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,...,l_k) 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, 1^n-m-1), where L is a descent-ascent list of size m. As a corollary we have β(L, 1^n-m-1) ∼ c(L)·E_n, where E_n = β(1^n-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^a,2,1^b) ∼ 2/π·E_n, 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^a, L₂, 1^b, L₃) is the product of three constants depending respectively on the lists L₁, L₂, and L₃, times the Euler number E_a+b+m+1, where m is the sum of the sizes of the L_i's.