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 can be sincerely glaring from his lifetime courses record and the varied citations his courses have bought during the last 3 a long time. My organization with him all started in 1982 while I got here to McMaster Univer sity. for the reason that then, i've been linked to him on many alternative concerns at expert in addition to cultural degrees; i've got benefited significantly from him on either those grounds. i've got loved a great deal being his colleague within the statistics crew right here at McMaster collage and in addition as his pal. whereas i love him for his honesty, sincerity and commitment, I savor greatly his kindness, modesty and broad-mindedness. other than our universal curiosity in arithmetic and information, we either have nice love for Indian classical song and dance. we've spent a number of many alternative topics linked to the Indian tune and hours discussing dance. I nonetheless bear 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 functions of combinatorial tools in likelihood and facts has develop into a really energetic and fertile region of analysis within the contemporary past.

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A number of remarkable formulas were discovered to solve most of these problems. But there are still some important open questions. The problem of turn enumeration of lattice paths was attacked in many different ways. However, there is a uniform approach which is able to handle all these problems, which is by encoding paths in terms of two-rowed arrays. 2), who probably was the first to count paths with respect to their turns, used to see turn enumeration problems. However, he did not use the combinatorics of tworowed arrays.

Niederhausen, H. (1996). Symmetric Sheffer sequences and their applications to lattice path counting, Journal of Statistical Planning and Inference, 54, 87-100. 15. P6lya, G. and Szego, G. (1972). Problems and Theorems in Analysis-I, New York: Springer-Verlag. 16. , Kahaner, D. and Odlyzko, A. (1973). On the foundations of combinatorial theory, VIII. Finite operator calculus, Journal Mathematical Analysis and Applications, 42, 684-760. 17. Sulanke, R. A. (1993). Refinements of the Narayana numbers, Bulletin of the ICA, 7,60-66.

If we can construct a solution under this hypothesis, then the assumption will be justified. Because Band Q are both delta operators, there exists a translation invariant and invertible operator T such that B = TQ [see Corollary 4 of Rota, Kahaner and Odlyzko (1973)]. 7) (any two translation invariant operators commute). Equivalently, 1- RT = (RT)CR-cSQc-1 and (RT)-C - (RT)I-c = R- cSQc-l. 4)]. 7). 1 Applications: Lattice paths with several step directions Counting (weighted) lattice paths with several step directions leads to recurrence relations of the form r dn(x) = dn(x - 1) +L 8 Pibn-1(X - ri) i=l +L O'ibn-c(X - 8i) i=l if the step vectors are (1,0), (rl, 1), ...