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

A Note on Major Sequences and External Activity in Trees

A bijection is given from major sequences of length n (a variant of parking functions) to trees on f0; : : : ; ng that maps a sequence with sum \Gamma n+1 2 \Delta + k to a tree with external activity k. Key Words: Major sequence, external activity, parking function, bijection Mathematical Reviews Subject Numbers: Primary 05A19; Secondary 05A15, 05C05, 05C30 the electronic journal of combinatorics 4 (no. 2) (1997), #R4 2 We present a bijection from major seqeunces (a variant of parking funtions) of length n to trees on f0; : : : ; ng that takes area to external activity. Our main tool is a decomposition of major sequences due to Kreweras [6]. An integer sequence S = (s 1 ; : : : ; s n ) is called a major sequence of length n [6] if its non-decreasing rearrangement (z 1 ; : : : ; z n ) satisfies z i i for all 1 i n and z n n: Another way to view (z 1 ; : : : ; z n ) is as a lattice path from (0; 0) to (n; n) that never drops below the main diagonal. In Figure 1 the top latt..