THE MAHONIAN PROBABILITY DISTRIBUTION ON WORDS IS ASYMPTOTICALLY NORMAL

E. RODNEY CANFIELD, SVANTE JANSON, AND DORON ZEILBERGER

This article is dedicated to Dennis Stanton, q-grandmaster and versatile unimodaliter (and log-concaviter)

Abstract. The Mahonian statistic is the number of inversions in a permutation of a multiset with a_{i} elements of type i, 1 ≤ i ≤ m. The counting function for this statistic is the q analog of the multinomial coefficient \binom{a_{1}+···+a_{m}}{a_{1},...a_{m}}, and the probability generating function is the normalization of the latter. We give two proofs that the distribution is asymptotically normal. The first is computer-assisted, based on the method of moments. The Maple package MahonianStat, available from the webpage of this article, can be used by the reader to perform experiments and calculations. Our second proof uses characteristic functions. We then take up the study of a local limit theorem to accompany our central limit theorem. Here our result is less general, and we must be content with a conjecture about further work. Our local limit theorem permits us to conclude that the coeffiecients of the q-multinomial are log-concave, provided one stays near the center (where the largest coefficients reside.)

1. Introduction

The most important discrete probability distribution, by far, is the Binomial distribution, B(n, p) for which we know everything explicitly, ℙ(X = i) (\binom{n}{i}p^{i}(1 − p)^{n−i}), the probability generating function ((pt + (1 − p))^{n}), the moment generating function ((pe^{t} + 1 − p)^{n}), etc. etc. Most importantly, it is asymptotically normal, which means that the normalized random variable

Z_{n} = \frac{X_{n} − np}{\sqrt{np(1 − p)}}

tends to the standard Normal distribution N(0, 1), as n → ∞.

Another important discrete distribution function is the Mahonian distribution, defined on the set of permutations on n objects, and describing,

Date: August 13, 2009; revised September 22, 2009. Accompanied by Maple package MahonianStat available from

http://www.math.rutgers.edu/∼zeilberg/mamarim/mamarimhtml/mahon.html.

The work of D. Zeilberger was supported in part by the United States of America National Science Foundation. The work of E. R. Canfield was supported in part by the NSA Mathematical Sciences Program.

inter-alia, the random variable “number of inversions”. (Recall that an inversion in a permutation π_{1}, …, π_{n} is a pair 1 ≤ i < j ≤ n such that π_{i} > π_{j}). Let us call this random variable M_{n}. The probability generating function, due to Netto, is given explicitly by:

F_{n}(q) = \frac{1}{n!} ∏_{i=1}^{n} \frac{1 − q^{i}}{1 − q} .  (1.1)

The formula (1.1) has a simple probabilistic interpretation (see Feller’s account in [3, Section X.6]): If Y_{j} is the number of i with 1 ≤ i < j and π_{i} > π_{j}, then

M_{n} = Y_{1} + ⋯ + Y_{n},  (1.2)

and Y_{1}, …, Y_{n} are independent random variables and Y_{j} is uniformly distributed on {0, …, j − 1}, as is easily seen by constructing π by inserting 1, …, n in this order at random positions; thus Y_{j} has probability generating function (1 − q^{j})/(j(1 − q)). It follows from (1.1) or (1.2) by simple calculations that the Mahonian distribution has mean and variance

𝔼 M_{n} = \frac{n(n − 1)}{4},  (1.3)
Var M_{n} = \frac{n(n − 1)(2n + 5)}{72} = \frac{2n^{3} + 3n^{2} − 5n}{72}.  (1.4)

Even though there is no explicit expression for the coefficients themselves (i.e. for the exact probabilitity that a permutation of n objects would have a certain number of inversions), it is a classical result (see [3, Section X.6]), that follows from an extended form of the Central Limit Theorem, that the normalized version

\frac{M_{n} − n(n − 1)/4}{\sqrt{(2n^{3} + 3n^{2} − 5n)/72}} ,

tends to N(0, 1), as n → ∞. So this sequence of probability distributions, too, is asymptotically normal.

But what about words, also known as multi-set permutations?. Permutations on n objects can be viewed as words in the alphabet {1, 2, …, n}, where each letter shows up exactly once. But what if we allow repetitions? I.e., we consider all words with a_{1} occurrences of 1, a_{2} occurrences of 2, …, a_{m} occurrences of m. (We assume throughout that m ≥ 2 and each a_{j} ≥ 1.) We all know that the number of such words is the multinomial coefficient

\binom{a_{1} + ⋯ + a_{m}}{a_{1}, …, a_{m}}

and many of us also know that the number of such words with exactly k inversions is the coefficient of q^{k} in the q-analog of the multinomial coefficient

\binom{a_{1} + ⋯ + a_{m}}{a_{1}, …, a_{m}}_{q} := \frac{[a_{1} + ⋯ + a_{m}]!}{[a_{1}]! ⋯ [a_{m}]!} ,  (1.5)

where [n]! := [1][2] ⋯ [n], and [n] := (1 − q^{n})/(1 − q); see [1, Theorem 3.6]. Assuming that all words are equally likely (the uniform distribution), the probability generating function is thus

F_{a_{1},...,a_{m}}(q) := \frac{(∏_{i=1}^{m} a_{i}!) · ∏_{i=1}^{a_{1}+···+a_{m}}(1 − q^{i})}{(a_{1} + ⋯ + a_{m})! ∏_{j=1}^{m} ∏_{i=1}^{a_{j}}(1 − q^{i})} = \frac{F_{a_{1}+···+a_{m}}(q)}{F_{a_{1}}(q) ⋯ F_{a_{m}}(q)}.

(1.6)

Indeed, this can be seen as follows. Let M_{a_{1},...,a_{m}} denote the number of inversions in a random word. If we distinguish the a_{i} occurrences of i by adding different fractional parts, in random order, the number of inversions will increase by Z_{i}, say, with the same distribution as M_{a_{i}}; further M_{a_{1},...,a_{m}} and Z_{1}, …, Z_{m} are independent. On the other hand, M_{a_{1},...,a_{m}}+Z_{1}+⋯+Z_{m} has the same distribution as M_{a_{1}+···+a_{m}}. Hence,

F_{a_{1},...,a_{m}}(q)F_{a_{1}}(q) ⋯ F_{a_{m}}(q) = F_{a_{1}+···+a_{m}}(q),  (1.7)

which is (1.6).

