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. We recover and generalize a result by Carlitz and Scoville, obtained in 1975, stating that the distribution of left-to-right maxima on down-up permutations of even length is given by ${(\sec (t\left)\right)}^q$. We also derive the joint distribution of the maxima (resp., minima) statistics, extending the scope of the respective results of Carlitz and Scoville, who obtain them in terms of certain systems of PDEs and recurrence relations. 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 André by introducing new q-analogues and $(p,q)$-analogues for the number of alternating permutations.

Additionally, we verify Callan's conjecture (2012) that up-down permutations of even length fixed by reverse and complement are counted by 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.