A new decomposition of derangements

Dongsu Kim and Jiang Zeng

January 28, 2001

Abstract

We give a new decomposition of derangements, which gives a direct interpretation of a formula for their generating function. This decomposition also works for counting derangements by number of excedances.

1 Introduction

A permutation π of [n]={1, 2, …, n} is a derangement, if π(i) ≠ i, for all i ∈ [n]. A value i ∈ [n] is an excedance of π if i < π(i). The number of excedances in π is denoted by exc π. Let D_{n} be the set of derangements of [n], and d_{n}(x) the polynomial

d_{n}(x) = ∑_{π∈D_{n}} x^{exc π}.

For example, d_{0}(x) = 1, d_{1}(x) = 0, d_{2}(x) = x, d_{3}(x) = x + x^{2}, d_{4}(x) = x + 7x^{2} + x^{3}. The generating function of d_{n}(x) can be written as follows [2, 5] :

∑_{n≥0} d_{n}(x)\frac{t^{n}}{n!} = \frac{1}{1 − ∑_{n≥2}(x + x^{2} + ⋯ + x^{n−1}) t^{n}/n!} .  (1)

Of course (1) can be proved by various methods, but, as pointed out by Gessel [4], it seems difficult to directly interpret (1) (even in the x = 1 case !) in terms of derangements. In [4] Gessel gave a direct proof of (1) in a different model with x = 1. His proof is actually based on a factorization of some D-permutations, and cannot be generalized in a straightforward way to prove (1). Our purpose is to give a decomposition of derangements which interprets (1) directly.

A sequence σ = s_{1}s_{2} ⋯ s_{k} of k distinct integers s_{1}, …, s_{k} is called a cycle of length k if s_{1} = min{s_{1}, …, s_{k}}. A cycle σ is called unimodal (resp. prime), if there exists i, 2 ≤ i ≤ k, such that s_{1} < ⋯ < s_{i−1} < s_{i} and s_{i} > s_{i+1} > ⋯ > s_{k} if i < k (resp. in addition, s_{i−1} < s_{k}). Hence each unimodal (resp. prime) cycle is of length ≥ 2.

Considering that s_{1} is the smallest in our case, this definition is consistent with the usual definition of ‘unimodal’. Clearly each cycle σ = s_{1} … s_{k} can be identified with the cyclic permutation σ′ of the set {s_{1}, …, s_{k}} by σ′(s_{i}) = s_{i+1} for i ∈ [k], with s_{k+1} = s_{1}. We let exc σ denote the number of excedances of the associated cyclic permutation σ′.

Let (l_{1}, …, l_{m}) be a composition of n. A P -decomposition of type (l_{1}, …, l_{m}) of [n] is a sequence of prime cycles τ = (τ_{1}, τ_{2}, …, τ_{m}) such that τ_{i} is of length l_{i} and the underlying sets of τ_{i}, i ∈ [m], form a partition of [n].

Define the excedance of τ as the total number of excedances in its prime cycles, i.e. exc τ = exc τ_{1} + ⋯ + exc τ_{m}, and weight τ by x^{exc τ}. It turns out that the right-hand side of (1) is the excedance generating function of P -decompositions. Indeed, since the weight of prime cycles on any l-set is x + x^{2} + ⋯ + x^{l−1}, the generating function of P -decompositions of type (l_{1}, …, l_{m}) is given by

\binom{l_{1} + ⋯ + l_{m}}{l_{1} , …, l_{m}} ∏_{i=1}^{m}(x + ⋯ + x^{l_{i}−1})\frac{t^{l_{1}+···+l_{m}}}{(l_{1} + ⋯ + l_{m})!}.

Summing on l_{1}, …, l_{m} ≥ 2 and m ≥ 0, we obtain the right hand side of (1).

In the next section we give an algorithm (or bijection), which maps each derangement into a P -decomposition with the same number of excedances, and thus prove (1). In section 3 we will apply a similar decomposition to give a direct interpretation of a generating function of Eulerian polynomials. Finally, in section 4 we indicate how to extend our algorithm to deal with similar problems in multipermutations.