By (1.6), we further have the factorization

F_{a_{1},...,a_{m}}(q) = ∏_{j=2}^{m} F_{A_{j−1},a_{j}}(q),  (1.8)

where A_{j} := a_{1} + ⋯ + a_{j}, which reduces the general case to the two-letter case.

Note that (1.6) shows that the distribution of M_{a_{1},...,a_{m}} is invariant if we permute a_{1}, …, a_{m}; a symmetry which is not obvious from the definition. Remark 1.1. The two-letter case is particularly interesting, since the unnormalized generating function

\binom{a + b}{a}F_{a,b}(q) = \frac{(1 − q^{a+b})(1 − q^{a+b−1}) ⋯ (1 − q^{a+1})}{(1 − q^{b})(1 − q^{b−1}) ⋯ (1 − q^{1})} = \frac{[a + b]!}{[a]! [b]!} ,

(the q-binomial coefficient in (1.5)) is the same as the generating function for the set of integer-partitions with largest part ≤ a and ≤ b parts, in other words the set of integer-partitions whose Ferrers diagram lies inside an a by b rectangle, where the random variable is the “number of dots” (i.e. the integer being partitioned). In other words, the number of such partitions of an integer n equal the number of words of a 1’s and b 2’s with n inversions. See Andrews [1, Section 3.4].

It is easy to see that the mean of M_{a_{1},...,a_{m}} is

μ(a_{1}, …, a_{m}) := 𝔼 M_{a_{1},...,a_{m}} = e_{2}(a_{1}, …, a_{m})/2

(here e_{k}(a_{1}, …, a_{m}) is the degree k elementary symmetric function), so considering the shifted random variable M_{a_{1},...,a_{m}} − μ(a_{1}, …, a_{m}), “number of inversions minus the mean”, we get that the probability generating function is

G_{a_{1},...,a_{m}}(q) := q^{−μ(a_{1},...,a_{m})}F_{a_{1},...,a_{m}}(q) = \frac{F_{a_{1},...,a_{m}}(q)}{q^{e_{2}(a_{1},...,a_{k})/2}}  (1.9)

By computing (q(qG)′)′ and plugging-in q = 1, or from (1.7) and (1.3)–(1.4), it is easy to see that the variance σ^{2} := Var M_{a_{1},...,a_{m}} is

σ^{2} = \frac{(e_{1} + 1)e_{2} − e_{3}}{12} .  (1.10)

(By σ we mean σ(a_{1}, …, a_{m}) and we omit the arguments (a_{1}, …, a_{m}) from the e_{i}’s.)

Let N := e_{1} = a_{1} + ⋯ + a_{m}, the length of the random word, and let a^{∗} := max_{j} a_{j} and N_{∗} := N − a^{∗}.

One main result of the present article is:

Theorem 1.2. Consider the random variable, M_{a_{1},...,a_{m}}, “number of inversions”, on the (uniform) sample space of words with a_{1} 1’s, a_{2} 2’s, …, a_{m} m’s. For any sequence of sequences (a_{1}, …, a_{m}) = (a_{1}^{(ν)}, …, a_{m^{(ν)}}^{(ν)}) such that N_{∗} := N − a^{∗} → ∞, the sequence of normalized random variables

X_{a_{1},...,a_{m}} = \frac{M_{a_{1},...,a_{m}} − μ(a_{1}, …, a_{m})}{σ(a_{1}, …, a_{m})} ,

tends to the standard normal distribution N (0, 1), as ν → ∞.

Theorem 1.2 includes both the case when m ≥ 2 is fixed, and the case when m → ∞. If m is fixed and a_{1} ≥ a_{2} ≥ ⋯ ≥ a_{m}, as may be assumed by symmetry, then the condition N_{∗} → ∞ is equivalent to a_{2} → ∞. In the case m → ∞, the assumption N_{∗} → ∞ is redundant, because N_{∗} ≥ m − 1.

Remark 1.3. The condition N_{∗} → ∞ is also necessary for asymptotic normality, see Section 5.

We give a short proof of this result using characteristic functions in Section 3. We give first in Section 2 another proof (at least of a special case) that is computer-assisted, using the Maple package MahonianStat available from the webpage of this article:

http://www.math.rutgers.edu/∼zeilberg/mamarim/mamarimhtml/mahon.html,

where one can also find sample input and output. This first proof uses the method of moments.

We conjecture that Theorem 1.2 can be refined to a local limit theorem as follows:

Conjecture 1.4. Uniformly for all a_{1}, …, a_{m} and all integers k,

ℙ(M_{a_{1},...,a_{m}} = k) = \frac{1}{\sqrt{2π}σ} (e^{−(k−μ)^{2}/(2σ^{2})} + O(\frac{1}{N_{∗}} )).  (1.11)

We have not been able to prove this conjecture in full generality, but we prove it under additional hypotheses on a_{1}, …, a_{m} in Section 4.

For the special case of the Mahonian random variable M_{n}, Louchard and Prodinger [5] have found (by the saddle point method) a sharper result including a second order term; they also give results for large deviations. It would be interesting to obtain such results for M_{a_{1},...,a_{m}} too.

2. A computer-inspired proof

We assume for simplicity that m is fixed, and that (a_{1}, …, a_{m}) = (ta_{1}^{0}, …, ta_{m}^{0}) for some fixed a_{1}^{0}, …, a_{m}^{0} and t → ∞.

We discover and prove the leading term in the asymptotic expansion, in t, for an arbitrary 2r-th moment, for the normalized random variable X_{a_{1},...,a_{m}} = (M_{a_{1},...,a_{m}} − μ)/σ, and show that it converges to the moment μ_{2r} = (2r)!/(2^{r}r!) of N (0, 1), for every r.

For the sake of exposition, we will only treat in detail the two-letter case, where we can find explicit expressions for the asymptotics of the 2r-th moment of M_{a_{1},a_{2}} − μ, for a_{1} = ta, a_{2} = tb with symbolic a, b, t and r to any desired (specific) order s (i.e. the leading coefficient t^{3r} as well as the terms involving t^{3r−1}, …, t^{3r−s}). A modified argument works for the general case, but we can only find the leading term, i.e. that

α_{2r} := 𝔼(X_{a_{1},...,a_{m}})^{2r} = \frac{(2r)!}{2^{r}r!} + O(t^{−1}) .

Of course the odd moments are all zero, since the distribution of M_{a_{1},...,a_{m}} is symmetric about μ.

