The Asymptotics of Almost Alternating Permutations

The goal of this paper is to study the asymptotic behavior of almost alternating permutations, that is, permutations that are alternating except for a finite number of exceptions. Let β l1 lk denote the number of permutations which consist of l1 ascents, l2 descents, l3 ascents, and so on. By combining the Viennot triangle and the boustrophedon transform, we obtain the exponential generating function for the numbers β L 1n−m−1 , where L is a descent–ascent list of size m. As a corollary we have β L 1n−m−1 ∼ c L · En, where En = β 1n−1 denotes the nth Euler number and c L is a constant depending on the list L. Using these results and inequalities due to Ehrenborg–Mahajan, we obtain β 1 2 1 ∼ 2/π · En, when min a b tends to infinity and where n = a+ b+ 3. From this result we obtain that the asymptotic behavior of β L1 1 L2 1 L3 is the product of three constants depending respectively on the lists L1, L2, and L3, times the Euler number Ea+b+m+1, where m is the sum of the sizes of the Li’s.  2002 Elsevier Science (USA)