2 Unimodal and prime decompositions

Given a derangement π of [n], we first factorize it into cycles of length ≥ 2 :

π = (C_{1}, …, C_{k}),

sorted in the decreasing order of their minima. For each cycle σ = s_{1}s_{2} ⋯ s_{k} we define the following U-algorithm to decompose it into a sequence of unimodal cycles. For the algorithm we set s_{k+1} = s_{1}. U-Algorithm

1. If σ is unimodal then U(σ) = (σ).

2. Otherwise, let i be the largest integer such that s_{i−1} > s_{i} < s_{i+1}, let j be the unique integer greater than i such that s_{j} > s_{i} > s_{j+1}, and set U(σ) = (U(σ_{1}), σ_{2}), where σ_{1} = s_{1} ⋯ s_{i−1}s_{j+1} ⋯ s_{k} and σ_{2} = s_{i}s_{i+1} ⋯ s_{j}, which is unimodal.

Example 2.1 Let σ = 1 8 4 7 12 14 11 9 13 10 6 3 5 2. The U-algorithm runs as follows:

σ → (U(1 8 4 7 12 14 11 9 13 10 6 2), 3 5)
→ (U(1 8 4 7 12 14 11 6 2), 9 13 10, 3 5)
→ (U(1 8 2), 4 7 12 14 11 6, 9 13 10, 3 5)
→ (1 8 2, 4 7 12 14 11 6, 9 13 10, 3 5).

We extend U to π by applying U to each of its cycles to obtain

U(π) = (U(C_{1}), U(C_{2}), …, U(C_{r})) = (u_{1}, …, u_{m}),

which is called the unimodal decomposition of π.

Note that the first cycle C_{1} of π corresponds to the segment (u_{1}, …, u_{i}), where i is the smallest integer satisfying min(u_{1}) > min(u_{i+1}), and the second to a segment of (u_{i+1}, …, u_{m}) in the same manner, etc., so that the underlying set of each cycle can be read off from the unimodal decomposition of π. The following result characterizes all the sequences of unimodal cycles obtained by the U-algorithm.

Lemma 2.2 A sequence of disjoint unimodal cycles, u = (u_{1}, …, u_{m}), is a unimodal decomposition of a derangement in D_{n} if and only if the underlying sets of u_{i}, i ∈ [m], form a partition of [n] and max(u_{i−1}) > min(u_{i}) for each i = 2, …, m.

Proof. Clearly it suffices to show the ‘if’ part. Without loss of generality we may assume that min(u_{1}) < min(u_{i}), for each i = 2, …, m. We build π step by step. Let π^{(1)} = u_{1}. For i > 1, assume that π^{(i−1)} has been built and that π^{(i−1)} = s_{1}s_{2} ⋯ s_{l}, where s_{1}, …, s_{l} is an appropriate rearrangement of elements in u_{1}, u_{2}, …, u_{i−1}. Let u_{i} = r_{1}r_{2} ⋯ r_{a}. Since max(u_{i−1}) > min(u_{i}), there is an integer j such that s_{j} > min(u_{i}), let j_{0} be the largest such integer and set π^{(i)} = s_{1}s_{2} ⋯ s_{j_{0}}r_{1}r_{2} ⋯ r_{a}s_{j_{0}+1} ⋯ s_{l}. Let π = π^{(m)}. Clearly U(π) = u.

For each unimodal cycle σ = s_{1}s_{2} ⋯ s_{k} we define the following V -algorithm to decompose it into a sequence of prime cycles. V -Algorithm

1. If σ is prime then V (σ) = (σ).

2. Otherwise, let j be the smallest integer such that s_{j} > s_{i} > s_{j+1} > s_{i−1} for some integer i greater than 1 and set V (σ) = (V (σ_{1}), σ_{2}), where σ_{1} = s_{1} ⋯ s_{i−1}s_{j+1} ⋯ s_{k} and σ_{2} = s_{i}s_{i+1} ⋯ s_{j}, which is prime.

We extend V -algorithm to U(π) by applying V to each of its components to obtain

V ∘ U(π) = (V (u_{1}), V (u_{2}), …, V (u_{m})) = (τ_{1}, …, τ_{m}),

which is called the prime decomposition of π.

