On the Adjacent Cycle Derangements

A derangement, that is, a permutation without fixed points, of a finite set is said to be an adjacent cycle when all its cycles are formed by a consecutive set of integers. In this paper we determine enumerative properties of these permutations using analytical and bijective proofs. Moreover a combinatorial interpretation in terms of linear species is provided. Finally we define and investigate the case of the adjacent cycle derangements of a multiset.


Introduction
Let n {1, 2, . . . , n}, and let S n denote the set of permutations of n . We will use both the one-line notation of a permutation of S n and its decomposition as a product of cycles. It will be convenient to write each cycle ending with its smallest element and order cycles by increasing smallest elements.
For example α 3176452 3721 654 . In 1 the authors define an adjacent cycle of a permutation to be a cycle in which the elements form a consecutive set of integers.
We generalize the notion defining a permutation of S n to be adjacent cycle a.c. for short when in every cycle the elements form a consecutive set of integers. In other words every cycle of k elements, having i as the smallest element, is represented by the sequence i 1, i 2, . . . , i k, i . In 2 it is proved that the adjacent cycle permutations of n turn out to be fixed by the bijection that associates to a permutation π of n the permutation π whose one-line form is obtained from the canonical form of π by removing all inner parentheses.
For example, let π 23146785 231 4 6785 . Notice that we omit commas between the integers in a cycle or in a sequence, unless it leads to ambiguity.

ISRN Discrete Mathematics
We will denote by P a n and D a n the set of adjacent cycle permutations and the set of adjacent cycle derangements of n , respectively, and p a n |P a n | and d a n |D a n | their cardinalities; moreover, D a k,n represents the set of adjacent cycle derangements having k cycles and d a k,n |D a k,n | its cardinality. We say that an adjacent cycle permutation α of n is extended to n 1, . . . , n h, where h ≥ 1, when n 1, . . . , n h are inserted in this order in the last cycle of α before its last element. Clearly we obtain an adjacent cycle permutation of n h .
For example, the extension of the permutation α 21 453 of 5 to 6, 7, 8 determines the new permutation 21 456783 of 8 .
Let F n denote the nth Fibonacci number, with initial conditions F 0 0, F 1 1. We now outline the contents of this paper. In Section 2 we determine properties of the adjacent cycle permutations and adjacent cycle derangements of n , using analytical and bijective proofs.
In Section 3 we prove a relation involving the Fibonacci numbers, which seems new. In Section 4 we present a combinatorial interpretation of the adjacent cycle permutations in terms of linear species; as a consequence we are able to calculate the enumerative results previously obtained in a direct way. In the last section we define and investigate the derangements of a multiset.

Adjacent Cycle Permutations and Adjacent Cycle Derangements
In this section we investigate the adjacent cycle permutations and adjacent cycle derangements of the set n , for a positive integer n. First we determine the number of adjacent cycle derangements having a fixed number of cycles.
Proof. An adjacent cycle derangement π of the set n with k cycles is defined by inserting k − 1 dividers in the positions between two consecutive integers of the sequence 1, 2, . . . , n , so that the numbers of integers before the first divider, between two consecutive dividers, and after last divider are greater than 1. As a consequence we have to exclude the positions between 1 and 2 and between n − 1 and n and the k − 2 positions that immediately follow every divider, but the last one; in other words, we have to exclude k among the n − 1 possible positions. Thus the number turns out to be equal to In the more general case of adjacent cycle permutations of n , the number p k,n of the adjacent cycle permutations having k cycles is obtained by choosing k − 1 among the n − 1 possible positions. See also 2 .
Proof. Note that the maximum number of cycles of an adjacent cycle derangement of n is obtained by setting the length of all cycles equal to 2, with possibly one cycle of length 3, if n is odd. Then the maximum value of k is n/2 , and therefore we obtain d a n n/2 Moreover d a n n/2

2.5
Thus d a n satisfies the same recursive relation of the Fibonacci numbers. It is immediate to verify that d a 1 0 and d a 2 1; then d a n F n−1 .
Another proof of 2.4 may be obtained in a bijective way. Theorem 2.4. For every integer n > 2 one has D a n ∼ D a n−1 D a n−2 .

2.6
Hence d a n d a n−1 d a n−2 , where the initial conditions satisfy d a Proof. Let π be an element of D a n . We have two cases to consider.
1 n belongs to a cycle of π with more than 2 elements. Then π is the extension to n of a permutation π ∈ D a n−1 . 2 n belongs to a cycle ρ having 2 elements. The π is the product of an a.c. derangement α ∈ D a n−2 and the cycle ρ n, n − 1 .
The sets D a n−1 and D a n−2 are clearly disjoint; then D a n ∼ D a n−1 D a n−2 . By equating the cardinalities we obtain d a n d a n−1 d a n−2 , where the initial conditions hold d a Another bijective proof of the results of Theorem 2.4 is obtained in the following theorem.

