A Note on the Stanley Distribution

Shalosh B. EKHAD^{1}

In the second of the three AMS colloquium talks, recently masterfully^{2} presented by Algebraic Combinatorics guru Richard Stanley, he talked about the random variable “length of the largest alternating (i.e. up-down) subsequence” in a random permutation of length n. This was in analogy with the celebrated random variable “length of the largest increasing subsequence”. For the latter, it was famously proved by Baik-Deift-Johansson that the limiting distribution (after it is centralized and divided by the standard deviation) is the intriguing Tracy-Widom distribution.

Stanley humorously narrated that when he investigated the limiting distribution for the former random variable, he was hoping that the limiting distribution would be equally interesting, and has already fantasized that he would be immortalized (along with Gauss, Poisson, Cauchy, and Tracy-Widom) by having an exotic new probability distribution named after him. To his dismay, it turned out (because of a general theorem of Robin Pemantle and Herb Wilf) that the limiting distribution is the utterly boring Gaussian (aka normal) distribution. So much for Stanley’s dream of immortality (of course, there is a Stanley-Reisner Ring, but “Stanley Distribution” has a better ring to it!).

Stanley found that the expectation μ_{n} and the variance σ_{n}^{2} are given by

μ_{n} = \frac{2}{3} n + \frac{1}{6} ; σ_{n}^{2} = \frac{8}{45} n − \frac{13}{180} (for n ≥ 4 ) ,

and he deduced (from the explicit generating function that he derived, using the above-mentioned Pemantle-Wilf theorem) that

Z_{n} := \frac{X_{n} − μ_{n}}{σ_{n}}

1 Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen

Rd., Piscataway, NJ 08854-8019, USA. c/o zeilberg at math dot rutgers dot edu , http://www.math.rutgers.edu/~zeilberg/ . Jan. 20, 2010. Exclusively published in the Personal Journal of Ekhad and Zeilberger, http://www.math.rutgers.edu/~zeilberg/pj.html , and arxiv.org .

2 The three colloquium talks delivered each year during the annual meeting of the American Mathematical Society

are supposed to be the highlights of the meeting, and, in the old days, every participant was expected to attend, since they were supposed (often successfully) to explain to non-specialists, in a lucid and engaging manner, some non-trivial mathematics, and were delivered by a great mathematician. Some of these series of talks were better than others, and the three lectures by Richard Stanley, last week, were amongst the very best ones (along with Gian-Carlo Rota’s 12 years ago, and a few other ones). Unfortunately, only about 400 participants (out of more than 5000 people!) attended the first lecture (where there was no conflict with shorter talks), and only about 150 attended the second and third ones. Shame on you, 2010 “mathematicians”!

Luckily, one can still retrieve the slides of these fascinating talks from Richard Stanley’s website: http://wwwmath.mit.edu/~rstan/transparencies/ids-ams.pdf , http://www-math.mit.edu/~rstan/transparencies/altperm-ams.pdf , http://www-math.mit.edu/~rstan/transparencies/redec-ams.pdf .

converge to the normal distribution, i.e.:

Z_{n} → N (in distribution).

This is equivalent to the statement that the moments of Z_{n} converge to the moments of the Gaussian distribution (0 for odd moments and 1 · 3 · . . . · (2r − 1) = \frac{(2r)!}{2^{r}r!} for the even, 2r-th, moment), to wit:

α_{2r}(Z_{n}) = \frac{(2r)!}{2^{r}r!} + o(1) ,

α_{2r+1}(Z_{n}) = o(1) .

So the leading terms for the asymptotics for the moments are indeed the boring, normal ones. But, thanks to the amazing Maple package HISTABRUT, written by my master, Doron Zeilberger, that will soon be released with an accompanying article explaining how to use it, one can get:

α_{2r}(Z_{n}) = \frac{(2r)!}{2^{r}r!} (1 + \frac{1}{1764} \frac{r (r − 1) (10 r − 713)}{n} + O(\frac{1}{n^{2}} )) ,

α_{2r+1}(Z_{n}) =

−\frac{\sqrt{10}}{43} \frac{(2r)!}{2^{r}r!} \frac{1}{\sqrt{n}} (r − 1 + \frac{1}{931392} \frac{(r − 1) (1760 r^{3} − 381744 r^{2} + 1430752 r + 150351)}{n} + O(\frac{1}{n^{2}} )) .

These formulas tell you how fast the moments of the (discrete) Stanley Distribution (Z_{n}) converge to those of the normal distribution.

Zeilberger’s Maple package HISTABRUT can yield even higher-order asymptotics, but who cares?. Also, I have to admit that the above formulas are not yet rigorous, but they are certainly rigorizable. So there exists a rigorous proof, but who cares?