CENTRAL AND LOCAL LIMIT THEOREMS FOR EXCEDANCES BY CONJUGACY CLASS AND BY DERANGEMENT

Lane Clark

Department of Mathematics, Southern Illinois University Carbondale, Carbondale, IL 62901

                           lclark@math.siu.edu

Received: 11/15/00, Revised: 9/14/01 Accepted: 1/31/02, Published: 2/8/02

Abstract

We give central and local limit theorems for the number of excedances of a uniformly distributed random permutation belonging to certain sequences of conjugacy classes and belonging to the sequence of derangements.

1. Introduction

For n ∈ ℙ and 0 ≤ k ≤ D_{n} ∈ ℕ, let b(n, k) ∈ [0, ∞) and B_{n} := b(n, 0)+⋯+b(n, D_{n}) > 0. We say the array {b(n, k) : n ≥ 1, 0 ≤ k ≤ D_{n}} satisfies a central limit theorem with mean μ_{n} and variance σ_{n}^{2} provided

lim_{n→∞} sup_{x∈ℝ} | ∑_{k≤⌊(x)_{n}⌋} \frac{b(n, k)}{B_{n}} − Φ(x)| = 0,  (1)

where (x)_{n} := xσ_{n}+μ_{n}. Equivalently, we say {b(n, k)} is asymptotically normal. Further, the array {b(n, k) : n ≥ 1, 0 ≤ k ≤ D_{n}} satisfies a local limit theorem on S ⊆ ℝ if and only if

lim_{n→∞} sup_{x∈S} |σ_{n} \frac{b(n, ⌊(x)_{n}⌋)}{B_{n}} − ϕ(x)| = 0.  (2)

In general, a central limit theorem for a sequence of random variables gives (1) from which (2) follows under certain conditions (see Theorems 1, 3 and 4 and Lemma 2). We note that (1) is equivalent to pointwise convergence in view of the uniform continuity of e^{−t^{2}/2} and [11; Theorem 1 of Section 9] (see Bender [1]).

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 2 (2002), #A03 2

A permutation of [n] is a bijection σ : [n] → [n] which we write as σ = (σ(1), …, σ(n)). We denote the set of permutations of [n] by 𝔖[n]. A permutation σ ∈ 𝔖[n] is a derangement of [n] provided σ(i) ≠ i for all i ∈ [n]. Let 𝔇[n] denote the set of derangements of [n].

Given a partition λ = (λ_{1} ≥ ⋯ ≥ λ_{p} ≥ 1) of n (= λ_{1} + ⋯ + λ_{p}) ∈ ℙ, let m_{i} = m_{i}(λ) := |{j ∈ [p] : λ_{j} = i}| for i ∈ [n]. Here we write λ = 1^{m_{1}}2^{m_{2}} ⋯ n^{m_{n}}. We say σ ∈ 𝔖[n] is of cycle type λ if and only if the decomposition of σ into a product of disjoint cycles has precisely m_{i}(λ) cycles of length i for i ∈ [n]. We denote the set of all permutations of [n] of cycle type λ by 𝔖_{λ}[n]. The 𝔖_{λ}[n], where λ is a partition of n, are the conjugacy classes of 𝔖[n]. It is readily seen (see [7]; p. 233]) that |𝔖_{λ}[n]| = n!/(1^{m_{1}}2^{m_{2}} ⋯ n^{m_{n}})(m_{1}! m_{2}! ⋯ m_{n}!).

We say i ∈ [n] is an excedance (respectively, a descent) of σ ∈ 𝔖[n] provided σ(i) > i (respectively, i ≠ n and σ(i) > σ(i + 1)). Let E(σ) (respectively, Des(σ)) denote the set of excedances (respectively, descents) of σ and e(σ) := |E(σ)| (respectively, d(σ) := |Des(σ)| + 1 and maj(σ) := ∑{i : i ∈ Des(σ)}).

