Doron Gepner’s Statistics on Words in {1, 2, 3}^{∗} is (Most Probably) Asymptotically Logistic

Doron ZEILBERGER

Dedicated to my friend and hero, Doron Gepner (b. March 31, 1956), on his 60^{th} birthday. “Doron le Doron me-Doron”.

Preface

I first met Doron Gepner in 1980, when he was a Physics graduate student at the Weizmann Institute of Science, and I was a young khoker bakhir. Already then Doron was a legend, since he was the first person in Israel, as far as I know, to have solved Rubik’s cube completely from scratch, using group-theoretical methods. I was so impressed that I asked him to present a guest-lecture in my graduate combinatorics class, and the students loved it.

Doron then went on to do seminal work in theoretical physics, that, unfortunately, is over my head. But the part that is really interesting to me is his current work, greatly generalizing the celebrated Rogers-Ramanujan identities, and giving lots of new insight. I am sure that this work will lead to many future gems.

The purpose of the present note is to present a present (Doron in Hebrew)^{1}, from one Doron to another, by paying an old debt. In 1987, when he was a postdoc at Princeton University, Doron introduced a new permutation statistic (see below), that came up in his work in string theory and conformal field theory. It so happened that at the time, my friend and collaborator, the eminent French combinatorialist, Dominique Foata, visited me. Foata is the world’s greatest expert on permutation (and word-) statistics (and coined the term!), so it was only natural that we both got intrigued and tried to investigate Gepner’s new statistics, that we christened gep, in analogy with the classical statistics inv, maj, and des (see below). We had some preliminary results, but not enough for a paper. This was due to the fact that my beloved servant, Shalosh B. Ekhad, was not yet born. Now, almost thirty years later, it is a good opportunity to revisit Doron Gepner’s difficult statistics and harness the full power of my silicon servant, and of Maple, to study it seriously.

Important note: All the results in this paper were gotten by using the Maple package GEPNER.txt, available, free of charge, from the url

http://www.math.rutgers.edu/~zeilberg/tokhniot/GEPNER.txt .

Sample input and output files may be gotten from the front of this article:

1 Exactly thirty years ago, on March 31, 1986, my wife Jane and I were at Doron Gepner’s 30th birthday party, (in Princeton) that Ida

(Doron’s wife) organized in their place. Another guest was a colleague of Doron, an Egyptian postdoc, and we pointed out to him that we have the same name, and that it means a “gift”, (presumably “God’s gift”), to which he retorted “ in your cases it seems to be the devil’s gifts”.

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/gepner.html .

A crash course on Permutation and Word Statistics

A permutation statistics is an integer-valued function on the set of permutations. The most famous one is the number of inversions, inv(π), (that shows up in the definition of the determinant of a square matrix)

inv( π_{1} … π_{n}) := ∑_{1≤i<j≤n} χ(π_{i} > π_{j}) ,

where χ(S) is 1 or 0, according to whether S is true or false, respectively.

Almost as famous is Major Percy Alexander MacMahon’s statistics, maj(π), called the “major index”

maj( π_{1} … π_{n}) := ∑_{i=1}^{n−1} i χ(π_{i} > π_{i+1}) .

The generating functions according to inv and maj are both given by [n]! := 1(1 + q)(1 + q + q^{2}) ⋯ (1 + q + … + q^{n−1}) = (1 − q) … (1 − q^{n})/(1 − q)^{n}) as proved by Netto and MacMahon respectively. In particular the permutation statistics inv and maj are equally distributed.

Permutations of length n may be viewed as words in the “alphabet” {1, 2, …, n} with exactly one occurrence of each letter. The above definitions of inv and maj make perfect sense when defined on words, of any length, in the same alphabet, where repetitions (and omissions) are welcome.

