Advances in Combinatorial Methods and Applications to by Ira M. Gessel, Sangwook Ree (auth.), N. Balakrishnan (eds.)

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 chance and facts. this can be truly obtrusive from his lifetime courses record and the varied citations his courses have got during the last 3 a long time. My organization with him started in 1982 while I got here to McMaster UniverĀ­ sity. because then, i've been linked to him on many various concerns at specialist in addition to cultural degrees; i've got benefited significantly from him on either those grounds. i've got loved greatly being his colleague within the facts workforce right here 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. apart from our universal curiosity in arithmetic and information, we either have nice love for Indian classical track and dance. we've spent quite a few many various 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 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 sector of study within the contemporary past.

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Example text

In general, s~) (m) equals the weighted sum of lattice paths from (0,0) to (m, n), where every left turn --+ bgets the weight f-t. Because of symmetry, the classical ballot problem has a simple solution for this kind of weighted paths: which is the weighted sum of lattice paths from (0,0) to (m, n) that stay below the line m = n - L. ) 1 1 1 1 0 1 + 3Jl 1 + 2Jl l+Jl 1 1 1 + 5Jl + 3Jl2 1 + 3Jl + Jl2 l+Jl 1 + 611, + 6Jl2 1 + 3Jl + Jt 3 + Jl2 1 + 6Jl + 6Jl2 + Jl3 0 ... w3(m) w2(m) wl(m) wo(m) 4 5 ...

L - 1, and r n (n - L) = 0 for all n 2: L. 6) indicates. It is given by 22 Heinrich Niederhausen This well-known solution is easily identified as a difference of two Sheffer polynomials for the backwards difference operator \7, and it is obviously zero at m = -1 for all n = 1, ... , L - 1. However, for n :2 L the initial values are attained because of a very special property: r n (n _ L) = (n + n - L) _ (n + : - L) = (n + n - L) _ (n + n - L) . n n L n n In other words, for nonnegative integers nand m, we can interchange the degree n with the argument m in the polynomial sn(m) := (n~m), and get again a polynomial sm(n) = (m~n) = sn(m).

We want to recall this expansion, because it is so often omitted in the literature. We saw that rn(m) = s~l)(m) - s~~2L(m + L) is the number of lattice paths below the line Heinrich Niederhausen 24 m = n - L, and reaching (m, n); a ballot number. A sum gives the number of such paths that also stay above m = n + K : tn(m) = 2)rn -i(K+L)(m + i(K + L)) - rm-i(K+L)-K(n + i(K + L) + K)) i~O It is amazing that tn(m) satisfies the recurrence, and both types of boundary values. The telescopic nature of this sum becomes essential if we verify the condition tn(n - L) = 0 for n 2: L.

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