Euler Numbers and Skew-Hooks
Richard M. Grassl · Mathematics Magazine · 1993
The Euler number En is the number of permutations of the integers 1,2,3,..., n that first rise, then alternately fall and rise. For n = 4 the permutations of 1,2,3,4 satisfying this property are 1324, 1423, 2314, 2413, and 3412. So we see that E4 = 5. If one throws in the term Eo = 1, the first eleven Euler numbers 1,1,1,2,5,16,61, 272,1385,7936,50521 are given as the beginning terms of sequence #587 in the N. J. A. Sloane Handbook of Integer Sequences '[9]. Such permutations, sometimes called up-down permutations, were first studied in 1879 by D. Andre [2]; they make extensive appearances in the literature, showing up in a variety of contexts. One frequently used recursive technique for finding En involves the recurrence relation En+-1 2= i (EjEn-i along with the initial conditions Eo = 1, E1 = 1. (The reader is invited to try to prove this recurrence.) One disadvantage in using this scheme, of course, is that in computing En+ 1, one needs to know all the values EO E1 E2,.., En. In this paper, we first highlight several settings in which the Euler numbers play prominent roles and then give a method for finding a relatively simple, nonrecursive formula for En) employing a technique that is accessible to most undergraduates.