In the two-letter case, the mean of M_{a,b} is simply ab/2, so the probability generating function for M_{a,b} − μ is, see (1.6),

G_{a,b}(q) = \frac{F_{a,b}(q)}{q^{ab/2}} = \frac{a!b!(1 − q^{a+b})(1 − q^{a+b−1}) ⋯ (1 − q^{a+1})}{q^{ab/2}(a + b)!(1 − q^{b})(1 − q^{b−1}) ⋯ (1 − q^{1})} .

Taking ratios, we have:

\frac{G_{a,b}(q)}{G_{a−1,b}(q)} = \frac{a(1 − q^{a+b})}{q^{b/2}(a + b)(1 − q^{a})} .  (2.1)

Recall that the binomial moments B_{r} := 𝔼[\binom{M_{a,b}−μ}{r}] are the Taylor coefficients of the probability generating function (in our case G_{a,b}(q)) around q = 1. Writing q = 1 + z, we have

G_{a,b}(1 + z) = ∑_{r=0}^{∞} B_{r}(a, b)z^{r} .

Note that B_{0}(a, b) = 1 and B_{1}(a, b) = 0. Let us call the expression on the right side of (2.1), with q replaced by 1 + z, P (a, b, z):

P (a, b, z) := \frac{a(1 − (1 + z)^{a+b})}{(1 + z)^{b/2}(a + b)(1 − (1 + z)^{a})} .

Maple can easily expand P (a, b, z) to any desired power of z, It starts out with

P (a, b, z) = 1 + \frac{1}{24} (2 a + b) bz^{2} − \frac{1}{24} (2 a + b) bz^{3}

− \frac{1}{5760} (8 a^{3} − 8 a^{2}b − 12 ab^{2} − 3 b^{3} − 440 a − 220 b) bz^{4} + …

note that the coefficients of all the powers of z are polynomials in (a, b).

So let us write

P (a, b, z) = ∑_{i=0}^{∞} p_{i}(a, b)z^{i} ,

where p_{i}(a, b) are certain polynomials that Maple can compute for any i, no matter how big.

Looking at the recurrence

G_{a,b}(1 + z) = P (a, b, z)G_{a−1,b}(1 + z) ,

and comparing coefficients of z^{r} on both sides, we get

B_{r}(a, b) − B_{r}(a − 1, b) = ∑_{s=1}^{r} B_{r−s}(a − 1, b)p_{s}(a, b) .  (2.2)

Assuming that we already know the polynomials B_{r−1}(a, b), B_{r−2}(a, b), …, B_{0}(a, b), the left side is a certain specific polynomial in a and b, that Maple can easily compute, and then B_{r}(a, b) is simply the indefinite sum of that polynomial, that Maple can do just as easily. So (2.2) enables us to get explicit expressions for the binomial moments B_{r}(a, b) for any (numeric) r.

But what about the general (symbolic) r? It is too much to hope for the full expression, but we can easily conjecture as many leading terms as we wish. We first conjecture, and then immediately prove by induction, that for r ≥ 1

B_{2r}(a, b) = \frac{1}{r!} (\frac{ab(a + b)}{24})^{r} + lower order terms
B_{2r+1}(a, b) = \frac{−1}{(r − 1)!} (\frac{ab(a + b)}{24})^{r} + lower order terms ,

where we can conjecture (by fitting polynomials in (a, b) to the data obtained from the numerical r’s) any (finite, specific) number of terms.

Once we have asymptotics, to any desired order, for the binomial moments, we can easily compute the moments μ_{r}(a, b) of M_{a,b} − μ themselves, for any desired specific r and asymptotically, to any desired order. We do that by using the expressions of the powers as linear combination of fallingfactorials (or equivalently binomials) in terms of Stirling numbers of the second kind, S(n, k). Note that for the asymptotic expressions to any desired order, we can still do it symbolically, since for any specific m, S(n, n − m) is a polynomial in n (that Maple can easily compute, symbolically, as a polynomial in n). In particular, the variance is:

σ^{2} = μ_{2}(a, b) = \frac{ab(a + b + 1)}{12} ,

in accordance with (1.10). In general we have μ_{2r+1}(a, b) = 0, of course, and the six leading terms of μ_{2r}(at, bt) can be found in the webpage of this article. From this, Maple finds that α_{2r}(at, bt) := μ_{2r}(at, bt)/μ_{2}(at, bt)^{r} are given asymptotically (for fixed a, b and t → ∞) by:

α_{2r}(at, bt) = \frac{(2r)!}{2^{r}r!} · (1 − \frac{r(r − 1) (b^{2} + ab + a^{2})}{5ab (a + b)} · \frac{1}{t}) + O(t^{−2}) .

In particular, as t → ∞, they converge to the famous moments of N (0, 1). QED.

2.1. The general case. To merely prove asymptotic normality, one does not need a computer, since we only need the leading terms. The above proof can be easily adapted to the general case (a_{1}, …, a_{m}) = (ta_{1}^{0}, …, ta_{m}^{0}). One simply uses induction on m, the number of different letters.

2.2. The Maple package MahonianStat. The Maple package MahonianStat, accompanying this article, has lots of features, that the readers can explore at their leisure. Once downloaded into a directory, one goes into a Maple session, and types read MahonianStat;. To get a list of the main procedures, type: ezra();. To get help with a specific procedure, type ezra(ProcedureName);. Let us just mention some of the more important procedures.

AsyAlphaW2tS(r,a,b,t,s): inputs symbols r,a,b,t and a positive integer s, and outputs the asymptotic expansion, to order s, for α_{2r} (=μ_{2r}/μ_{2}^{r})

ithMomWktE(r,e,t): the r-th moment about the mean of the number of inversions of a_{1}t 1’s, …, a_{m}t m’s in terms of the elementary symmetric functions, in a_{1}, …, a_{m}. Here r is a specific (numeric) positive integer, but e and t are symbolic.

AppxWk(L,x): Using the asymptotics implied by the asymptotic normality of the (normalized) random variable under consideration, finds an approximate value for the number of words with L[1] 1’s, L[2] 2’s, …, L[m] m’s with exactly x inversions. For example, try: AppxWk([100,100,100],15000);

For the two-lettered case, one can get better approximations, by procedure BetterAppxW2, that uses improved limit-distributions, using more terms in the probability density function.

The webpage of this article has some sample input and output.