Let W(a_{1}, …, a_{n}) be the set of words in the alphabet {1, …, n} with a_{1} occurrences of 1, a_{2} occurrences of 2, …, a_{n} occurrences of n. MacMahon proved (Theorems 3.6 and 3.7 in [A])

∑_{w∈W(a_{1},...,a_{n})} q^{inv(w)} = \frac{[a_{1} + … + a_{n}]!}{[a_{1}]! … [a_{n}]!} ,

∑_{w∈W(a_{1},...,a_{n})} q^{maj(w)} = \frac{[a_{1} + … + a_{n}]!}{[a_{1}]! … [a_{n}]!} ,

(where, as mentioned above, [m]! := (1 − q)(1 − q^{2}) ⋯ (1 − q^{m})/(1 − q)^{m}).

In particular they are still equally distributed, and Dominique Foata ([F]) gave a gorgeous bijective proof.

Asymptotic Normality

Most (but not all) combinatorial statistics (naturally parametrized by one or more integer parameters), for example tossing a (fair or loaded) coin n times and observing the number of Heads, are asymptotically normal, that means that (for the sake of simplicity let’s only consider the one parameter case), if you call the sequence X_{n}, figure out its mean, μ_{n}, (usually extremely easy) (aka as average, aka as expectation), call it μ_{n}, and its variance, m_{2}(n) (also, usually, fairly easy), and define the standardized sequence of random variables

Z_{n} := \frac{X_{n} − μ_{n}}{\sqrt{m_{2}(n)}} ,

then the sequence {Z_{n}}, converges, in distribution, to the good-old normal distribution (aka Gaussian distribution) whose probability density function is, famously, e^{−x^{2}/2}/\sqrt{2π}. A good way to prove this is to discover explicit expressions for the moments (about the mean), m_{r}(n) (or at least the leading terms), and prove that the standardized moments, m_{r}(n)/m_{2}(n)^{r/2} tend, as n goes to infinity, to the moments of the standard normal distribution, that equal 0 when r is odd and 1 · 3 ⋯ (r − 1), when r is even. This approach can be often taught to a computer, see [Z1][Z2]. The asymptotic normality of inv, for the two-lettered case, was first proved by Mann and Whitney [MW], and for the general case by Persi Diaconis[D] (and reproved in [CJZ]).

Enter Doron Gepner’s Statistics

One way to look at an inversion in a word w_{1} … w_{m} is as the number of pairs of letters w_{i}w_{j} with 1 ≤ i < j ≤ m, whose reduction is the permutation of length 2, 21, the only odd permutation of length 2.

This leads naturally to the analog for three-letter subwords. Given a word w = w_{1} … w_{m} (in any alphabet) consider the set of triples 1 ≤ i < j < k ≤ m such that w_{i}w_{j}w_{k} reduces to one of the three odd permutations of length 3, namely one of the members of {132, 213, 321}.

And indeed, this came up (naturally!), in Doron Gepner’s[G] deep work in conformal field theory, and leads to the following definition.

Definition. The Gepner statistics on a word w = w_{1} … w_{m} on any finite, totally ordered, alphabet, denoted by gep(w), is defined by

gep(w_{1} … w_{m}) := ∑_{1≤i<j<k≤m} χ(w_{i} < w_{k} < w_{j} OR w_{j} < w_{i} < w_{k} OR w_{k} < w_{j} < w_{i}) .

It seems that the Gepner statistics, gep, is much harder to study than the inversion number,inv. and it is extremely unlikely that a ‘nice’ (or even ‘ugly’) explicit formula for the generating function exists. But it is still interesting to find out whether it is asymptotically normal, and if not, to determine the limiting distribution.

Failing an explicit formula for the generating function, the best that we can hope for is an algorithm to compute the first few terms of the generating functions for the Gepner statistics. Using the method of [Z1] and [Z2], we can then infer explicit (rigorously proved!) polynomial expressions for the first few moments, and try to see what is going on. It is easy to see by a (fully rigorous) handwaving argument that each moment is always some polynomial, and it is also easy to bound the degree of the r-th moment to be 3r (in the cases considered here). The degree of the polynomial expression, in n, of the r-th moment-about-the mean is lower, in fact, in the two cases considered below, it happens to be 2r. Since a polynomial of degree d is uniquely determined by d + 1 distinct values, an experimental-yet-rigorous approach would be to find numerical values and then “fit the data”.

