Counting lattice chains and Delannoy paths in higher dimensions |
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Authors: | John S Caughman Charles L Dunn Nancy Ann Neudauer Colin L Starr |
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Institution: | aFariborz Maseeh Department of Mathematics & Statistics, Portland State University, Box 751, Portland, OR 97202, United States;bDepartment of Mathematics, Linfield College, 900 SE Baker Street, McMinnville, OR 97128, United States;cDepartment of Mathematics & Computer Science, Pacific University, Forest Grove, OR 97116, United States;dDepartment of Mathematics, Willamette University, Salem, OR 97301, United States |
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Abstract: | Lattice chains and Delannoy paths represent two different ways to progress through a lattice. We use elementary combinatorial arguments to derive new expressions for the number of chains and the number of Delannoy paths in a lattice of arbitrary finite dimension. Specifically, fix nonnegative integers n1,…,nd, and let L denote the lattice of points (a1,…,ad)∈Zd that satisfy 0≤ai≤ni for 1≤i≤d. We prove that the number of chains in L is given by where . We also show that the number of Delannoy paths in L equals Setting ni=n (for all i) in these expressions yields a new proof of a recent result of Duchi and Sulanke 9] relating the total number of chains to the central Delannoy numbers. We also give a novel derivation of the generating functions for these numbers in arbitrary dimension. |
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Keywords: | Lattice chain Delannoy number Generating function |
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