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

Sri Gopal Mohanty has made pioneering contributions to lattice course counting and its purposes to likelihood and facts. this can be in actual fact glaring from his lifetime guides checklist and the various citations his courses have obtained over the last 3 many years. My organization with him begun in 1982 whilst I got here to McMaster Univer sity. considering that then, i've been linked to him on many alternative matters at expert in addition to cultural degrees; i've got benefited enormously from him on either those grounds. i've got loved greatly being his colleague within the facts workforce the following at McMaster college and likewise as his good friend. whereas i love him for his honesty, sincerity and commitment, I relish greatly his kindness, modesty and broad-mindedness. other than our universal curiosity in arithmetic and facts, we either have nice love for Indian classical track and dance. we now have spent various many various matters linked to the Indian track and hours discussing dance. I nonetheless be mindful 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 equipment in chance and information has turn into a truly energetic and fertile region of analysis within the fresh past.

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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. 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)].

The purpose of this survey article is to show that two-rowed arrays allow to handle turn enumeration in a purely combinatorial way. The combinatorics of tworowed arrays is able to explain all the existing formulas in a conceptual way. " Another purpose of this survey is to show the wide diversity of connections The Enumeration of Lattice Paths 31 and applications in other fields like combinatorics, representation theory, and q-series. Moreover, it is not unreasonable to expect that the recent subject of tum enumeration of nonintersecting lattice paths will also have its applications in probability, statistics, or physics.

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 . Let Fr(u) = 2::i=oCiUi . + • iCi . 1. We know that N r t r - M r , where Mr contains only negative powers of t. 1 The coefficients of M r , as a power series in nonnegative integers.