Permutations Generated by a Depth 2 Stack and an Infinite Stack in Series are Algebraic

We prove that the class of permutations generated by passing an ordered sequence 12 . . . n through a stack of depth 2 and an infinite stack in series is in bijection with an unambiguous context-free language, where a permutation of length n is encoded by a string of length 3n. It follows that the sequence counting the number of permutations of each length has an algebraic generating function. We use the explicit context-free grammar to compute the generating function: ∑ n>0 cnt n = (1 + q) ( 1 + 5q − q2 − q3 − (1− q) √ (1− q2)(1− 4q − q2) )