Abstract
Generalizing the notion of up-down permutations, the author considers sequences σ = (a 1, a 2, , α N) of length N = s 2 + s 2 ++ s n, where α i ∈ {1, 2,n } and the element j occurs exactly s j times. The repeated elements of a are labeled i, i′, i″, and it is assumed that they occur in a m natural order. Generating functions for the number of up-down sequences are constructed. Making use of the generating functions, explicit formulas for the number of up-down sequences are obtained.
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