Due to the complexity of the Gepner statistics (and our finite time) we will only consider the cases of permutations, i.e. W(1, …, 1) (with n 1s), and words in the three-letter alphabet {1, 2, 3} with n occurrences of each of the letters 1, 2, and 3, i.e. W(n, n, n).

First Surprise: The Gepner Statistics on Permutations is NOT Asymptotically Normal

Define the Gepner polynomials of the first kind, G_{n}(q), to be the polynomial

G_{n}(q) := ∑_{π∈S_{n}} q^{gep(π)} .

Here are the first 8 Gepner polynomials

G_{1}(q) = 1 , G_{2}(q) = 2 , G_{3}(q) = 3q + 3 , G_{4}(q) = 4q^{4} + 16q^{2} + 4 ,
G_{5}(q) = 5 q^{10} + 25 q^{7} + 25 q^{6} + 10 q^{5} + 25 q^{4} + 25 q^{3} + 5 ,
G_{6}(q) = 6 q^{20} + 36 q^{16} + 72 q^{14} + 138 q^{12} + 216 q^{10} + 138 q^{8} + 72 q^{6} + 36 q^{4} + 6 ,

G_{7}(q) = 7 q^{35} + 49 q^{30} + 98 q^{27} + 49 q^{26} + 98 q^{25} + 98 q^{24} + 147 q^{23} + 441 q^{22} + 308 q^{21} + 196 q^{20}

+490 q^{19} + 539 q^{18} + 539 q^{17} + 490 q^{16} + 196 q^{15} + 308 q^{14} + 441 q^{13} + 147 q^{12} + 98 q^{11} + 98 q^{10}

+49 q^{9} + 98 q^{8} + 49 q^{5} + 7 ,

G_{8}(q) = 8 q^{56} + 64 q^{50} + 128 q^{46} + 288 q^{44} + 128 q^{42} + 896 q^{40} + 1344 q^{38} + 1600 q^{36} + 3200 q^{34}

+4792 q^{32} + 4352 q^{30} + 6720 q^{28} + 4352 q^{26} + 4792 q^{24} + 3200 q^{22} + 1600 q^{20} + 1344 q^{18} + 896 q^{16}

+128 q^{14} + 288 q^{12} + 128 q^{10} + 64 q^{6} + 8 .

By looking at the Taylor expansion around q = 1, one can get the so-called factorial moments, and from them the usual moments, and from them the moments about the mean (see [Z1]), and Shalosh B. Ekhad found that the average is

μ_{1}(n) = \frac{1}{12} n (n − 1) (n − 2) ,

(of course, this is obvious, since it must be \binom{n}{3}/2 [why?]). Less trivially (but still humanly doable), it found that the variance, m_{2}(n), is given by the following polynomial expression

m_{2}(n) = \frac{1}{72} n^{2} (n − 1) (n − 2) ,

in particular, the standard deviation is O(n^{2}), while the average is O(n^{3}), so we have, at least, concentration about the mean.

Of course, by symmetry, m_{3}(n) = 0, but much less trivially, we have the following rigorously proved formula for the fourth moment

m_{4}(n) = \frac{1}{43200} n^{3} (n − 1) (n − 2) (29 n^{3} − 71 n^{2} − 26 n − 16) .

