By Ira M. Gessel, Sangwook Ree (auth.), N. Balakrishnan (eds.)

Sri Gopal Mohanty has made pioneering contributions to lattice direction counting and its functions to likelihood and records. this is often truly obvious from his lifetime courses record and the various citations his courses have obtained over the last 3 many years. My organization with him begun in 1982 while I got here to McMaster Univer sity. seeing that then, i've been linked to him on many alternative matters at specialist in addition to cultural degrees; i've got benefited tremendously from him on either those grounds. i've got loved a great deal being his colleague within the facts staff right here at McMaster college and in addition as his pal. whereas i like him for his honesty, sincerity and commitment, I enjoy a great deal his kindness, modesty and broad-mindedness. apart from our universal curiosity in arithmetic and data, we either have nice love for Indian classical song and dance. we have now spent quite a few many various matters linked to the Indian song and hours discussing dance. I nonetheless have in mind fondly the lengthy force (to Amherst, Massachusetts) I had many years in the past with him and his spouse, Shantimayee, and all of the hearty discussions we had in the course of that trip. Combinatorics and purposes of combinatorial tools in chance and records has develop into a really lively and fertile zone of analysis within the contemporary past.

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1 A path from r to k with weight f6° steps equal to 1, and length k - r + ]0 + 2]1 + .... 1 ... has k - r +]1 + 212 + ... Let ]-1 be the number of steps equal to 1. Since the path is from r to k, we have r + ]-1 - 0]0 - 1]1 - 212 - ... = k, and the first assertion follows. Then the length of the path is ]-1 + 11 + 12 + ... = k - r +]0 + 2]1 + . .. • PROOF. Ira M. Gessel and Sangwook Ree 8 We now fix r throughout the rest of this section. Let G(n, k) be the sum of the weights of all good paths of length n from r to k.

G. (1979). York: Academic Press. 27 Lattice Path Counting and Applications, New 8. Niederhausen, H. (1979). Lattice paths with three step directions, Congressus Numerantium, 14, 753-774. 9. Niederhausen, H. (1981). Sheffer polynomials for computing exact Kolmogorov-Smirnov and Renyi type distributions, Annals of Statistics, 9, 923944. 10. Niederhausen, H. (1982). How many paths cross at least 1 given lattice points? Congressus Numerantium, 36, 161-173. 11. Niederhausen, H. (1986). The enumeration ofrestricted random walks by Sheffer polynomials with applications to statistics, Journal of Statistical Planning and Inference, 14, 95-114.

For K = L, the formula was derived by Koroljuk (1955). See Fray and Roselle (1971) for another proof of the general case. In the two-sided one-sample case, the distribution must be expanded in terms of xn In!. Because of its lack of symmetry, no closed form is known. 5 Geometric Sheffer Sequences A Sheffer sequence {sn} n>O is geometric if So == 1 and if there exists a pair of constants a and a such that Sn(an) for all n = aSn-1(an) = 1, 2, .... 5). The following theorem explains why they are called "geometric".