Article

Some Enumerations for Parking Functions

07/2008;
Source: arXiv

ABSTRACT In this paper, let $\mathcal{P}_{n,n+k;\leq n+k}$ (resp. $\mathcal{P}_{n;\leq s}$) denote the set of parking functions $\alpha=(a_1,...,a_n)$ of length $n$ with $n+k$ (respe. $n$)parking spaces satisfying $1\leq a_i\leq n+k$ (resp. $1\leq a_i\leq s$) for all $i$. Let $p_{n,n+k;\leq n+k}=|\mathcal{P}_{n,n+k;\leq n+k}|$ and $p_{n;\leq s}=|\mathcal{P}_{n;\leq s}|$. Let $\mathcal{P}_{n;\leq s}^l$ denote the set of parking functions $\alpha=(a_1,...,a_n)\in\mathcal{P}_{n;\leq s}$ such that $a_1=l$ and $p_{n;\leq s}^l=|\mathcal{P}_{n;\leq s}^l|$. We derive some formulas and recurrence relations for the sequences $p_{n,n+k;\leq n+k}$, $p_{n;\leq s}$ and $p_{n;\leq s}^l$ and give the generating functions for these sequences. We also study the asymptotic behavior for these sequences.

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Article: On the Enumeration of Generalized Parking Functions
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ABSTRACT: Let x = (x1 , x2 , . . . , xn) # N n . Define a x-parking function to be a sequence (a1 , a2 , . . . , an) of positive integers whose non-decreasing rearrangement b1 # b2 # # bn satisfies b i # x1 + +x i . Let Pn (x) denote the number of x-parking functions. We discuss the enumerations of such generalized parking functions. In particular, We give the explicit formulas and present two proofs, one by combinatorial argument, and one by recurrence, for the number of x-parking functions for x = (a, n-m-1 z }| { b, . . . , b, c, m-1 z }| { 0, . . . , 0) and x = (a, n-m-1 z }| { b, . . . , b, m z }| { c, . . . , c). 1 Introduction The notion of parking function was introduced by Konheim and Weiss [3] in their studying of an occupancy problem in computer science. Then the subject has been studied by numerous mathematicians. For example, Foata and Riordan [1], and Francon [2] constructed bijections from the set of parking functions to the set of acyclic functions on ...
06/2000;
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Article: Taylor expansions for Catalan and Motzkin numbers
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ABSTRACT: In this paper we introduce two new expansions for the generating functions of Catalan numbers and Motzkin numbers. The novelty of the expansions comes from writing the Taylor remainder as a functional of the generating function. We give combinatorial interpretations of the coefficients of these two expansions and derive several new results. These findings can be used to prove some old formulae associated with Catalan and Motzkin numbers. In particular, our expansion for Catalan number provides a simple proof of the classic Chung–Feller theorem; similar result for the Motzkin paths with flaws is also given.
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Article: Ordered k-flaw Preferences Sets
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ABSTRACT: In this paper, we focus on ordered $k$-flaw preference sets. Let $\mathcal{OP}_{n,\geq k}$ denote the set of ordered preference sets of length $n$ with at least $k$ flaws and $\mathcal{S}_{n,k}=\{(x_1,...,x_{n-k})\mid x_1+x_2+... +x_{n-k}=n+k, x_i\in\mathbb{N}\}$. We obtain a bijection from the sets $\mathcal{OP}_{n,\geq k}$ to $\mathcal{S}_{n,k}$. Let $\mathcal{OP}_{n,k}$ denote the set of ordered preference sets of length $n$ with exactly $k$ flaws. An $(n,k)$-\emph{flaw path} is a lattice path starting at $(0,0)$ and ending at $(2n,0)$ with only two kinds of steps--rise step: $U=(1,1)$ and fall step: $D=(1,-1)$ lying on the line $y = -k$ and touching this line. Let $\mathcal{D}_{n,k}$ denote the set of $(n, k)$-flaw paths. Also we establish a bijection between the sets $\mathcal{OP}_{n,k}$ and $\mathcal{D}_{n,k}$. Let $op_{n,\geq k,\leq l}^m$ $(op_{n, k, =l}^m)$ denote the number of preference sets $\alpha=(a_1,...,a_n)$ with at least $k$ (exact) flaws and leading term $m$ satisfying $a_i\leq l$ for any $i$ $(\max\{a_i\mid 1\leq i\leq n\}=l)$, respectively. With the benefit of these bijections, we obtain the explicit formulas for $op_{n,\geq k,\leq l}^m$. Furthermore, we give the explicit formulas for $op_{n, k, =l}^m$. We derive some recurrence relations of the sequence formed by ordered $k$-flaw preference sets of length $n$ with leading term $m$. Using these recurrence relations, we obtain the generating functions of some corresponding $k$-flaw preference sets.
07/2008;