From this it follows that the standardized fourth moment, aka as the kurtosis, equals

α_{4}(n) = \frac{\frac{1}{43200} n^{3} (n − 1) (n − 2) (29 n^{3} − 71 n^{2} − 26 n − 16)}{(\frac{1}{72} n^{2} (n − 1) (n − 2))^{2}}

and taking the limit as n → ∞, we get that the limiting kurtosis is

\frac{87}{25} = 3.48 ,

and not 3, so the Gepner statistics is not asymptotically normal, and happens to be leptokurtic (statisticians’ big word for saying that the kurtosis exceeds 3). We have no clue about the limiting distribution, and would love to know it.

The difficulty, from a computational point of view, is that we have no efficient way to get more Gepner polynomials than by very naive brute force. My former PhD student, Brian Nakamura([N]), found more efficient algorithms for related problems, but this needs more work. On the other hand, for the case below, where we look at the Gepner statistics defined on W(n, n, n), we can do much better than brute force, and squeeze out explicit expressions for more moments.

Second Surprise: The Gepner Statistics on Words in {1, 2, 3}^{∗} is Asymptotically Logistic

Define the Gepner polynomials of the second kind, g_{n}(q), to be the polynomial

g_{n}(q) := ∑_{w∈W(n,n,n)} q^{gep(w)} .

(Recall that W(n, n, n) is the set of words with n occurrences each of the letters 1, 2, and 3).

Here are the first five Gepner polynomials of the second kind.

g_{1}(q) = 3q + 3 , g_{2}(q) = 6 q^{8} + 21 q^{6} + 36 q^{4} + 21 q^{2} + 6 ,

g_{3}(q) = 9 q^{27} + 27 q^{24} + 108 q^{21} + 264 q^{18} + 432 q^{15} + 432 q^{12} + 264 q^{9} + 108 q^{6} + 27 q^{3} + 9 , g_{4}(q) = 12 q^{64} + 36 q^{60} + 108 q^{56} + 336 q^{52} + 870 q^{48} + 2016 q^{44} + 4041 q^{40} + 6252 q^{36} + 7308 q^{32} ,

+6252 q^{28} + 4041 q^{24} + 2016 q^{20} + 870 q^{16} + 336 q^{12} + 108 q^{8} + 36 q^{4} + 12 ,

g_{5}(q) = 15 q^{125}+45 q^{120}+135 q^{115}+330 q^{110}+900 q^{105}+2115 q^{100}+4710 q^{95}+10230 q^{90}+21195 q^{85}+40290 q^{80} +69423 q^{75}+102780 q^{70}+126210 q^{65}+126210 q^{60}+102780 q^{55}+69423 q^{50}+40290 q^{45}+21195 q^{40}+10230 q^{35}

+4710 q^{30} + 2115 q^{25} + 900 q^{20} + 330 q^{15} + 135 q^{10} + 45 q^{5} + 15 .

How to get More Gepner Polynomials? Consider, for motivation, computing the generating function, according to inv, of the words in W(a_{1}, a_{2}, a_{3}), i.e. how can we generate many terms of the polynomials

f(a_{1}, a_{2}, a_{3}) := ∑_{w∈W(a_{1},a_{2},a_{3})} q^{inv(w)} .

Suppose that we did not know MacMahon’s explicit formula [a_{1} + a_{2} + a_{3}]!/([a_{1}]![a_{2}]![a_{3}]!) for it. A natural approach would be to use Dynamical Programming and try to express f(a_{1}, a_{2}, a_{3}) in terms of f with smaller arguments. Note that for w′ ∈ W(a_{1} − 1, a_{2}, a_{3}),

inv(w′1) = inv(w′) + a_{2} + a_{3} ,

since appending a 1 to w′ creates a_{2} new inversions due to the a_{2} 2s, and a_{3} new inversions due to the a_{3} 3’s. Similarly, for w′ ∈ W(a_{1}, a_{2} − 1, a_{3}),

inv(w′2) = inv(w′) + a_{3} ,

since appending a 2 to w′ creates a_{3} new inversions due to the a_{3} 3’s. Finally, for w′ ∈ W(a_{1}, a_{2}, a_{3} − 1),