3. A general proof of Theorem 1.2

We have an exact formula (1.10) for the variance σ^{2} of M_{a_{1},...,a_{m}}. We first show that σ^{2} is always of the order Θ(N^{2}N_{∗}). Lemma 3.1. For any a_{1}, …, a_{m},

\frac{N^{2}N_{∗}}{36} ≤ σ^{2} ≤ \frac{(N + 1)NN_{∗}}{12} ≤ \frac{N^{2}N_{∗}}{6} .

Proof. For the upper bounds we assume, by symmetry, that a_{1} ≥ ⋯ ≥ a_{m}. Then a^{∗} = a_{1} and

e_{2} = a_{1} ∑_{j=2}^{m} a_{j} + a_{2} ∑_{j=3}^{m} a_{j} + ⋯ ≤ N ∑_{j=2}^{m} a_{j} = NN_{∗}.

Since e_{1} = N, (1.10) yields the upper bounds.

For the lower bound, we first observe that 2e_{2}e_{1} − 6e_{3} ≥ 0 (since this difference can be written as a sum of certain a_{j}a_{k}a_{l}). Hence e_{3} ≤ e_{1}e_{2}/3 and (1.10) yields

12σ^{2} ≥ e_{1}e_{2} − e_{3} ≥ \frac{2}{3}e_{1}e_{2}.

Further,

2e_{2} = ∑_{j=1}^{m} a_{j}(N − a_{j}) ≥ ∑_{j=1}^{m} a_{j}(N − a^{∗}) = NN_{∗},

and the lower bound follows. □

Proof of Theorem 1.2. From (1.6) follows the identity

F_{n_{1},n_{2}}(e^{iθ}) = ∏_{j=1}^{n_{2}} \frac{(e^{i(n_{1}+j)θ} − 1)/(i(n_{1} + j)θ)}{(e^{ijθ} − 1)/(ijθ)}.  (3.1)

By Taylor’s series

log \frac{e^{z} − 1}{z} = z/2 + z^{2}/24 + O(z^{4}), |z| ≤ 1,

and we substitute this expansion into the identity (3.1) to conclude:

F_{n_{1},n_{2}}(e^{iθ}) = exp (in_{1}n_{2}θ/2 − n_{1}n_{2}(n_{1} + n_{2} + 1)θ^{2}/24 + O(n_{2}n_{1}^{4}θ^{4})) , (3.2)

uniformly for n_{1} ≥ n_{2} ≥ 1 and |θ| ≤ (n_{1} + n_{2})^{−1}.

We use the factorization (1.8). By symmetry, we may assume a_{1} ≥ a_{2} ≥ ⋯ ≥ a_{m}, and then A_{j−1} ≥ a_{j−1} ≥ a_{j} for each j. Thus (3.2) yields, uniformly for q = e^{iθ} with |θ| ≤ N^{−1},

F_{a_{1},...,a_{m}}(q) = ∏_{j=2}^{m} F_{A_{j−1},a_{j}}(q)

= exp (∑_{j=2}^{m}(iA_{j−1}a_{j}θ/2 − A_{j−1}a_{j}(A_{j} + 1)θ^{2}/24 + O(a_{j}A_{j−1}^{4}θ^{4}))) .

Here, the sums of the coefficients of θ and θ^{2} are easily evaluated, but we do not have to do that since they have to equal iμ and −σ^{2}/2, respectively. Further,

∑_{j=2}^{m} A_{j−1}^{4}a_{j} ≤ N^{4} ∑_{j=2}^{m} a_{j} = N^{4}N_{∗}.  (3.3)

Consequently, if |θ| ≤ N^{−1},

F_{a_{1},...,a_{m}}(e^{iθ}) = exp(iμθ − σ^{2}θ^{2}/2 + O(N^{4}N_{∗}θ^{4}))  (3.4)

and, by (1.9),

G_{a_{1},...,a_{m}}(e^{iθ}) = exp(−σ^{2}θ^{2}/2 + O(N^{4}N_{∗}θ^{4})).  (3.5)

Let θ = t/σ. For any fixed t, by Lemma 3.1,

|Nt/σ| = O(N_{∗}^{−1/2}) = o(1),

so |θ| ≤ N^{−1} if ν is large enough. Hence, by (3.5) and Lemma 3.1,

G_{a_{1},...,a_{m}}(e^{it/σ}) = exp (−\frac{t^{2}}{2} + O(\frac{N^{4}N_{∗}t^{4}}{N^{4}N_{∗}^{2}} )) = exp(−t^{2}/2 + o(1)),

and Theorem 1.2 follows by the continuity theorem [4, Theorem XV.3.2]. □

4. The local limit theorem

“If one can prove a central limit theorem for a sequence a_{n}(k) of numbers arising in enumeration, then one has a qualitative feel for their behavior. A local limit theorem is better because it provides asymptotic information about a_{n}(k) …,” [2]. In this section we prove that the relation (1.11) holds uniformly over certain very general, albeit not unrestricted, sets of tuples a = (a_{1}, …, a_{m}). The exact statement is given below in Theorem 4.5.

As explained in Bender [2], there are two standard conditions for passage from a central to a local limit theorem: (1) if the sequence in question is unimodal, then one has a local limit theorem for n in the set {|n − μ| ≥ ϵσ}, ϵ > 0; (2) if the sequence in question is log-concave, then one has a local limit theorem for all n. Our sequence, the coefficients of the q-multinomial, is in fact unimodal, as first shown by Schur [7] using invariant theory, and later by O’Hara [6] using combinatorics. Unfortunately, the ensuing local limit theorem fails to cover the most interesting coefficients, the largest ones, near the mean μ. However, our polynomials are manifestly not log-concave as is seen by inspecting the first three coefficients (assuming n_{1}, n_{2} ≥ 2)

\binom{n_{1} + n_{2}}{n_{1}}_{q} = 1 + q + 2q^{2} + · · · .

The question arises might the coefficients be log-concave near the mean, and here is a small table of empirical values: (c[j] = [q^{j}]\binom{2n}{n}_{q})

\begin{matrix}n & (c[n^{2}/2 − 1])^{2} − c[n^{2}/2] × c[n^{2}/2 − 2] \\ 2 & -1 \\ 4 & -7 \\ 6 & -165 \\ 8 & -1529 \\ 10 & 44160 \\ 12 & 7715737 \\ 14 & 905559058 \\ 16 & 101507214165 \\ 18 & 11955335854893 \\ 20 & 1501943866215277 \end{matrix}

Based on this scant evidence, we speculate that some sort of log-concavity theorem is true, but that its proper statement is complicated by describing the appropriate range of a and j. Thus, we use neither of the two standard methods mentioned above for proving our local limit theorem. (Later, we shall see that our theorem has implications for log-concavity.) Instead, we use another standard method, direct integration (Fourier inversion) of the characteristic function, or equivalently of the probability generating function F (q) for q = e^{iθ} on the unit circle. We begin with one such estimate for rather small θ. Lemma 4.1. There exists a constant τ > 0 such that for any a_{1}, …, a_{m} and |θ| ≤ τ/N,

|F_{a_{1},...,a_{m}}(e^{iθ})| = |G_{a_{1},...,a_{m}}(e^{iθ})| ≤ e^{−σ^{2}θ^{2}/4}.

