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The major counting of nonintersecting lattice paths and generating functions for tableaux. (English) Zbl 0830.05003

Mem. Am. Math. Soc. 552, 109 p. (1995).
The author develops a theory of counting nonintersecting lattice paths by the major index and generalizations of it which he calls strange major indices. He obtains determinantal expressions involving \(q\)-binomials for the corresponding generating functions for families of nonintersecting lattice paths with given starting points and given final points, where the starting points lie on a line parallel to \(x+ y= 0\). The original motivation of doing major counting lay in a problem of Choi and Gouyou- Beauchamps which is solved in this paper. The author gives an explicit product for the generating function for tableaux with \(p\) odd rows, with at most \(C\) columns, and with parts between 1 and \(n\). Further, he computes the generating function for the same kind of tableaux which in addition have only odd parts. He also obtains a closed form for the generating function for symmetric plane partitions with at most \(n\) rows, with parts between 1 and \(C\), and with \(p\) odd entries on the main diagonal. By summing with respect to \(p\) he obtains new proofs of the Bender-Knuth and MacMahon conjectures, which were first proved by Andrews, Gordon and Macdonald. The link between nonintersecting lattice paths and tableaux is given by variations of the Knuth correspondence.
Reviewer: J.Cigler (Wien)

MSC:

05A15 Exact enumeration problems, generating functions
05A10 Factorials, binomial coefficients, combinatorial functions
05A17 Combinatorial aspects of partitions of integers
05A30 \(q\)-calculus and related topics
05E10 Combinatorial aspects of representation theory
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