  In 1916, MacMahon showed that permutations in $S_n$ with a fixed descent set
$I$ are enumerated by a polynomial $d_I(n)$. Diaz-Lopez, Harris, Insko, Omar,
and Sagan recently revived interest in this descent polynomial, and suggested
the direction of studying such enumerative questions for other consecutive
patterns (descents being the consecutive pattern $21$). Zhu studied this
question for the consecutive pattern $321$. We continue this line of work by
studying the case of any consecutive pattern of the form $k,k-1,\ldots,1$,
which we call a $k$-descent. In this paper, we reduce the problem of
determining the asymptotic number of permutations with a certain $k$-descent
set to computing an explicit integral. We also prove an equidistribution
theorem, showing that any two sparse $k$-descent sets are equally likely.
  Counting the number of $k$-descent-avoiding permutations while conditioning
on the length $n$ and first element $m$ simultaneously, one obtains a number
triangle $f_k(m,n)$ with some useful properties. For $k=3$, the $m=1$ and $m=n$
diagonals are OEIS sequences A049774 and A080635. We prove a $k$th difference
recurrence relation for entries of this number triangle. This also leads to an
$O(n^2)$ algorithm for computing $k$-descent functions.
  Along the way to these results, we prove an explicit formula for the
distribution of first elements of $k$-descent-avoiding permutations, as well as
for the joint distribution of first and last elements. We also develop an
understanding of discrete order statistics. In our approach, we combine
algebraic, analytic, and probabilistic tools. A number of open problems are
stated at the end.