inv(w′3) = inv(w′) ,

since appending a 3 to w′ does not create any new inversions. Since W(a_{1}, a_{2}, a_{3}) = W(a_{1} − 1, a_{2}, a_{3})1 ∪ W(a_{1}, a_{2} − 1, a_{3})2 ∪ W(a_{1}, a_{2}, a_{3} − 1)3 , we have

f(a_{1}, a_{2}, a_{3}) := ∑_{w∈W(a_{1},a_{2},a_{3})} q^{inv(w)}

= ∑_{w′∈W(a_{1}−1,a_{2},a_{3})} q^{inv(w′)+a_{2}+a_{3}} + ∑_{w′∈W(a_{1},a_{2}−1,a_{3})} q^{inv(w′)+a_{3}} ∑_{w′∈W(a_{1},a_{2},a_{3}−1)} q^{inv(w′)}
q^{a_{2}+a_{3}}f(a_{1} − 1, a_{2}, a_{3}) + q^{a_{3}}f(a_{1}, a_{2} − 1, a_{3}) + f(a_{1}, a_{2}, a_{3} − 1) .

We have just established the recurrence

f(a_{1}, a_{2}, a_{3}) = q^{a_{2}+a_{3}}f(a_{1} − 1, a_{2}, a_{3}) + q^{a_{3}}f(a_{1}, a_{2} − 1, a_{3}) + f(a_{1}, a_{2}, a_{3} − 1) ,

that would enable us to crank out many f(n, n, n).

In fact, this recurrence can be also used to prove MacMahon’s formula, by verifying that [a_{1} + a_{2} +a_{3}]!/([a_{1}]![a_{2}]![a_{3}]!) also satisfies the same recurrence (a trivial high-school algebra verification, even for humans), and that the boundary conditions agree.

How can we generalize this argument for computing

F (a_{1}, a_{2}, a_{3}) := ∑_{w∈W(a_{1},a_{2},a_{3})} q^{gep(w)} .

Let’s use, once again the decomposition

W(a_{1}, a_{2}, a_{3}) = W(a_{1} − 1, a_{2}, a_{3})1 ∪ W(a_{1}, a_{2} − 1, a_{3})2 ∪ W(a_{1}, a_{2}, a_{3} − 1)3 .

Let’s consider a member, w, of W(a_{1}, a_{2}, a_{3}), that ends with the letter 1, so w = w′1 for w′ ∈ W(a_{1} − 1, a_{2}, a_{3}). Appending 1 to w′ introduces new contributions to the gep statistics due to all the occurrences of the pairs 32.

Let’s consider a member, w, of W(a_{1}, a_{2}, a_{3}), that ends with the letter 2, so w = w′2 for w′ ∈ W(a_{1}, a_{2} − 1, a_{3}). Appending 2 to w′ introduces new contributions to the gep statistics due to the occurrences of the pairs 13.

Let’s consider a member, w, of W(a_{1}, a_{2}, a_{3}), that ends with the letter 3, so w = w′3 for w′ ∈ W(a_{1}, a_{2}, a_{3} − 1). Appending 3 to w′ introduces new contributions to the gep statistics due to the occurrences of the pairs 21.

This forces us to keep track of the number of occurrences of the pairs 32, 13, and 21.

So we have

gep(w′1)−gep(w′) = #32(w′) , gep(w′2)−gep(w′) = #13(w′) , gep(w′3)−gep(w′) = #21(w′) .

We also have

#32(w′1) − #32(w′) = 0 , #32(w′2) − #32(w′) = a_{3} , #32(w′3) − #32(w′) = 0 ,
#13(w′1) − #13(w′) = 0 , #13(w′2) − #13(w′) = 0 , #13(w′3) − #13(w′) = a_{1} ,
#21(w′1) − #21(w′) = a_{2} , #21(w′2) − #21(w′) = 0 , #21(w′3) − #21(w′) = 0 .

