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 can be in actual fact obvious from his lifetime guides checklist and the varied citations his guides have obtained during the last 3 a long time. My organization with him begun in 1982 while I got here to McMaster Univer sity. in view that then, i've been linked to him on many alternative matters 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 data crew the following at McMaster collage and in addition as his buddy. whereas i like him for his honesty, sincerity and commitment, I savor greatly his kindness, modesty and broad-mindedness. other than our universal curiosity in arithmetic and facts, we either have nice love for Indian classical song and dance. we've spent a variety 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 functions of combinatorial equipment in likelihood and records has turn into a really energetic and fertile region of analysis within the contemporary past.

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Tf'(t) f(v) - f(t) ~ = ~ Nrv -r . 12) have positive coefficients when expanded as power series in v-I and C 1 . First, we have f(v)-f(t) _ t v, = i=-1 L Ii 00 (v-i_C i ) _t v, = 1- (1 1 1) L fi fi + 2ti - 1 + ... + it . i=1 v, v, V , 00 Thus, (v - t)/{f(v) - f(t)} has nonnegative coefficients. 13) is zero for i = -1 and i = O. 13) are nonnegative. 5 • Examples Let us now return to the problem discussed in the first section: given positive integers rand p, count paths in the plane with steps (1,0) and (0,1), from (r,O) to (m, n), where m > pn, that never touch the line x = py.

9) f6° fi! " where k - r + jo + 2jl + ... ) {I0 + '/, = if k = rand n = th . 10) only when k = rand jo = jl = ... = O. This proves (iii). 10) once we show that g( -1, k) = 0 for all k. 8) that g(n, k) = 0 for n < -r. 10) by induction on n that g(n, k) = 0 for all negative n. Thus, (ii) holds. ial of f. PROOF. It follows from the definition of Faber polynomials that d Fr () -t -d f = tr r t . + negatIve powers of t. 3, it is sufficient to show that ~t Fr(f)/(1 - f) 10 Ira M. Gessel and Sangwook Ree is the derivative of some Laurent series in t, since this will imply that it has no term in C 1 .

These are then put together and ordered into Z = (Zl' Z2,' .. ,Z2n) according to size. The run statistics counts the number of maximal consecutive subsequences in Z the members of which belong to just one of the sets X or y. Thus, if n = 5, and if Z = (XI, YI, Y 2, Y 3 , X2, X 3 , Y 4 , X 4 , X5, Y5), then the number of runs in Z is 6. The one-sided Kolmogorov-Smirnov statistic D;tn is defined by where ai is the number of occurrences of X/s in the initial segment ZI, Z2,' .. ,Zi of Z, while bi is the number of occurrences of Yj's in this initial segment.