Proof. Suppose that 0 < |θ| ≤ τ/N. Then, using Lemma 3.1,

\frac{N^{4}N_{∗}θ^{4}}{σ^{2}θ^{2}} ≤ \frac{N^{2}N_{∗}τ^{2}}{σ^{2}} ≤ 36τ^{2},

so if τ is chosen small enough, the error term O(N^{4}N_{∗}θ^{4}) in (3.4) and (3.5) is ≤ σ^{2}θ^{2}/4, and thus the result follows from (3.5). □

We let in the sequel τ denote this constant. We may assume 0 < τ ≤ 1. Lemma 4.2. Uniformly, for all a_{1}, …, a_{m} and all integers k,

|ℙ(M_{a_{1},...,a_{m}} = k) − \frac{1}{\sqrt{2π}σ} e^{−(k−μ)^{2}/(2σ^{2})}| ≤ ∫_{τ/N}^{π} |F_{a_{1},...,a_{m}}(e^{iθ})| dθ+O(\frac{1}{σN_{∗}} ).

Proof. For any integer k,

ℙ(M_{a_{1},...,a_{m}} = k) − \frac{1}{\sqrt{2π}σ} e^{−(k−μ)^{2}/(2σ^{2})}

= \frac{1}{2π} ∫_{−π}^{π} F_{a_{1},...,a_{m}}(e^{iθ})e^{−ikθ} dθ − \frac{1}{2π} ∫_{−∞}^{∞} e^{−σ^{2}θ^{2}/2}e^{−i(k−μ)θ} dθ

= \frac{1}{2π} ∫_{|θ|≤τ/N}(G_{a_{1},...,a_{m}}(e^{iθ}) − e^{σ^{2}θ^{2}/2})e^{−i(k−μ)θ} dθ

+ \frac{1}{2π} ∫_{τ/N≤|θ|≤π} F_{a_{1},...,a_{m}}(e^{iθ})e^{−ikθ} dθ

− \frac{1}{2π} ∫_{|θ|≥τ/N} e^{−σ^{2}θ^{2}/2}e^{−i(k−μ)θ} dθ

=: I_{1} + I_{2} + I_{3}.

By (3.5) and the inequality |e^{w} − 1| ≤ |w| max(1, |e^{w}|) we find for |θ| ≤ τ/N, using Lemma 4.1,

\begin{aligned}|G_{a_{1},...,a_{m}}(e^{iθ}) − e^{−σ^{2}θ^{2}/2}| & ≤ O(N^{4}N_{∗}θ^{4}) max (e^{−σ^{2}θ^{2}/2}, |G_{a_{1},...,a_{m}}(e^{iθ})|) \\ & = O(N^{4}N_{∗}θ^{4}e^{−σ^{2}θ^{2}/4}). \end{aligned}

Integrating, we find

\begin{aligned}|I_{1}| & ≤ ∫_{|θ|≤τ/N} |G_{a_{1},...,a_{m}}(e^{iθ}) − e^{−σ^{2}θ^{2}/2}| dθ \\ & ≤ O(N^{4}N_{∗}) ∫_{−∞}^{∞} θ^{4}e^{−σ^{2}θ^{2}/4} dθ = O(N^{4}N_{∗}σ^{−5}) = O(\frac{1}{σN_{∗}} ). \end{aligned}

Further, again using Lemma 4.1,

|I_{3}| ≤ ∫_{τ/N}^{∞} e^{−σ^{2}θ^{2}/2} dθ ≤ 3σ^{−1}e^{−(στ/N)^{2}/2} ≤ \frac{6}{σ(στ/N)^{2}} = O(\frac{1}{σN_{∗}} ).

Finally, |I_{2}| ≤ ∫_{τ/N}^{π} |F_{a_{1},...,a_{m}}(e^{iθ})| dθ. □

In order to verify Conjecture 1.4, it thus suffices to show that the integral ∫_{τ/N}^{π} |F_{a_{1},...,a_{m}}(e^{iθ})| dθ in Lemma 4.2 is O(\frac{1}{σN_{∗}} ).

Remark 4.3. For example, an estimate

F_{a_{1},...,a_{m}}(e^{iθ}) = O(\frac{1}{σ^{3}θ^{3}} ),  0 < θ ≤ π,  (4.1)

is sufficient for (1.11). We conjecture that this estimate (4.1) holds when N_{∗} ≥ 6, say. Note that it does not hold for very small N_{∗}: taking θ = π we have, for even n_{1}, F_{n_{1},1}(−1) = 1/(n_{1} + 1) = 1/N, and the same holds for F_{n_{1},2}(−1).

Note further that even the weaker estimate

F_{a_{1},...,a_{m}}(e^{iθ}) = O(\frac{1}{σ^{2}θ^{2}} ),  0 < θ ≤ π,  (4.2)

would be enough to prove (1.11) with the weaker error term O(N_{∗}^{−1/2}).

We obtain a partial proof of Conjecture 1.4 using the following lemma. Lemma 4.4. For a given τ ∈ (0, 1] there exists c = c(τ) > 0 such that

|F_{n_{1},n_{2}}(e^{iθ})| ≤ e^{−cn_{2}}  (4.3)

for n_{1} ≥ n_{2} ≥ 1 and τ/(n_{1} + n_{2}) ≤ |θ| ≤ π.

More generally, for any a_{1}, …, a_{m} and τ/N ≤ |θ| ≤ π,

|F_{a_{1},...,a_{m}}(e^{iθ})| ≤ e^{−cN_{∗}}.  (4.4)

Proof. We prove first (4.3). For positive integer n define

f_{n}(y, q) = ∏_{j=0}^{n}(1 − yq^{j})^{−1}.

