and show that these polynomials enumerate various enriched structures of vector parking functions. In the second part, we study increasing parking sequences and their representation via a special class of lattice paths. We also study two notions of invariance in parking sequences and prove some interesting results for a number of cases where the sequence of car sizes have special properties