The structure of the unimodal decomposition of π can be easily obtained from its prime decomposition. The first unimodal cycle in U(π) corresponds to the segment (τ_{1}, …, τ_{i}), where i is the smallest integer satisfying max(τ_{i}) > min(τ_{i+1}), and the second to a segment of (τ_{i+1}, …, τ_{m}) in the same manner, etc.

Example 2.3 Let σ be the same as the preceding example, whose unimodal decomposition is U(σ) = (1 8 2, 4 7 12 14 11 6, 9 13 10, 3 5). Note that only the second cycle in U(σ) is not prime. The V -algorithm applied to the second cycle runs as follows :

4 7 12 14 11 6 → (V (4 7 11 6), 12 14) → (4 6, 7 11, 12 14).

Therefore V ∘ U(σ) = (1 8 2, 4 6, 7 11, 12 14, 9 13 10, 3 5).

It is clear that the composition V ∘ U maps any derangement of [n] into a P - decomposition of [n]. The following result shows that this mapping is bijective.

Theorem 2.4 Any P -decomposition of [n] is the prime decomposition of a unique derangement in D_{n}.

Proof. Let τ = (τ_{1}, τ_{2}, …, τ_{m}) be a P -decomposition of [n]. We first construct a sequence of unimodal cycles as follows: starting from the right, if there is any pair of adjacent τ_{i} and τ_{i+1} such that max(τ_{i}) < min(τ_{i+1}), then we insert the elements of τ_{i+1} in τ_{i} just before the maximum of τ_{i} and obtain a new cycle τ_{i} ∗τ_{i+1}. Repeat this process with (τ_{1}, …, τ_{i} ∗ τ_{i+1}, …, τ_{m}), until there are no more such pairs. By lemma 2.2, the resulting sequence σ is a unimodal decomposition of some π ∈ D_{n}, i.e. U(π) = σ. It follows that V ∘ U(π) = V (σ) = τ.

From the U-algorithm it is clear that the number of excedances in a cycle is the same as the sum of excedances in each unimodal component. Also the prime decomposition has the same property. Thus we have proved (1).

3 Application to Eulerian polynomials

If instead of derangements we let A_{n}(x) denote the sum of x^{exc π} for all permutations π of [n], then the polynomials xA_{n}(x) are the well-known Eulerian polynomials and have several other combinatorial interpretations in addition to counting permutations by number of excedances [6]. By virtue of classical theory of generating functions we see immediately that A_{n}(x) are related to d_{n}(x) by the following :

∑_{n≥0} A_{n}(x)\frac{t^{n}}{n!} = e^{t} ∑_{n≥0} d_{n}(x) \frac{t^{n}}{n!} .

Hence it follows from (1) that

∑_{n≥0} A_{n}(x)\frac{t^{n}}{n!} = \frac{1}{1 − ∑_{n≥1}(x − 1)^{n−1} t^{n}/n!} .  (2)

A similar proof can be given for (2), but in this case a weight-preserving sign-reversing involution is needed.

A sequence σ = a_{1}a_{2} … a_{k} of k distinct integers a_{1}, a_{2}, …, a_{k} is called unimodal if k = 1 or k ≥ 2 and there exists an integer i, 1 ≤ i ≤ k, such that a_{1} < a_{2} < ⋯ < a_{i} and a_{i} > a_{i+1} > ⋯ > a_{k} if i < k. This is the usual definition of ‘unimodal’. We define the weight of the unimodal sequence σ by x^{i−1}(−1)^{k−i}, i.e. an ascent is given x and a descent −1.

A U-decomposition (resp. I-decomposition) of [n] is a sequence of unimodal (resp. increasing) sequences (τ_{1}, τ_{2}, …, τ_{m}) such that the underlying sets of τ_{i}, i ∈ [m], form a partition of [n] (resp. in addition, for i > 1, if τ_{i} is a singleton then it is greater than the last entry of τ_{i−1}). Hence the right side of (2) is the generating function of U-decompositions.

We now set up a weight-preserving sign-reversing involution on the U-decompositions to reduce the above generating function to that of I-decompositions. Given a Udecomposition π = (π_{1}, π_{2}, …, π_{l}), we call an integer k attachable, if k forms a singleton, i.e. π_{i} = k for some i > 1, and k is smaller than the last entry of π_{i−1}; detachable, if there exists π_{j} whose last entry is k and whose penultimate entry is greater than k. The involution is then defined by detaching or attaching the smallest attachable or detachable integer (if any). It is clear that π is fixed if and only if π is an I-decomposition.