Given a partition λ = 1^{m_{1}}2^{m_{2}} ⋯ n^{m_{n}} of n ≥ 2, let a_{λ}(n, k) := |{σ ∈ 𝔖_{λ}[n] : |Des(σ)| = k}| and b_{λ}(n, k) := |{σ ∈ 𝔖_{λ}[n] : e(σ) = k}| for k ∈ ℕ. From [8; Theorem B], a_{λ}(n, 1) = ∏_{i=1}^{n} \binom{f_{i2}+m_{i}−1}{m_{i}} where f_{i2} = ∑_{d|i} μ(d)2^{i/d}/i for i ∈ P when m_{1} ≠ n and n ≥ 2. Here, a_{λ}(n, 1) ∈ ℙ since f_{i2} ∈ ℙ upon appealing to its definition, while, a_{λ}(n, 1) = (n−b+1)f_{b2} when λ = 1^{n−b}b with 2 ≤ b ≤ n. From [4; Theorem 3.1], b_{λ}(n, 1) is the coefficient of x in Q_{λ,n}(x) (see Section 3). Properties of the Eulerian polynomials imply that b_{λ}(n, 1) = 0 when m_{2} + ⋯ + m_{n} ≥ 2 and b_{λ}(n, 1) = \binom{n}{b} when λ = 1^{n−b}b with 2 ≤ b ≤ n. Hence, a_{λ}(n, 1) ≠ b_{λ}(n, 1) when m_{2}+⋯+m_{n} ≥ 2 and a_{λ}(n, 1) ≠ b_{λ}(n, 1) for all but at most b−1 integers n when λ = 1^{n−b}b with 2 ≤ b ≤ n. Consequently, descents and excedances are not equidistributed over conjugacy classes in general. As is well known (see [14; p. 23]), they are equidistributed over 𝔖[n].

In connection with the Betti numbers of certain varieties, Stanley [15] was interested in the symmetry and unimodality of the coefficients of some polynomials obtained by enumerating a set of permutations according to the number of excedances. Brenti [3],

[4] showed these excedance polynomials are symmetric and unimodal when the set is a conjugacy class or is the set of derangements thus generalizing a conjecture and answering a question of [15]. Fulman [10] gave a central limit theorem for the coefficients of polynomials obtained by enumerating permutations belonging to certain sequences of conjugacy classes according to the number of descents. Our discussion in the previous paragraph shows that descents and excedances are not equidistributed over the conjugacy classes considered in [10] (see the remark after Theorem 3). We give both central and local limit theorems for the coefficients of excedance polynomials over certain sequences of conjugacy classes, including those of [10], and over the sequence of derangements (see Theorems 3 and 4). A more precise statement of the results and techniques is given in the next paragraph.

Using the method of moments, Fulman [10] recently gave a central limit theorem for d(σ) with μ_{n} = (n − 1)/2, σ_{n}^{2} = (n − 1)/12 and for maj(σ) with μ_{n} = \binom{n}{2}/2, σ_{n}^{2} = n(n − 1)(2n + 5)/72 when σ is a uniformly distributed random permutation in 𝔖_{λ(n)}[n] for certain sequences λ(n) of cycle types. In this paper, we give central and local limit theorems for e(σ) with various μ_{n}, σ_{n}^{2} when σ is a uniformly distributed random permutation in 𝔖_{λ(n)}[n] for certain sequences λ(n) of cycle types, including those of [10], and for e(σ) with μ_{n} = (n − 1)/2 + o(1), σ_{n}^{2} = 25n/12 + o(1) when σ is a uniformly distributed random permutation in 𝔇[n] (see Theorems 3 and 4). Of course our results immediately give asymptotic formulas for the number of such permutations with a certain number of excedances. We use the method of Harper [12]. A slight extension of this method requiring only a nice factorization of the polynomials over ℝ[x] is given in the next section (see Theorem 1). We refer the reader to the excellent survey of Pólya frequency sequences by Pitman [13].