Defining the family of polynomials in q and the three catalytic variables t_{32}, t_{13}, and t_{21}:

F(a_{1}, a_{2}, a_{3}; q, t_{32}, t_{13}, t_{21}) := ∑_{w∈W(a_{1},a_{2},a_{3})} q^{gep(w)}t_{32}^{#32(w)}t_{13}^{#13(w)}t_{21}^{#21(w)} ,

we get the functional-recurrence equation

F(a_{1}, a_{2}, a_{3}; q, t_{32}, t_{13}, t_{21}) =

t_{21}^{a_{2}}·F(a_{1}−1, a_{2}, a_{3}; q, qt_{32}, t_{13}, t_{21})+t_{32}^{a_{3}}·F(a_{1}, a_{2}−1, a_{3}; q, t_{32}, qt_{13}, t_{21})+t_{13}^{a_{1}}·F(a_{1}, a_{2}, a_{3}−1; q, t_{32}, t_{13}, qt_{21}) .

At the end of the day, we plug-in t_{32} = 1, t_{13} = 1, t_{21} = 1, and get our object of desire

F (a_{1}, a_{2}, a_{3}; q) = F(a_{1}, a_{2}, a_{3}; q, 1, 1, 1) .

and finally, we can get quite a few Gepner polynomials of the second kind from

g_{n}(q) = F (n, n, n; q) .

But we can do better! If we are only interested in, say, the first 20 moments, then we don’t need the full polynomials. Since the first r factorial (and hence true) moments can be gotten from the first r Taylor coefficients, about q = 1, of g_{n}(q), we can do a change variables

q = 1 + p , t_{32} = 1 + x , t_{13} = 1 + y , t_{21} = 1 + z ,

and define

H(a_{1}, a_{2}, a_{3}; p, x, y, z) := F(a_{1}, a_{2}, a_{3}; 1 + p, 1 + x, 1 + y, 1 + z) .

We get a corresponding functional recurrence equation for H, and now at each step, we can truncate, and only retain the terms in p, x, y, z of total degree ≤ r, and of course, at the end of the day, plug-in x = 0, y = 0, z = 0. This enabled us to get many more truncated Gepner polynomials, and enabled us to find explicit expressions for the first 12 moments. This leads to

Theorem: The average of the Gepner statistics, defined on the set of words in {1, 2, 3} with n occurrences each of 1, 2, and 3, is (of course)

μ_{n} = \frac{n^{3}}{2} .

The variance is

m_{2}(n) = \frac{n^{4}}{4} .

Of course, all the odd moments are zero. We also have, for the fourth through 12th moments we have the following polynomial expressions.

m_{4}(n) = \frac{1}{80} n^{7} (−16 + 21 n) .

m_{6}(n) = \frac{1}{448} n^{9} (279 n^{3} − 656 n^{2} + 512 n − 128) .

m_{8}(n) = \frac{1}{1280} n^{11} (3429 n^{5} − 16480 n^{4} + 32512 n^{3} − 32512 n^{2} + 16384 n − 3328) .

m_{10}(n) = \frac{1}{11264} n^{13} (3 n − 4) ·

(68985 n^{6} − 469716 n^{5} + 1391760 n^{4} − 2251584 n^{3} + 2072832 n^{2} − 1022208 n + 209920) .

m_{12}(n) = n^{15}(\frac{343717911}{1863680} n^{9} − \frac{265635477}{116480} n^{8}

+\frac{93771627}{7280} n^{7}−\frac{45012561}{1040} n^{6}+\frac{43093563}{455} n^{5}−\frac{77940021}{560} n^{4}+\frac{62196357}{455} n^{3}−\frac{39213532}{455} n^{2}+\frac{14354176}{455} n−\frac{463661}{91}) .

