Using the Entringer numbers to count the alternating permutations according a new parameter (Q2724866)

An up-down (resp. down-up) permutation of order \(n\) is a permutation \((a_1,\dots, a_n)\) of \(\{1,\dots, n\}\) such that for every \(i\), \(1\leq i\leq n-1\), \(a_i< a_{i+1}\) if and only if \(i\) is odd (resp. even). An alternating permutation is either an up-down or a down-up permutation. The Entringer number \({\mathcal E}^k_n\) [\textit{R. C. Entringer}, A combinatorial interpretation of the Euler and Bernoulli numbers, Nieuw Arch. Wiskd., III. Ser. 14, 241-246 (1966; Zbl 0145.01402)] is the number of up-down permutations of order \(n+1\) starting with \(n-k+1\) and also the number of down-up permutations of order \(n+1\) starting with \(k+1\).NEWLINENEWLINENEWLINEIn the article under review, the deviation of a permutation \((a_1,\dots, a_n)\) is defined to be \(a_1- a_n\) and its separation to be \(|a_1- a_n|\) and it is proved that the number of alternating permutations of order \(n\) and deviation \(d\) is equal to the number of up-down (or down-up) permutations of order \(n\) and separation \(d\) which, in turn, is equal to \({\mathcal E}^{n-1}_{n-d}\) if \(n\) is even and to \((n-d){\mathcal E}^{n-2}_{n-1-d}\) if \(n\) is odd.