Let ℕ denote the nonnegative integers; ℙ the positive integers; ℝ_{0} the nonnegative real numbers and ℝ the real numbers. The collection of all polynomials in an indeterminate x whose coefficients are in the set A is denoted by A[x]. For n ∈ ℙ, [n] := {1, …, n}. The cardinality of a set A is denoted by |A|. We denote the largest integer at most x by ⌊x⌋.

The expectation of a random variable (r.v.) X is denoted by E(X) and its variance by Var(X). We write X_{n} →^{d} X when the sequence X_{n} of r.v.s converges in distribution to the r.v. X. For x ∈ ℝ, let

ϕ(x) := \frac{1}{\sqrt{2π}} e^{−x^{2}/2} and Φ(x) := ∫_{−∞}^{x} ϕ(t) dt .

We write N(0, 1) for a normally distributed r.v. with mean 0 and variance 1. We refer the reader to Comtet [7] for combinatorics and Durrett [9] for probability.

2. General Results

For completeness, we first formalize a slight extension of the method of Harper [12] requiring only a nice factorization of the polynomials over ℝ[x], which we will use in the next section.

For n ∈ ℙ, let Q_{n}(x) := P_{n,1}(x) ⋯ P_{n,N(n)}(x) where P_{n,j}(x) := ∑_{k=0}^{d_{n,j}} a(n, j, k)x^{k} ∈ ℝ_{0}[x] − {0} for 1 ≤ j ≤ N(n) ∈ ℙ. Let X_{n,1}, …, X_{n,N(n)} be row-independent r.v.s with

Pr (X_{n,j} = k) = \frac{a(n, j, k)}{P_{n,j}(1)} (1 ≤ j ≤ N(n), 0 ≤ k ≤ d_{n,j})

(which, of course, exist), hence,

E(X_{n,j}) = \frac{P′_{n,j}(1)}{P_{n,j}(1)} and E(X_{n,j}^{2}) = \frac{P′_{n,j}(1) + P′′_{n,j}(1)}{P_{n,j}(1)} (1 ≤ j ≤ N(n)).

Let

X_{n} := X_{n,1} + ⋯ + X_{n,N(n)}

hence,

μ_{n} := E(X_{n}) = \frac{Q′_{n}(1)}{Q_{n}(1)} and σ_{n}^{2} := Var(X_{n}) = \frac{Q′_{n}(1)}{Q_{n}(1)} + \frac{Q′′_{n}(1)}{Q_{n}(1)} − (\frac{Q′_{n}(1)}{Q_{n}(1)})^{2} . (3)

If Q_{n}(x) := ∑_{k=0}^{D_{n}} b(n, k)x^{k} where D_{n} := d_{n,1} + ⋯ + d_{n,N(n)}, then by row-independence

Pr (X_{n} = k) = \frac{b(n, k)}{Q_{n}(1)} (0 ≤ k ≤ D_{n}) .

Let

Y_{n,j} := \frac{X_{n,j} − E(X_{n,j})}{σ_{n}} and Y_{n} := Y_{n,1} + ⋯ + Y_{n,N(m)} = \frac{X_{n} − μ_{n}}{σ_{n}} .

Then Y_{n,j} assumes the values (k−E(X_{n,j}))/σ_{n} with probabilities a(n, j, k)/P_{n,j}(1) for 0 ≤ k ≤ d_{n,j}. Let M_{n} := max_{1≤j≤N(n)} {E(X_{n,j}), |d_{n,j}−E(X_{n,j})|} and G_{n,j}(x) := Pr (Y_{n,j} ≤ x) be the distribution function of Y_{n,j} for 1 ≤ j ≤ N(n). For all ϵ > 0,

lim_{n→∞} ∑_{j=1}^{N(n)} ∫_{|y|>ϵ} y^{2}dG_{n,j}(y) = 0 ,

provided lim_{n→∞} M_{n}/σ_{n} = 0. By the Lindeberg-Feller Theorem (see [9; pp. 98–101]), Y_{n} →^{d} N(0, 1). We have proved the following central limit theorem which can be expressed in terms of the b(n, k), X_{n} or Y_{n} since

Pr (Y_{n} ≤ x) = Pr (X_{n} ≤ ⌊(x)_{n}⌋) = ∑_{k≤⌊(x)_{n}⌋} \frac{b(n, k)}{Q_{n}(1)}

for all x ∈ ℝ where (x)_{n} := xσ_{n} + μ_{n}.

