A Generalization of the Catalan Numbers

Transcription

1 Journal of Integer Sequences, Vol 6 (203, Article 368 A Generalization of the Catalan Numbers Reza Kahkeshani Department of Pure Mathematics Faculty of Mathematical Sciences University of Kashan Kashan Iran kahkeshanireza@kashanuacir Abstract In this paper, we generalize the Catalan number C n to the (m,nth Catalan number C(m,n using a combinatorial description, as follows: the number of paths in R m from the origin to the point ( n,,n,(m n with m kinds of moves such that the path m never rises above the hyperplane x m x + +x m Introduction Catalan numbers (A00008 are a very prominent sequence of numbers that arises in a wide varity of combinatorial situations [, 2] Stanley [0] gave a list of 66 different combinatorial descriptionsofcatalannumbersandheaddedtothelistsomemore[] Someofthespecific instances are The number of movements in xy-plane from (0,0 to (n,n with two kinds of moves R : (x,y (x+,y, U : (x,y (x,y +, such that the path never rises above the line y x Triangulations of a convex (n+2-gon into n triangles by n diagonals that do not intersect in their interiors

2 Binary parenthesizations of a string of n + letters Binary trees with n vertices The solution to these problems is the nth Catalan number C n ( 2n, n+ n and the sequence C 0,C,C 2,,C n, is called the Catalan sequence There have been many attempts to generalize the Catalan numbers Probably the most important generalization consists of the k-ary numbers or k-catalan numbers, defined by C k n kn+ ( kn+ n (k n+ ( kn n n ( kn n where k,n N Clearly, Cn 2 C n The k-good paths (below the line y kx from (0, to ( n,(k n, staircase tilings and k-ary trees are structures known to be enumerated by k-ary numbers [5, 6, 8, 0] Moreover, Kim [7, Thm 2] showed that Cn k is the number of partitions of n(k +2 polygon by (k +-gon where all vertices of all (k +-gon lie on the vertices of n(k +2 polygon Gould [3] developed a generalization that has both the Catalan numbers and the k-catalan numbers as special cases, defined as A n (a,b a a+bn ( a+bn and showed the following convolution formula for these sequences: n A k (a,ba n k (c,b A n (a+c,b k0 These numbers are also known as the Rothe numbers [9] and Rothe-Hagen coefficients of the first type [4] Clearly, A n (,2 C n and A n (,k Cn k We know that one of the interpretations of the Catalan numbers is the movements in R 2 with two kinds of moves such that the path never rises above the line y x In this paper, a new generalization of the Catalan numbers using this interpretation is introduced Consider m kinds of moves in R m such that they are one unit parallel to the positive axes We show that the number of paths from the origin to the point ( n,,n,(m n using these moves m such that the path never rises above the hyperplane x m x + +x m is n(m + n, ( 2n(m n,,n,n(m m We call this number the (m,nth Catalan number C(m,n Clearly, C(2,n is the ordinary nth Catalan number C n 2,

3 2 Generalization In this section, we prove the our main theorem We show that the generalized Catalan numbers C(m,n are given by ( 2n(m n(m + n,,n,n(m m Theorem Let R m be the m-dimensional vector space Consider R : (x,x 2,,x m (x +,x 2,,x m, R 2 : (x,x 2,,x m (x,x 2 +,,x m, R m : (x,x 2,,x m (x,x 2,,x m +, be m kinds of moves in R m (ie, R i denotes the move one unit parallel to the x i -axis in the positive direction Then the number of paths from 0 (0,,0 to the point N : ( m n,,n,(m n using the moves R,R 2,,R m such that the path never rises above the m hyperplane x m x + +x m is ( 2n(m n(m + n,,n,n(m m Proof We call a path from 0 to N of n R s, n R 2 s,, n R m s, and (m n R m s acceptable if the path never rises above the hyperplane x m x + +x m and unacceptable otherwise Let A m n and Un m denote the number of acceptable and unacceptable paths, respectively It is easy to see that each path from 0 to N corresponds to an arrangement of n R s, n R 2 s,, n R m s, and (m n R m s Then ( 2n(m! A m n +Un m n! m ( n(m! Now, consider an unacceptable path and its arrangement r,r 2,,r 2n(m, where r i {R,R 2,,R m } indicates the ith step of the path Since the path rises above the hyperplane, there is a first t such that the number of R m s in r,,r t exceeds the sum of the numbers R,R 2,,R m Moreover, r t R m We only change r t+,,r 2n(m the part of the path after the crossing in the arrangement as follows: Mark all the positions of the R m s in that part of the path and fill those positions with the sequence (in order consisting of all but the last of the non-r m s Then replace those non-r m s that have been used in 3