For 0 ≤ R < 1, we have (e.g. by Taylor expansions) e^{2R} ≤ \frac{1+R}{1−R}, and thus

e^{4R} ≤ \frac{(1+R)^{2}}{(1−R)^{2}} = 1 + \frac{4R}{(1−R)^{2}} . Hence, by convexity, for any real ζ,

e^{2R(1−cos ζ)} ≤ 1 + \frac{2R(1 − cos ζ)}{(1 − R)^{2}} = \frac{1 + R^{2} − 2R cos ζ}{(1 − R)^{2}} = \frac{|1 − Re^{iζ}|^{2}}{(1 − R)^{2}} ,

and thus

|(1 − Re^{iζ})^{−1}| ≤ (1 − R)^{−1} exp (−R(1 − cos ζ)) .

Consequently, by a simple trigonometric identity, for any real ϕ and θ,

|f_{n_{1}}(Re^{iϕ}, e^{iθ})| ≤ (1 − R)^{−n_{1}−1}

× exp (−R(n_{1} + 1 − cos(ϕ + \frac{n_{1}}{2} θ)\frac{sin(n_{1} + 1)θ/2}{sin θ/2}))

≤ (1 − R)^{−n_{1}−1} × exp (R(−n_{1} − 1 + \frac{sin(n_{1} + 1)θ/2}{sin θ/2})) .

The function g(θ) = g_{n}(θ) := \frac{sin n(θ/2)}{sin(θ/2)}, where n ≥ 1, is an even function of θ; is decreasing for 0 ≤ θ ≤ π/n, as can be verified by calculating g′; and satisfies |g(θ)| ≤ g(π/n) for π/n ≤ |θ| ≤ π. Further, for n ≥ 2 and 0 ≤ |θ| ≤ π/n,

g_{n}(θ) = 2\frac{sin(nθ/4)}{sin(θ/2)} cos(nθ/4) = 2g_{n/2}(θ) cos(nθ/4) ≤ n cos(nθ/4)

≤ n(1 − \frac{n^{2}θ^{2}}{40} ).

Let θ_{0} = τ(n_{1} + n_{2})^{−1} < π/(n_{1} + 1). For θ_{0} ≤ |θ| ≤ π we thus have

|g_{n_{1}+1}(θ)| ≤ g_{n_{1}+1}(θ_{0}) ≤ n_{1} + 1 − \frac{n_{1}^{3}θ_{0}^{2}}{40} ;

whence, for 0 ≤ R < 1, the estimate above yields

|f_{n_{1}}(Re^{iϕ}, e^{iθ})| ≤ (1 − R)^{−n_{1}−1} exp (−Rn_{1}^{3}θ_{0}^{2}/40) .  (4.5)

Combinatorially we know that [y^{ℓ}q^{n}]f_{n_{1}}(y, q) is the number of partitions of n having at most ℓ parts no one of which exceeds n_{1}. As said in Remark 1.1, this equals [q^{n}]\binom{n_{1}+ℓ}{n_{1}}F_{n_{1}+ℓ}(q). Hence, using Cauchy’s integral formula, for any R > 0,

\binom{n_{1} + n_{2}}{n_{1}}F_{n_{1},n_{2}}(q) = [y^{n_{2}}]f_{n_{1}}(y, q) = \frac{1}{2πi} ∫_{|y|=R} f_{n_{1}}(y, q)\frac{dy}{y^{n_{2}+1}} .

Consequently, (4.5) implies that for θ_{0} ≤ |θ| ≤ π and 0 < R < 1,

\binom{n_{1} + n_{2}}{n_{1}}|F_{n_{1},n_{2}}(q)| ≤ (1 − R)^{−n_{1}−1}R^{−n_{2}} exp (−Rn_{1}^{3}θ_{0}^{2}/40) .

Now choose R = ρ := n_{2}/(n_{1} + n_{2}) ≤ 1/2. By Stirling’s formula,

\binom{n_{1} + n_{2}}{n_{1}} = Ω(n_{2}^{−1/2}) (1 − ρ)^{−n_{1}−1} ρ^{−n_{2}}

and thus, for θ_{0} ≤ |θ| ≤ π,

|F_{n_{1},n_{2}}(q)| ≤ O(n_{2}^{1/2}) exp (−ρn_{1}^{3}θ_{0}^{2}/40) = O(n_{2}^{1/2}) exp (−Ω(n_{2})) .

This shows (4.3) for n_{2} sufficiently large. To handle the remaining finitely many values of n_{2} we shall show: for each n_{2} ≥ 1 and τ ∈ (0, 1], there exists δ > 0 such that

|F_{n_{1},n_{2}}(e^{iθ})| ≤ 1 − δ  (4.6)

for all n_{1} ≥ n_{2} and τ/(n_{1} + n_{2}) ≤ |θ| ≤ π. To do this, we use

|F_{n_{1},n_{2}}(e^{iθ})| = ∏_{j=1}^{n_{2}} \frac{j}{n_{1} + j} |\frac{sin(n_{1} + j)θ/2}{sin(jθ/2)}|

= ∏_{j=1}^{n_{2}} \frac{j sin(θ/2)}{sin jθ/2} · ∏_{j=1}^{n_{2}} \frac{|g_{n_{1}+j}(θ)|}{n_{1} + j} = Π_{1} · Π_{2} . (4.7)

Let N = n_{1} + n_{2} and τ/N ≤ |θ| ≤ π. For n ≥ n_{1} + 1 we have |nθ| ≥ n_{1}|θ| ≥ N|θ|/2 ≥ τ/2, and thus the estimates above show that

|g_{n}(θ)| ≤ g_{n}(τ/2n) ≤ n(1 − τ^{2}/160) .

Hence the final product Π_{2} in (4.7) is bounded by 1 − τ^{2}/160 < 1. The product Π_{1} is a continuous function of θ, and equals 1 for θ = 0; hence |Π_{1}| ≤ 1 + τ^{2}/200 for |θ| ≤ ε, where ε > 0 is sufficiently small. (Recall that n_{2} now is fixed.) This proves (4.6) for |θ| ≤ ε.

