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On lattice path counting by major and descents. (English) Zbl 0981.05510

Summary: \(n\)-dimensional lattice paths which do not touch the hyperplanes \(x(i)-x(i+1)=-1\), \(i=1,2,...,(n-1)\) and \(x(n)-x(1)=-1-K\) are enumerated by MacMahon’s major index and variations of the major index. A formula involving determinants is obtained. For \(n=2\) we also present a formula for counting these lattice paths simultaneously by major and descents.
This paper is a summary of the articles that appeared in: Eur. J. Comb. 14, 43-51 (1993; Zbl 0777.05008), Discrete Math. 126, 195-208 (1994; Zbl 0790.60014).

MSC:

05A15 Exact enumeration problems, generating functions
60C05 Combinatorial probability
62E15 Exact distribution theory in statistics
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