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By Alfonso García Pérez

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From Corollary 1 we see that <-i)*-4(°)) is equal to the (1, d)th element of Mk \ Some minor simplification helps us to complete the proof. Taking into account both restrictions, let gk(A(n)\B(n)) be the number of paths with upper restriction ^(n) and lower restriction 4. Higher Dimensional Paths 51 B(n). We know that x(r) < a(r). However the nondecreasing condition on 6(r) leads us to x(r) > b(ru . . , rt_ l5 rt + 1, ri+ u . . , rk) if the path has moved from the ( r l 5 . . , rf_ l5 r^ + 1, r £ + 1 5 .

Stocks, D. , Lattice paths in E3 with diagonal steps, Canad. Math. Bull. 10 (1967), 653-658. , A generalization of the ballot problem and its application in the theory of queues, J. Amer. Statist. Assoc. 57 (1962), 327-337. , Ballot Problems, Z. Wahrsch. Verxv. Gebiete 1 (1962), 154-158. , Combinatorial Methods in the Theory oj Stochastic Process. Wiley, New York, 1967. Thrall, R. M. A combinatorial problem, Michigan Math. J. 1 (1952), 81-88. Whittaker, E. , and Watson, G. , A Course of Modern Analysis, 4th ed.

Nu such that the initial point of the path segment on each of the planes is joined by the terminal point of the path segment in the preceding plane by a step along the X r a x i s . Thus any path can be uniquely represented by the matrix ( x(0,0) . . x(0, 7 l ) \ * ( W i ) . . x(l,j 2 ) o * W - , 0 \ x(nl9jni) ... x(nun2)/ with x(nun2) = n0, 0 = j 0

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