A symmetrical Eulerian identity

Abstract

Eulerian numbers, introduced by Euler in 1736 [5], while not as ubiquitous as the more familiar Bernoulli numbers, Stirling numbers, harmonic numbers, or binomial coefficients, nevertheless arise in a variety of contexts in enumerative combinatorics, for example, in the enumeration of permutations with a given number of descents [7]. Because the recurrence for Eulerian numbers is a bit more complicated than for many other families of special numbers, and because they increase in size rather rapidly, it was stated in [7] that, “We don’t expect the Eulerian numbers to satisfy as many simple identities.” Nevertheless, the following identity is rather elegant and appears to be new.