Proposition 2.6. Let n be a positive integer. Then
1.

2.8
Moreover d a n 1 d a n d a Proof. Let α ∈ D a n , and let α n i, where 3 ≤ i ≤ n − 1. Then α is the product of an a.c. derangement of {1, 2, . . . , i − 1} and the cycle i 1, i 2, . . . , n, i . The cases i 2 and i n are excluded because they imply a cycle of length 1. In particular D a 1 {γ}, where γ 2, 3, . . . , n, 1 .
The sets D a j , 2 ≤ j ≤ n − 1 and {γ} are disjoint; then equating the cardinalities we obtain d a n 1 d a n−1 d a n−2 · · · d a 2 1. In a similar way we have d a n and, by subtraction, last result.

Fibonacci Numbers
In this section we determine a relation involving the Fibonacci numbers. Denote by A i the set of a.c. permutations of n fixing the integer i.
Proof. For i equal to 1 or to n an a.c. permutation π of A i is the product of the unitary cycle i and an a.c. permutation of a set of cardinality n − 1. Then by Proposition 2.2 the number of these permutations is 2 n−2 .
For i / ∈ {1, n} the permutation π is the product of an a.c. permutation of the set i − 1 by the cycle i and an a.c. permutation of the set {i 1, . . . , n} having cardinality n − i. Then the number of these permutations is Theorem 3.2. For every positive integer n one has the identity where s 0 denotes the number of all the adjacent cycle permutations of n .
Proof. The identity follows from the inclusion-exclusion principle, the relations 3.1 and 3.2 , and the condition that s 0 2 n−1 , by Proposition 2.2.

A Combinatorial Interpretation
In this section we give an interpretation of the adjacent cycle permutations and the adjacent cycle derangements of a finite set in terms of linear species of particular linear partitions. See 3 for the fundamental notions of the theory of species. We recall that a linearly ordered set or a linear order L, ≤ is a set L together with a linear order relation ≤. We will write simply L for L, ≤ and |L| for its cardinality.
A k-interval of L is a subset of the form having exactly k elements. Let Lin be the category of finite linearly ordered sets and order bijections, and let Set be the category of finite sets and functions.
A linear species of combinatorial structures is a functor S : Lin → Set.
The cardinality of a linear species S is the formal geometric series where 0 ∅ and n {1, 2, . . . , n} for n > 0. A linear partition π of a finite linear order L is a family of nonempty disjoint intervals, called blocks, whose union is L. For an integer h, we will say that a block B is an h-block of π when B has cardinality h.
Let S and T be two linear species, with T ∅ ∅. The composition S • T is the species equivalent to giving a partition π of L, a structure of species T on each block of π, and a structure of species S on π.
The geometric species or uniform species G is the species defined by G L { * } a singleton , for all finite linear orders L.

ISRN Discrete Mathematics
The cardinality of G is the geometric series The linear species G − 1 is defined by for every L ∈ Lin. Its cardinality is the series For an integer h, the h-geometric species G h is defined as the singleton on every linear order L of cardinality h and as the empty set in all other cases: Thus the cardinality of G h is the series G h t t h .

The Linear Species of Adjacent Cycle Permutations
Let P be the linear species of the linear partitions. Giving a linear partition of a finite linear order L is equivalent to giving a partition π of L, a structure of a nonempty linear order on each block of π, and a geometric structure on π. Therefore we have the isomorphism Then, passing to the cardinalities and setting |P | P t Σp n t n , the relation 4.7 becomes Expanding the series, we obtain p n 2 n−1 for n > 0 1 for n 0.

4.9
From the definition it follows that an adjacent cycle permutation of n corresponds to a linear partition of the linear order L 1, 2, . . . , n in which the blocks are nonempty.