Theorem 1. Suppose lim_{n→∞} M_{n}/σ_{n} = 0. For each x ∈ ℝ,

lim_{n→∞} ∑_{k≤⌊(x)_{n}⌋} \frac{b(n, k)}{Q_{n}(1)} = Φ(x)

where (x)_{n} = xσ_{n} + μ_{n}. (As mentioned in the introduction, this is equivalent to (1).)

Remark. If d_{n,j} ≤ 1 for all 1 ≤ j ≤ N(n), then M_{n} ≤ 1 and σ_{n} → ∞ as n → ∞ suffices (Harper’s method).

A sequence a(0), …, a(D) of real numbers is log-concave provided a^{2}(j) ≥ a(j−1)a(j+ 1) for all 1 ≤ j ≤ D − 1. It has no internal zeros if and only if there exist no indices 0 ≤ i < j < k ≤ D with a_{i}, a_{k} ≠ 0 but a_{j} = 0. The array {b(n, k) : n ≥ 1, 0 ≤ k ≤ D_{n}} of real numbers has property P provided the sequence b(n, 0), …, b(n, D_{n}) has property P for all n ∈ ℙ. We require the following result of Canfield [5; Theorem II].

Lemma 2 (Canfield [5]). Suppose the array {b(n, k) : n ≥ 1, 0 ≤ k ≤ D_{n}} satisfies (1) and is log-concave with no internal zeros where σ_{n} → ∞ as n → ∞. Then the array {b(n, k) : n ≥ 1, 0 ≤ k ≤ D_{n}} satisfies (2) on S = ℝ.

3. Applications to Excedances

For n ∈ ℙ, the polynomial

A_{n}(x) := ∑_{σ∈𝔖[n]}x^{d(σ)} := ∑_{k=1}^{n} A(n, k)x^{k} ∈ ℕ[x]

is called the nth Eulerian polynomial where A_{0}(x) := 1. The Eulerian numbers A(n, k) satisfy the recurrence relation

A(n, k) = (n − k + 1)A(n − 1, k − 1) + kA(n − 1, k) (n, k ≥ 2)  (4)

with the boundary conditions A(n, 1) = 1 (n ≥ 1) and A(n, k) = 0 (1 ≤ n < k) (see [7; pp. 240–246]). Then (4) together with A_{0}(x) = 1 gives

A_{n}(x) = nxA_{n−1}(x) + (x − x^{2})A′_{n−1}(x) (n ≥ 1).

Hence,

A′_{n}(x) = nA_{n−1}(x) + (nx − 2x + 1)A′_{n−1}(x) + (x − x^{2})A′′_{n−1}(x) (n ≥ 1)

and, upon iteration together with A_{n}(1) = n! (n ≥ 0), we have

A′_{n}(1) = \frac{(n + 1)!}{2}  (n ≥ 1)  (5)

while,

A′′_{n}(x) = (2n − 2)A′_{n−1}(x) + (nx − 4x + 2)A′′_{n−1}(x) + (x − x^{2})A′′′_{n−1}(x) (n ≥ 1)

and, upon iteration together with (5), we have

A′′_{n}(1) = (n + 1)! \frac{3n − 2}{12}  (n ≥ 2).  (6)

We first consider excedances by conjugacy class.

Excedances by Conjugacy Class. For a partition λ = (λ_{1}, …, λ_{p}) of n ∈ ℙ, let

Q_{λ,n}(x) := ∑_{σ∈𝔖_{λ}[n]}x^{e(σ)} ∈ ℕ[x] .

Brenti [4; Theorem 3.1] showed that all

Q_{λ,n}(x) = |𝔖_{λ}[n]| ∏_{i=1}^{p} \frac{A_{λ_{i}−1}(x)}{(λ_{i} − 1)!} .

