Distribution of maxima and minima statistics on alternating permutations, Springer numbers, and avoidance of flat POPs

In this paper, we find distributions of the left-to-right maxima, right-to-left maxima, left-to-right minima and right-to-left-minima statistics on up-down and down-up permutations of even and odd lengths. For instance, we show that the distribution of right-to-left maxima on up-down permutations of even length is given by $(\sec (t))^{q}$. We also derive the joint distribution of the maxima (resp., minima) statistics. To accomplish this, we generalize a result of Kitaev and Remmel by deriving joint distributions involving non-maxima (resp., non-minima) statistics. Consequently, we refine classic enumeration results of Andr\'e by introducing new $q$-analogues and $(p,q)$-analogues for the number of alternating permutations. Additionally, we verify Callan's conjecture (2012) that the number of up-down permutations of even length fixed by reverse and complement equals the Springer numbers, thereby offering another combinatorial interpretation of these numbers. Furthermore, we propose two $q$-analogues and a $(p,q)$-analogue of the Springer numbers. Lastly, we enumerate alternating permutations that avoid certain flat partially ordered patterns (POPs), where the only minimum or maximum elements are labeled by the largest or smallest numbers.