The Linear Species of the Adjacent Cycle Permutations with h Cycles
Let P h be the linear species of the linear partitions with h blocks, and let p h,n be the cardinality of P h L when |L| n. Giving a structure of species P h on a finite linear order L is equivalent to giving a partition π of L, a structure of linear order with at least 1 element on each block of π, and a structure of a linear order with h elements on π. Therefore we have Passing to the cardinalities we have

4.13
For k h n, we obtain

The Linear Species of Adjacent Cycle Derangements
An adjacent cycle derangement of n is an a.c. permutation without fixed points.
The linear species Δ a of the a.c. derangements of a finite set corresponds to giving a finite linear order L, a partition π of L, a structure of linear order of cardinality at least 2 on each block of π of L, and a geometric structure on π. Therefore Passing to the cardinalities we obtain Σ n≥0 d a n t n t

4.16
As t/ 1 − t − t 2 Σ k≥0 F k · t k is the series of the Fibonacci numbers with initial conditions F 0 0, F 1 1, then d a n F n−1 . 4.17

The Linear Species of the Adjacent Cycle Derangements with h Cycles
Let Δ a h be the linear species of the a.c. derangements with h cycles, and let d a h,n be the cardinality of Δ a h L when |L| n. In order to give a structure of species Δ a h on a finite linear order L one has to give a partition π of L, a structure of linear order with at least 2 elements on each block of π, and a structure of a linear order with h elements on π. Therefore we have Passing to the cardinalities we have

Adjacent Cycle Derangements of a Multiset
Let M {1 a 1 2 a 2 · · · m a m } be a multiset, where a i , the multiplicity of the element i, satisfies the condition a i ≥ 1. Recall, 5, page 24 , that the intercalation product α β of two permutations α and β of M is obtained by the two-line notation of α and β, setting beside these two-line representations and sorting the columns into nondecreasing order of the top line. It is known that every permutation π of M has a unique representation as the intercalation of cycles π x 11 · · · x 1n 1 y 1 x 21 · · · x 2n 2 y 2 · · · x i1 · · · x tn t y t , 5.1 where t ≥ 0, In other words in this representation, said canonical, the smallest element of each cycle is at the end of the cycle and is smaller than every other element; moreover the sequence 5.2 of last elements is in nondecreasing order. Now if we drop the parentheses and the 's in this representation, we obtain a sequence which represents a permutation π in one-line form. Namely the one-line representation of π is x 11 · · · x 1n 1 y 1 x 21 · · · x 2n 2 y 2 · · · x i1 · · · x tn t y t .

5.4
This procedure determines a function g : M → M such that g π π , which turns out to be a bijection.
In 2 it is proved that a permutation π is g-fixed if and only if in every cycle the elements form a nondecreasing sequence of integers but the last element which is lesser than every other in the cycle. In the case in which M is a set we recognize the notion of adjacent cycle permutation. Now we define a permutation of a multiset adjacent cycle when in every cycle two consecutive elements a, b satisfy the condition that either b a 1 or b a.
In particular an a.c. permutation without 1-cycles is an a.c. derangement.
In In the case of the set n we obtain Proposition 2.6. In Proposition 2.2 it is proved that the number of a.c. permutations of n is 2 n−1 ; now we prove that this number coincides with the number of a.c. derangements of particular multisets. Proof. Let α be an a.c. permutation of the ordered set {1, 2, . . . , m − 1}, having k cycles; then α is obtained by inserting k − 1 dividers in the positions between an integer and its consecutive in the sequence 1, 2, . . . , m − 1 . Let f be the map that associates to α the a.c. permutation α of M such that if in α there is a divider between t and t 1 then in α a divider is inserted before last and last but one of the elements t. Because the multiplicity of every element of M, but the first and possibly the last one, is greater than 1, then between two distinct dividers there are at least two distinct elements and the permutation turns out to be a derangement. Conversely, let β be an a.c. derangement of M having k cycles. We may represent β by inserting k − 1 suitable dividers in the positions before last element of the sequence i, . . . , i, having a i elements, 1 ≤ i ≤ m. In correspondence of every possible divider before last i we insert a divider between i and i − 1 in the sequence 1, 2, . . . , m − 1 . Thus we obtain an a.c. permutation of the set m − 1 having k cycles. This implies that there is a bijection between D a M and P m − 1 . Passing to the cardinalities we obtain the enumerative result.
Note that the result of Theorem 5.3 does not depend on the multiplicities, but only on the condition that they are greater than 1 but a 1  As particular case we obtain the following result.  Then d a M F k 2 .
Proof. The set of a.c. derangements of M is partitioned into the sets P 1 , . . . , P k , where P i , 1 ≤ i ≤ k, is the set of the derangements which contain the cycle 2 a−1 1 extended to the i elements 2, b 1 , . . . , b i−1 , and one of the derangements of the set {b i , . . . , b k }. The case i 1 corresponds to the derangement 2 a 1 . Thus the number of a.c. derangements is F k F k−1 · · · F 2 F 1 F 1 . Then the result follows.