By Bruck R.H.

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**Sample text**

Permutations as Linear Orders. 15 The Eulerian polynomials have several interesting properties that can be proved by purely combinatorial means. We postpone the study of those properties until the next subsection. For now, we will explore the connection between these polynomials and some inﬁnite generating functions. 21 For all positive integers n, the nth Eulerian polynomial has the alternative description An (x) = (1 − x)n+1 in xi . i≥0 Note that Euler ﬁrst deﬁned the polynomials An (x) in the above form.

For instance, I(5613427) = 5613472. It is then straightforward to verify that I either increases the number of alternating runs by one, or it decreases it by one. Therefore, I is just the involution we were looking for. /4 pairs created by the involution I, and P (p) is the permutation in P that has the smaller number of alternating runs. /4 permutations p, the following occurs. The number r(p) equals the number t1 (q) of the permutation P (q) in the pair P that is in the same pair as p and ends in an ascent.

N PROOF Let a0 , a1 , · · · , an be our sequence, and let P (x) = k=0 ak xk . Then for all roots (x, y) of the polynomial Q(x, y) = nk=0 ak xk y n−k , the ratio (x/y) must be real. (Otherwise x/y would be a non-real root of P (x)). Therefore, by Rolle’s Theorem, this also holds for the partial derivatives ∂Q/∂x and ∂Q/∂y. Iterating this argument, we see that the polynomial ∂ a+b Q/∂xa ∂y b also has real zeros, if a + b ≤ n − 1. In particular, this is true in the special case when a = j − 1, and b = n − j − 1, for some ﬁxed j.

### A survey of binary systems by Bruck R.H.

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