Of course, Q_{λ,n}(1) = |𝔖_{λ}[n]|. Then

Q′_{λ,n}(x) = Q_{λ,n}(x) ∑_{i=1}^{p} \frac{A′_{λ_{i}−1}(x)}{A_{λ_{i}−1}(x)}

so that (5) gives

Q′_{λ,n}(1) = Q_{λ,n}(1) \frac{n − m_{1}(λ)}{2}

and

Q′′_{λ,n}(x) = Q_{λ,n}(x) \{(∑_{i=1}^{p} \frac{A′_{λ_{i}−1}(x)}{A_{λ_{i}−1}(x)})^{2} + ∑_{i=1}^{p} \frac{A′′_{λ_{i}−1}(x)}{A_{λ_{i}−1}(x)} − ∑_{i=1}^{p} (\frac{A′_{λ_{i}−1}(x)}{A_{λ_{i}−1}(x)})^{2}\}

so that (5) and (6) give

Q′′_{λ,n}(1) = Q_{λ,n}(1) \{\frac{n^{2}}{4} − \frac{m_{1}(λ)}{12}(6n − 5) + \frac{m_{1}^{2}(λ)}{4} − \frac{5n}{12} − \frac{m_{2}(λ)}{6}\} .

Hence, (3) gives

μ_{λ,n} = \frac{n − m_{1}(λ)}{2} and σ_{λ,n}^{2} = \frac{n − m_{1}(λ) − 2m_{2}(λ)}{12} .

Since A_{n}(x) has degree of A_{n}(x) nonpositive real zeros for n ∈ ℕ (see [7; p. 292]), each Q_{λ,n}(x) has degree of Q_{λ,n}(x) nonpositive real zeros. Hence, M_{λ,n} ≤ 1 and the coefficients of all Q_{λ,n}(x) are log-concave with no internal zeros (see [2; Theorem 1.2.1]).

Now Q_{λ,n}(x) = ∑_{k=0}^{n−1} b_{λ}(n, k)x^{k} where b_{λ}(n, k) is the number of σ ∈ 𝔖_{λ}[n] with e(σ) = k. Let Z_{λ,n}(σ) = e(σ) where σ is chosen randomly from 𝔖_{λ}[n] according to a uniform distribution. Then Pr (Z_{λ,n} = k) = b_{λ}(n, k)/|𝔖_{λ}[n]| (0 ≤ k ≤ n − 1) so that

X_{λ,n} =^{d} Z_{λ,n}. Hence, E(Z_{λ,n}) = μ_{λ,n} and Var(Z_{λ,n}) = σ_{λ,n}^{2}.

For any sequence λ(n) of partitions of n, now let

Q_{n}(x) := Q_{λ(n),n}(x) := ∑_{k=0}^{n−1} b(n, k)x^{k}

with μ_{n} := μ_{λ(n),n} and σ_{n}^{2} := σ_{λ(n),n}^{2}. As a consequence of Theorem 1 and Lemma 2, we have the following results for the number of excedances of σ ∈ 𝔖_{λ(n)}[n] which can be expressed in terms of the b(n, k), X_{λ(n),n}, Y_{λ(n),n} or Z_{λ(n),n}.

Theorem 3. For any sequence λ(n) of partitions of n with σ_{n} → ∞ as n → ∞, the array {b(n, k) : n ≥ 1, 0 ≤ k ≤ n − 1} satisfies both a central limit theorem and a local limit theorem on ℝ with the above μ_{n} and σ_{n}^{2}.

Remark. We do not require, as in [10], all m_{i}(λ(n)) → 0 as n → ∞ for our results to hold. If, however, all m_{i}(λ(n)) → 0 as n → ∞, then σ_{n} → ∞ as n → ∞, and we have both a central limit theorem and a local limit theorem on ℝ with μ_{n} = n/2 and σ_{n}^{2} = n/12 (so not quite the same asymptotic distribution as d(σ) in [10]). Here, descents and excedances are not equidistributed over 𝔖_{λ(n)}[n] for all sufficiently large n.