For larger |θ| we use the factorization

F_{n_{1},n_{2}}(q) = \binom{n_{1} + n_{2}}{n_{2}}^{−1} \frac{(1 − q^{n_{1}+1}) ⋯ (1 − q^{n_{1}+n_{2}})}{(1 − q) ⋯ (1 − q^{n_{2}})}.  (4.8)

Let 0 < |θ_{0}| ≤ π, and suppose that k ≥ 0 of the factors in the denominator of (4.8) vanish at q = q_{0} := e^{iθ_{0}}. Then 0 ≤ k ≤ n_{2} − 1, since 1 − q_{0} ≠ 0. There are at least k factors in the numerator of (4.8) that vanish at q_{0} (since F is a polynomial, and all factors have simple roots only); for q = e^{iθ}, each of these factors is bounded by N|q − q_{0}| ≤ N|θ − θ_{0}| while every factor is bounded by 2; hence the numerator of (4.8) is O(N^{k}|θ − θ_{0}|^{k}). Let J be an interval around θ_{0} such that the denominator of (4.8) does not vanish at any q = e^{iθ} ≠ q_{0} with θ ∈ J̄; then the denominator is Θ(|θ − θ_{0}|^{k}) for θ ∈ J. Finally, the binomial coefficient in (4.8) is Θ(n_{1}^{n_{2}}).

Combining these estimates, we see that uniformly for θ ∈ J,

|F_{n_{1},n_{2}}(e^{iθ})| = O(\frac{N^{k}|θ − θ_{0}|^{k}}{n_{1}^{n_{2}}|θ − θ_{0}|^{k}} ) = O(n_{1}^{k−n_{2}}) = O(n_{1}^{−1}).

Since the set ε ≤ |θ| ≤ π may by covered by a finite number of such interval J, |F_{n_{1},n_{2}}(e^{iθ})| = O(n_{1}^{−1}) uniformly for ε ≤ |θ| ≤ π. Consequently (4.6) holds for all such θ if n_{1} is sufficiently large.

It remains to verify (4.6) for each fixed n_{2} and a finite number of n_{1}; in other words, that for each n_{2} ≥ 1 and n_{1} ≥ n_{2}, there exists δ > 0 such that (4.6) holds. To see this, note that the events M_{n_{1},n_{2}} = 0 and M_{n_{1},n_{2}} = 1 both have positive probability. It follows that |F_{n_{1},n_{2}}(e^{iθ})| < 1 for every θ with 0 < |θ| ≤ π, and (4.6) follows. This completes the proof of (4.3).

To prove (4.4), we assume as we may that a_{1} ≥ ⋯ ≥ a_{m} and use the factorization (1.8). Let J be the first index such that a_{2} + ⋯ + a_{J} ≥ N_{∗}/2. For j ≥ J, then A_{j−1} + a_{j} = A_{j} ≥ A_{J} ≥ a_{1} + N_{∗}/2 ≥ N/2, and thus A_{j}|θ| ≥ N|θ|/2 ≥ τ/2; hence (4.3) yields

|F_{A_{j−1},a_{j}}(e^{iθ})| ≤ e^{−c(τ/2)a_{j}}.

We thus obtain from (1.8), since each F_{n_{1},n_{2}} is a probability generating function and thus is bounded by 1 on the unit circle,

|F_{a_{1},...,a_{m}}(e^{iθ})| = ∏_{j=2}^{m} |F_{A_{j−1},a_{j}}(e^{iθ})| ≤ ∏_{j=J}^{m} e^{−c(τ/2)a_{j}} ≤ e^{−c(τ/2)N_{∗}/2},

because ∑_{j=J}^{m} a_{j} ≥ N_{∗}/2. This proves (4.4) (redefining c(τ)). □

Theorem 4.5. There exists a positive constant c such that for every C, the following is true. Uniformly for all a_{1}, …, a_{m} such that a^{∗} ≤ Ce^{cN_{∗}} and all integers k,

ℙ(M_{a_{1},...,a_{m}} = k) = \frac{1}{\sqrt{2π}σ} (e^{−(k−μ)^{2}/(2σ^{2})} + O(\frac{1}{N_{∗}} )).  (4.9)

Proof. Let c_{1} = c(τ) be the constant in Lemma 4.4. Then, Lemmas 4.2 and 4.4 yield

ℙ(M_{a_{1},...,a_{m}} = k) = \frac{1}{\sqrt{2π}σ} (e^{−(k−μ)^{2}/(2σ^{2})} + O(\frac{1}{N_{∗}} + σe^{−c_{1}N_{∗}})).

For any fixed c < c_{1} we have, using Lemma 3.1, σN_{∗}e^{−c_{1}N_{∗}} = O(Ne^{−cN_{∗}}) and thus

ℙ(M_{a_{1},...,a_{m}} = k) = \frac{1}{\sqrt{2π}σ} (e^{−(k−μ)^{2}/(2σ^{2})} + O(\frac{1 + Ne^{−cN_{∗}}}{N_{∗}} )).

The result follows, since Ne^{−cN_{∗}} = a^{∗}e^{−cN_{∗}}+N_{∗}e^{−cN_{∗}} = a^{∗}e^{−cN_{∗}}+O(1). □

4.1. Log-concavity. Let us review the proof of Theorem 4.5 with the intention of greater accuracy. The goal is to prove log-concavity in some range. For concreteness, let a = (n, n). Then σ^{2} is of order n^{3}, and for sufficient accuracy we take the Taylor series in the exponent of (3.2) out to O(θ^{10}). This yields, for some polynomials p_{k}(n) of degree k + 1,

F_{n,n}(e^{iθ}) = exp(iμθ − σ^{2}θ^{2}/2 + p_{4}(n)θ^{4} + p_{6}(n)θ^{6} + p_{8}(n)θ^{8} + O(n^{11}θ^{10}))
= e^{iμθ−σ^{2}θ^{2}/2}(1 + p_{4}(n)θ^{4} + p_{6}(n)θ^{6} + p_{8}(n)θ^{8}

+ \frac{1}{2}p_{4}^{2}(n)θ^{8} + p_{4}(n)p_{6}(n)θ^{10} + \frac{1}{6}p_{4}^{3}(n)θ^{12} + O(n^{11}θ^{10}))

Arguing as in the proof of Lemma 4.2 but using this estimate instead of (3.5) for |θ| ≤ τ/N, one easily obtains, after the substitution θ = t/σ, for any k and with x := (k − μ)/σ,

ℙ(M_{n,n} = k) = \frac{1}{2π} ∫_{−∞}^{∞} e^{−t^{2}/2−itx}(1 + \frac{p_{4}(n)}{σ^{4}}t^{4} + ⋯ + \frac{p_{4}^{3}(n)}{6σ^{12}}t^{12}) \frac{dt}{σ} + O(n^{−4}σ^{−1}).