4 the replacement with R m s Here is an example: let m 3, n 2 and the path be given by R R 3 R 2 R 3 R 3 R R 2 R 3 Then t 5, and the part of the path to be modified is R R 2 R 3 There is just one position of the R m s, so we replace R 3 with R, and then fill the R R 2 with R 3 R 3 to obtain the modified sequence R R 3 R 2 R 3 R 3 R 3 R 3 R The resulting arrangement r,r 2,,r 2n(m is an arrangement of (m n+ R m s, n R s,, n R i s, n R i+ s,, n R m s, and n R i s for a i m It is not difficult to see that this process is reversible: r,r 2,,r 2n(m,,,,R m rr m s, r R,,R m s,,,,r m rr m s, r R,,R m s,,, (m n rr m s, (m n r+r,,r m s,,, (m n r+r m s, (m n rr,,r m s r,r 2,,r 2n(m Hence, there are as many unacceptable arrangements as there are arrangements of(m n+ R m s, n R s,, n R i s, n R i+ s,, n R m s, and n R i s for a i m Then ( 2n(m! Un m (m n! m 2 (n! ( n(m +! So, ( ( 2n(m! 2n(m! A m n n! m ( n(m! (m n! m 2 (n! ( n(m +! ( 2n(m! ( n! m 2 (n! ( n(m! n m n(m + ( 2n(m! n! m ( n(m +! ( 2n(m n(m + n,,n,n(m m We denote A m n in the above proof by C(m,n The first few generalized Catalan numbers are evaluated to be 4

5 n\m Acknowledgments The author would like to thank the referee for his/her valuable comments and suggestions which have improved the clarity of the proof of the Theorem This work is partially supported by the University of Kashan under grant number /3 References [] R A Brualdi, Introductory Combinatorics, 5th ed, Prentice-Hall, 2009 [2] R P Grimaldi, Discrete and Combinatorial Mathematics, 5th ed, Addison-Wesley, 2004 [3] H W Gould, Combinatorial Identities, Morgantown, West Virginia, 972 [4] H W Gould, Fundamentals of Series, eight tables based on seven unpublished manuscript notebooks ( , edited and compiled by J Quaintance, May 200 Available at [5] S Heubach, N Y Li, and T Mansour, Staircase tilings and k-catalan structures, Discrete Math 308 (2008, [6] P Hilton and J Pedersen, Catalan numbers, their generalization, and their uses, Math Intelligencer 3 (2 (99, [7] D Kim, On the (n, k-th Catalan numbers, Commun Korean Math Soc 23 (2008, [8] I Pak, Reduced decompositions of permutations in terms of star transpositions, generalized Catalan numbers and k-ary trees, Discrete Math 204 (999,

6 [9] S L Richardson, Jr, Enumeration of the generalized Catalan numbers, MSc Thesis, Eberly College of Arts and Sciences, West Virginia University, Morgantown, West Virginia, 2005 [0] R P Stanley, Enumerative Combinatorics, Vol 2, Cambridge University Press, 999 [] R P Stanley, Catalan addendum, preprint, May Available at Mathematics Subject Classification: Primary 05A9; Secondary 05A0, 05A5 Keywords: Catalan number, path (Concerned with sequence A00008 Received March 6 203; revised version received July Published in Journal of Integer Sequences, July Return to Journal of Integer Sequences home page 6