We next consider excedances by derangement (which is not a conjugacy class). For n ∈ ℕ, let s_{n} := ∑_{k=0}^{n}(−1)^{k}/k! .

Excedances by Derangement. For n ∈ ℙ, let

Q_{n}(x) := ∑_{σ∈𝔇[n]}x^{e(σ)} ∈ ℕ[x]

and Q_{0}(x) := 1. Brenti [3; proof of Proposition 5] showed that

\frac{A_{n}(x)}{x} = ∑_{m=0}^{n} \binom{n}{m}Q_{m}(x) (n ≥ 1) .

By inversion (see [7; pp. 143–144])

Q_{n}(x) = (−1)^{n} + ∑_{m=1}^{n}(−1)^{n−m}\binom{n}{m}\frac{A_{m}(x)}{x}  (n ≥ 1) .  (7)

For n ≥ 3, (5), (6) and (7) give

Q_{n}(1) = n! s_{n} ,
Q′_{n}(1) = \frac{n!}{2}\{(n − 1)s_{n−1} + s_{n−2}\} ,
Q′′_{n}(1) = \frac{n!}{12} \{(3n^{2} + 13n + 10)s_{n−2} + (6n + 13)s_{n−3} + \frac{3(−1)^{n−3}}{(n − 3)!} \} .

Hence, (3) and s_{n−r}/s_{n} = 1 + o(n^{−1}) for r = 1, 2, 3 give

μ_{n} = \frac{n − 1}{2} + o(1) and σ_{n}^{2} = \frac{25n}{12} + o(1) as n → ∞ .

Zhang [16] proved a conjecture of Brenti [3; p. 1140] by showing that Q_{n}(x) has degree of Q_{n}(x) distinct nonpositive real zeros for n ∈ ℙ. Hence, M_{n} ≤ 1 and the coefficients of Q_{n}(x) are log-concave with no internal zeros for n ∈ ℙ.

Now Q_{n}(x) = ∑_{k=0}^{n−1} b(n, k)x^{k} where b(n, k) is the number of σ ∈ 𝔇[n] with e(σ) = k. Let Z_{n}(σ) = e(σ) where σ is chosen randomly from 𝔇[n] according to a uniform distribution. Then Pr (Z_{n} = k) = b(n, k)/|𝔇[n]| (0 ≤ k ≤ n − 1) so that X_{n} =^{d} Z_{n}. Hence, E(Z_{n}) = μ_{n} and Var(Z_{n}) = σ_{n}^{2} where σ_{n} → ∞ as n → ∞. As a consequence of Theorem 1 and Lemma 2, we have the following results for the number of excedances of σ ∈ 𝔇[n] which can be expressed in terms of the b(n, k), X_{n}, Y_{n} or Z_{n}.

Theorem 4. The array {b(n, k) : n ≥ 1, 0 ≤ k ≤ n − 1} satisfies both a central limit theorem and a local limit theorem on ℝ with the above μ_{n} and σ_{n}^{2}.

For n ∈ ℙ, we note for completeness that

\frac{A_{n}(x)}{x} = ∑_{σ∈𝔖[n]}x^{e(σ)} := ∑_{k=0}^{n−1} b(n, k)x^{k} ∈ ℕ[x] .

Hence, we have a central limit theorem for the array {b(n, k) : n ≥ 1, 0 ≤ k ≤ n − 1} with μ_{n} = (n − 1)/2 and σ_{n}^{2} = (n + 1)/12. Since the coefficients of A_{n}(x)/x are logconcave with no internal zeros, we also have a local limit theorem on ℝ. (Compare with the analogous results of [6] for A_{n}(x) with μ_{n} = (n + 1)/2 and σ_{n}^{2} = (n + 1)/12.)

Acknowledgment. I wish to thank the referee for comments and suggestions which have led to an improved version of this paper.

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