Enumeration of up-down sequences

AbstractGeneralizing the notion of up-down permutations, the author considers sequences σ = (a1, a2, , αN) of length N = s2 + s2 ++ sn, where αi ∈ {1, 2,n } and the element j occurs exactly sj 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