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

Sri Gopal Mohanty has made pioneering contributions to lattice direction counting and its purposes to likelihood and facts. this is often in actual fact glaring from his lifetime courses checklist and the varied citations his courses have obtained during the last 3 a long time. My organization with him begun in 1982 whilst I got here to McMaster Univer sity. seeing that then, i've been linked to him on many various concerns at expert in addition to cultural degrees; i've got benefited enormously from him on either those grounds. i've got loved a great deal being his colleague within the information crew the following at McMaster collage and in addition as his buddy. whereas i like him for his honesty, sincerity and commitment, I enjoy a great deal his kindness, modesty and broad-mindedness. other than our universal curiosity in arithmetic and information, we either have nice love for Indian classical tune and dance. now we have spent a number of many alternative matters linked to the Indian song and hours discussing dance. I nonetheless take into account fondly the lengthy force (to Amherst, Massachusetts) I had many years in the past with him and his spouse, Shantimayee, and the entire hearty discussions we had in the course of that trip. Combinatorics and purposes of combinatorial equipment in likelihood and information has develop into a really energetic and fertile zone of analysis within the fresh past.

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**Sample text**

I... JO 1 i+jO+2jl +3h+ .. =r 1,. Jo· J1· .... Counting Paths Let r be a positive integer and let k and n be nonnegative integers. Let S be a subset of the set {l,0,-1,-2,···}. We call the elements of S steps. We want to count sequences (Sl, S2, ... , sn) of elements of S such that every partial sum r + Sl + 82 + ... + 8i is positive and r + 81 + 82 + ... + 8 n = k. We call such a sequence of steps a good path of length n from r to k. 1 are all equivalent to specializations of the case S = {1, -p} for various values of p, r, and k.

I)i (r -. Pi) f~(t + fp/tpr-(p+l)i-l r -P't i::;r/(p+1) = 1- L (_I)i(r-~i-l)f~(t+fp/tpr-(p+l)i-l. 1, we substitute x for t and xPy for fp. i-l)(XPy)i(x+yr-(p+l)i-l. 4). 16) equal Nr /(1 - x - y) and let Nr = xr - Mr. 1:, fp = xPy, and fi = 0 for i i- p. 16) that Nr and Mr a,Ee homogeneous of degree r in x and y, they are determined by the sums Lr N r and Lr Mr. 1: + (p - 2)x 2 + ... + x p- 1 . 1 - y(1 + x + x 2 + ... 1. We can also obtain a simple explicit formula when p = 2. 18) gives f fMr=y (2+x) = xiyi+l r=l l-y(l+x) i,j=O [(~) + (i~l)l.

With relatively general initial conditions. 7) Qbn = Rbn- l + Sbn- c for all n = 0,1, ... , power series of order 0 in D x ), and Q is a delta operator. An example for R could be Rbn(x) = L:i=l Pibn(X - rd for some given constants PI, ... , Pk and rl, ... , rk. 7), we mean an expansion of bn(x) in terms of the basic sequence {qn} of Q. Suppose the unknown solution {b n } is the basic sequence for some delta operator B. If we can construct a solution under this hypothesis, then the assumption will be justified.