Now let’s standardize and take the limit as n goes to infinity. The limiting kurtosis is

κ_{4} = lim_{n→∞} \frac{m_{4}(n)}{m_{2}(n)^{2}} = \frac{21}{5} = 4.2 ,

so now the Gepner statistics is even more leptokurtic.

The standardized sixth moment tends to

κ_{6} = lim_{n→∞} \frac{m_{6}(n)}{m_{2}(n)^{3}} = \frac{279}{7} .

The standardized eighth moment tends to

κ_{8} = lim_{n→∞} \frac{m_{8}(n)}{m_{2}(n)^{4}} = \frac{3429}{5} .

Similarly, κ_{10} = 206955/11 and κ_{12} = 343717911/455. A quick google search for “21/5, 279/7 , 3429/5” revealed that these are the 4th, 6th, and 8th moments of the standard Logistic distribution, whose probability density function is

\frac{π}{4\sqrt{3}}sech^{2}(\frac{πx}{2\sqrt{3}}) ,

and whose moments are (2^{n} − 2) · |B_{n}| · 3^{n/2}, where B_{n} are the Bernoulli numbers.

So the first 12 moments perfectly agree, and I am sure that they all do, and hence I can assert with great confidence the interesting

Fact: Doron Gepner’s Statistics on the set of words W(n, n, n) has mean \frac{n^{3}}{2} , variance \frac{n^{4}}{4} and its limiting (scaled) distribution is the logistic distribution.

Let me conclude with

YOM HULEDET SHISHIM SAMEACH, DORON! .

Acknowledgment

I wish to thank my beloved servant, Shalosh B. Ekhad, for its diligent computations, and to Dominique Foata for helpful discussions way back in 1987.

References

[A] George Andrews, “The Theory of Partitions”, Cambridge University Press, 1984. Originally published, in 1976, by Addison-Wesley.

[BZ] Andrew Baxter and Doron Zeilberger, The Number of Inversions and the Major Index of Permutations are Asymptotically Joint-Independently Normal, The Personal Journal of Shalosh B. Ekhad and Doron Zeilberger, Feb. 4, 2011,

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/invmaj.html .

[CJZ] E. Rodney Canfield, Svante Janson, and Doron Zeilberger, The Mahonian Probability Distribution on Words is Asymptotically Normal, Advances in Applied Mathematics 46 (2011), 109-124.

[D] Persi Diaconis, “Group Representations in Probability and Statistics”, Institute of Mathematical Statistics, Hayward, CA, 1988.

[F] Dominique Foata, On the Netto inversion number of a sequence, Proceedings of the American Mathematical Society 19 (1968), 236-240.

[G] Doron Gepner, private communication, 1987.

[MW] H.B. Mann and D.R. Whitney, On a test whether one of two random variables is stochastically larger than other, Annals of Mathematical Statistics 18 (1947), 50-60.

[N] Brian Nakamura, “Computational Methods in Permutation Patterns”, PhD thesis, Rutgers University,

http://www.math.rutgers.edu/~zeilberg/Theses/BrianNakamuraThesis.pdf .

[Z1] Doron Zeilberger, The Automatic Central Limit Theorems Generator (and Much More!) , in:”Advances in Combinatorial Mathematics: Proceedings of the Waterloo Workshop in Computer Algebra 2008 in honor of Georgy P. Egorychev”, chapter 8, pp. 165-174, (I.Kotsireas, E.Zima, eds. Springer Verlag, 2009),

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/georgy.html .

[Z2] Doron Zeilberger, HISTABRUT:A Maple Package for Symbol-Crunching in Probability theory, The Personal Journal of Shalosh B. Ekhad and Doron Zeilberger,

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/histabrut.html .

Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA.

zeilberg at math dot rutgers dot edu ; http://www.math.rutgers.edu/~zeilberg/ .

Exclusively published in The Personal Journal of Shalosh B. Ekhad and Doron Zeilberger

( http://www.math.rutgers.edu/~zeilberg/pj.html) and arxiv.org .

March 31, 2016