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 chance and information. this can be essentially obtrusive from his lifetime guides record and the varied citations his guides have bought over the last 3 a long time. My organization with him all started in 1982 while I got here to McMaster Univer sity. considering then, i've been linked to him on many alternative matters at expert in addition to cultural degrees; i've got benefited vastly from him on either those grounds. i've got loved greatly being his colleague within the data staff the following at McMaster college and in addition as his pal. whereas i like him for his honesty, sincerity and commitment, I take pleasure in a great deal his kindness, modesty and broad-mindedness. other than our universal curiosity in arithmetic and records, we either have nice love for Indian classical track and dance. we've spent quite a few many various topics linked to the Indian track and hours discussing dance. I nonetheless consider 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 tools in likelihood and records has develop into a really energetic and fertile sector of analysis within the fresh past.

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**Example text**

Bn-i(x - ai - c). 5) Piecewise affine boundaries Suppose we want to solve the system with initial values first along the line Xn = an + c given by Yn = Tn(an + c) for all n = 0, ... ,L - 1, and thereafter on the line Xn = an + c given by Yn = Tn(an + c) Lattice Path Enumeration and Umbral Calculus 19 for all n = L, ... , where a,a,c,c and L are all given constants. 5), the beginning of the sequence can be calculated along the second line as _. z+c-c j=O C for all i = 0, ... ,L - . _ az - aJ + C - _.

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.

We work in the ring of formal Laurent series C[[t, fo, hit, 12/t 2, .. ]]. Let F(u) be a polynomial in u of degree r such that F(f) = t + negative powers of t. T Lattice Paths and Faber Polynomials 7 We say that F(u) is a Faber polynomial of f. It is easy to prove by induction that there is exactly one Faber polynomial Fr (u) of degree r, which we call the rth Faber polynomial of f. I). Schiffer (1948) gave the generating function ) v- r 1og f (v) - u -_ - L~ F r (u -. 6) may be rewritten in terms of formal power series as log{h(w) - uw} = - r L Fr(u)~ .