Asymptotics of descent functions

TL;DR

This paper investigates the asymptotic behavior of descent functions for permutations with specific consecutive pattern constraints, providing explicit formulas, recurrence relations, and probabilistic insights.

Contribution

It extends the study of descent polynomials to general $k$-descents, deriving asymptotic formulas, integral representations, and algorithms for counting permutations avoiding these patterns.

Findings

Derived explicit asymptotic formulas for $k$-descent counts.

Established a recurrence relation for the $k$-descent number triangle.

Developed an $O(n^2)$ algorithm for computing $k$-descent functions.

Abstract

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.