On the other hand, given a permutation π of [n], we can factorize it into ordered cycles π = (s_{1}, …, s_{r}, c_{1}, …, c_{t}), where s_{1}, …, s_{r} are the singletons ordered in increasing order and c_{1}, …, c_{t} the cycles of length ≥ 2 ordered in decreasing order of their minima. Applying V ∘ U algorithm to each cycle c_{i} we obtain

π = (s_{1}, …, s_{r}, V ∘ U(c_{1}), …, V ∘ U(c_{t})) = (π_{1}, …, π_{m}),

where each π_{i} is a prime or singleton cycle. Since each prime cycle a_{1} … a_{k−1}a_{k} … a_{l} with a_{1} < ⋯ < a_{k−1} < a_{l} < a_{l−1} < ⋯ < a_{k} is in one-to-one correspondence with a sequence of increasing segments, (a_{1}a_{2} ⋯ a_{k−1}a_{l}, a_{l−1}, a_{l−2}, …, a_{k}), which has no attachable or detachable element, we see that π is in one-to-one correspondence with an I-decomposition of [n]. Note that the singletons in π correspond to the singletons to the left of the first increasing sequence of length greater than one in an I-decomposition.

Therefore both sides of (2) are the generating functions of I-decompositions.

4 Remarks

Our decompositions work also for permutations of a multiset {1^{n_{1}}, 2^{n_{2}}, …, m^{n_{m}}}. More precisely, let w = w_{1}w_{2} … w_{n} be such a permutation and δ(w) = p_{1}p_{2} … p_{n} the nondecreasing rearrangement of the letters in w, where n = n_{1} + ⋯ + n_{m}. Then w is a multiderangement if p_{i} ≠ w_{i} for each i = 1, …, n, while the statistic of excedance of w is defined by exc w = #{i : w_{i} > p_{i}}. Let R(n) be the set of all such permutations and define

d_{n}(x) = ∑_{w∈R(n)} x^{exc w}.

Using Foata’s factorization of multipermutations (see [3]) we can factorize each multiderangement as a product of cycles of length at least 2, combining with our two decompositions we get the following result:

∑_{n_{1},...,n_{m}≥0} d_{n}(x)x_{1}^{n_{1}} … x_{m}^{n_{m}} = \frac{1}{1 − x e_{2} − (x + x^{2}) e_{3} − ⋯ − (x + x^{2} + ⋯ + x^{m−1}) e_{m}} ,

where e_{i} (2 ≤ i ≤ m) is the i-th elementary symmetric function of x_{1}, …, x_{m}. The above result seems to be first proved by Askey and Ismail [1] using MacMahon’s Master Theorem.

Acknowledgment

This work was partially supported by KOSEF: 971-0106-038-2.

References

[1] Askey, (R.) and Mourad, (I.), Permutation problems and special functions, Canad. J. Math. 28 (1976), no. 4, 853–874.

[2] Brenti, (F.), Unimodal polynomials arising from symmetric functions, Proc. Amer. Math. Soc. 108 (1990), no. 4, 1133–1141.

[3] Cartier, (P.) and Foata, (D.), Problèmes combinatoires de commutation et réarrangements, Lecture Notes in Mathematics 85, Springer-Verlag, Berlin, 1969.

[4] Gessel, (I.M.), A coloring problem, Amer. Math. Monthly 98 (1991), no. 6, 530–533.

[5] Roselle, (D.P.), Permutations by number of rises and successions, Proc. Amer. Math. Soc. 19 (1968), 8–16.

[6] Stanley, (R.P.), Enumerative Combinatorics, Volume 1, Cambridge University Press, 1997.

Dongsu Kim, Department of Mathematics, KAIST, Taejon 305-701, Korea Email address: dskim@math.kaist.ac.kr

Jiang Zeng, Institut Girard Desargues, Université Claude Bernard (Lyon I), France Email address: zeng@desargues.univ-lyon1.fr