Letting φ(x) := (2π)^{−1/2}e^{−x^{2}/2} denote the normal density function, and φ^{(j)} its derivatives, we obtain by Fourier inversion

\begin{aligned}ℙ(M_{n,n} = k) & = σ^{−1}(φ(x) + \frac{p_{4}(n)}{σ^{4}}φ^{(4)}(x) + ⋯ + \frac{p_{4}^{3}(n)}{6σ^{12}}φ^{(12)}(x) + O(n^{−4})) \\ & = \frac{1}{\sqrt{2π}σ} e^{−x^{2}/2}(1 + Q(n, x) + O(n^{−4})), \end{aligned} ; (4.10)

where Q(n, x) is σ^{−12} times a certain polynomial in n and x of degree 17 in n; thus for x = O(1) we have Q(n, x) = O(n^{−1}) and similarly, for derivatives with respect to x, Q′(n, x) = O(n^{−1}) and Q′′(n, x) = O(n^{−1}). (Q(n, x) can easily be computed explicitly using computer algebra, but we do not have to do it.)

Replacing k by k ± 1 in (4.10) we find, for x = O(1),

ℙ(M_{n,n} = k ± 1) = \frac{1}{\sqrt{2π}σ} e^{−(x±σ^{−1})^{2}/2}(1 + Q(n, x) ± σ^{−1}Q′(n, x) + O(n^{−4})),

and thus

ℙ(M_{n,n} = k − 1) ℙ(M_{n,n} = k + 1)

= \frac{1}{2πσ^{2}} e^{−x^{2}−σ^{−2}}((1 + Q(n, x))^{2} − σ^{−2}Q′(n, x)^{2} + O(n^{−4})).

= e^{−σ^{−2}} ℙ(M_{n,n} = k)^{2}(1 + O(n^{−4})).

Hence, for x = O(1), i.e., k = μ + O(σ),

\begin{aligned}ℙ(M_{n,n} = k)^{2} & − ℙ(M_{n,n} = k − 1) ℙ(M_{n,n} = k + 1) \\ & = (σ^{−2} + O(n^{−4})) ℙ(M_{n,n} = k)^{2} = \frac{1}{2πσ^{4}} e^{−x^{2}}(1 + O(n^{−1})). \end{aligned} ; (4.11)

In particular, this is positive for large n. This gives: Theorem 4.6 (A log-concavity result). For each constant C we have n_{0} such that for n ≥ n_{0} and |j − μ| ≤ Cσ

c_{j}^{2} ≥ c_{j−1}c_{j+1},

where

c_{j} := [q^{j}]\binom{2n}{n}_{q} = \binom{2n}{n} ℙ(M_{n,n} = j).

We note that the “mysterious” numbers appearing in our earlier table for the choice j = n^{2}/2 − 1 are asymptotically

\frac{1}{2πσ^{4}} \binom{2n}{n}^{2} ∼ \frac{18}{πn^{6}} \binom{2n}{n}^{2} ∼ \frac{18}{π^{2}}n^{−7}2^{4n}.

Remark 4.7. This argument for log-concavity in the central region does not use any special properties of the distribution; although we needed several terms in the asymptotic expansion above, it was only to see that they are sufficiently smooth, and the main term in the final result (4.11) comes from the main term e^{−x^{2}/2}/(\sqrt{2π}σ) in (4.9). What we have shown is just that the convergence to the log-concave Gaussian function in the local limit theorem is sufficiently regular for the log-concavity of the limit to transfer to ℙ(M_{n,n} = k) for k = μ + O(σ) and sufficiently large n.

5. Final comments

Suppose that N_{∗} ↛ ∞. We may, as usual, assume that a_{1} ≥ ⋯ ≥ a_{m}. By considering a subsequence (if necessary), we may assume that N_{∗} := N − a^{∗} = a_{2} + ⋯ + a_{m} is a constant; this entails that m is bounded, so by again considering a subsequence, we may assume that m and a_{2}, …, a_{m} are constant. We thus study the case when a_{1} → ∞ with fixed a_{2}, …, a_{m}.

In this case, the number of inversions between indices 2, …, m is O(1), which is asymptotically negligible. Ignoring these, we can thus consider the random word as N_{∗} letters 2, …, m inserted in a_{1} 1’s, and the number of inversions is the sum of their positions, counted from the end. It follows easily, either probabilistically or by calculating the characteristic function from (1.6), that M_{a_{1},...,a_{m}}/N, or equivalently M_{a_{1},...,a_{m}}/a_{1}, converges in distribution to the sum ∑_{j=1}^{N_{∗}} U_{j} of N_{∗} independent random variables U_{j} with the uniform distribution on [0, 1]. Equivalently, since σ^{2} ∼ n_{1}^{2}N_{∗}/12 ∼ N^{2}N_{∗}/12,

\frac{M_{a_{1},...,a_{m}} − μ(a_{1}, …, a_{m})}{σ(a_{1}, …, a_{m})} ⟶^{d} \sqrt{\frac{12}{N_{∗}}} ∑_{j=1}^{N_{∗}}(U_{j} − \frac{1}{2}),

where ⟶^{d} denotes convergence in distribution. This limit is clearly not normal for any finite N_{∗}. (However, its distribution is close to standard normal for large N_{∗}. Note that it is normalized to mean 0 and variance 1.)

References

[1] G. E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.

[2] E. A. Bender, Central and local limit theorems applied to asymptotic enumeration, J. Combinatorial Theory Ser. A 15 (1973) 91–111.

[3] W. Feller, An Introduction to Probability Theory and Its Application, volume I, third edition, Wiley, New York, 1968.

[4] W. Feller, An Introduction to Probability Theory and its Applications, volume II, 2nd ed., Wiley, New York, 1971.

[5] G. Louchard and H. Prodinger, The number of inversions in permutations: a saddle point approach. J. Integer Seq. 6 (2003) Article 03.2.8.

[6] K. M. O’Hara, Unimodality of Gaussian coefficients: a constructive proof, Journal of Combinatorial Theory, Series A 53 (1990) 29–52.

[7] I. Schur, Vorlesungen über Invariantentheorie, edited by H. Grunsky, Springer-Verlag, Berlin, 1968.

Computer Science Department, University of Georgia, Athens, GA 30602-7404, USA

E-mail address: erc [At] cs [Dot] uga [Dot] edu

Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden

E-mail address: svante.janson [At] math [Dot] uu [Dot] se
URL: http://www.math.uu.se/∼svante/

Mathematics Department, Rutgers University (New Brunswick), Piscataway, NJ 08854, USA

E-mail address: zeilberg [At] math [Dot] rutgers [Dot] edu