#### Abstract

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.

#### 1. Introduction

Let , and let denote the set of permutations of . We will use both the one-line notation of a permutation of 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 .

In  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 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 elements, having as the smallest element, is represented by the sequence .

In  it is proved that the adjacent cycle permutations of turn out to be fixed by the bijection that associates to a permutation of the permutation whose one-line form is obtained from the canonical form of by removing all inner parentheses.

For example, let .

Notice that we omit commas between the integers in a cycle or in a sequence, unless it leads to ambiguity.

We will denote by and the set of adjacent cycle permutations and the set of adjacent cycle derangements of , respectively, and and their cardinalities; moreover, represents the set of adjacent cycle derangements having cycles and its cardinality.

We say that an adjacent cycle permutation of is extended to , where , when are inserted in this order in the last cycle of before its last element. Clearly we obtain an adjacent cycle permutation of .

For example, the extension of the permutation of to determines the new permutation of .

Let denote the th Fibonacci number, with initial conditions , .

We now outline the contents of this paper. In Section 2 we determine properties of the adjacent cycle permutations and adjacent cycle derangements of , 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.

In this section we investigate the adjacent cycle permutations and adjacent cycle derangements of the set , for a positive integer . First we determine the number of adjacent cycle derangements having a fixed number of cycles.

Theorem 2.1. For positive integers and such that one has

Proof. An adjacent cycle derangement of the set with cycles is defined by inserting dividers in the positions between two consecutive integers of the sequence , so that the numbers of integers before the first divider, between two consecutive dividers, and after last divider are greater than . As a consequence we have to exclude the positions between and and between and and the positions that immediately follow every divider, but the last one; in other words, we have to exclude among the possible positions. Thus the number turns out to be equal to

In the more general case of adjacent cycle permutations of , the number of the adjacent cycle permutations having cycles is obtained by choosing among the possible positions. See also .

Proposition 2.2. The number of adjacent cycle permutations of having cycles, where , is , and the number of all adjacent cycle permutations is .

Proposition 2.3. For every positive integer one has

Proof. Note that the maximum number of cycles of an adjacent cycle derangement of is obtained by setting the length of all cycles equal to , with possibly one cycle of length , if is odd. Then the maximum value of is , and therefore we obtain
Moreover Thus satisfies the same recursive relation of the Fibonacci numbers. It is immediate to verify that and ; then .

Another proof of (2.4) may be obtained in a bijective way.

Theorem 2.4. For every integer one has Hence , where the initial conditions satisfy and ; thus .

Proof. Let be an element of . We have two cases to consider.(1)belongs to a cycle of with more than elements. Then is the extension to of a permutation . (2) belongs to a cycle having 2 elements. The is the product of an a.c. derangement and the cycle .
The sets and are clearly disjoint; then . By equating the cardinalities we obtain , where the initial conditions hold and . Then .

Example 2.5. Let us consider the case of the set .
The a.c. permutations are the following permutations represented in canonical form: The a.c. derangements are (2 1)(4 3), (2 3 4 1). Their number coincides with the Fibonacci number .
The a.c. permutations having a fixed number of cycles, for instance 2, are , while the number of a.c. derangements having cycles is .

Another bijective proof of the results of Theorem 2.4 is obtained in the following theorem.

Proposition 2.6. Let be a positive integer. Then Moreover , with .

Proof. Let , and let , where . Then is the product of an a.c. derangement of and the cycle . The cases and are excluded because they imply a cycle of length . In particular , where .
The sets , and are disjoint; then equating the cardinalities we obtain .
In a similar way we have and, by subtraction, last result.

#### 3. Fibonacci Numbers

In this section we determine a relation involving the Fibonacci numbers.

Denote by the set of a.c. permutations of fixing the integer .

Proposition 3.1. For every positive ,

Proof. For equal to or to an a.c. permutation of is the product of the unitary cycle and an a.c. permutation of a set of cardinality . Then by Proposition 2.2 the number of these permutations is .
For the permutation is the product of an a.c. permutation of the set by the cycle and an a.c. permutation of the set having cardinality . Then the number of these permutations is .

Denote

Theorem 3.2. For every positive integer one has the identity where denotes the number of all the adjacent cycle permutations of .

Proof. The identity follows from the inclusion-exclusion principle, the relations (3.1) and (3.2), and the condition that , by Proposition 2.2.

As an example consider the case . Then , , , , .

Then .

#### 4. 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  for the fundamental notions of the theory of species.

We recall that a linearly ordered set or a linear order is a set together with a linear order relation . We will write simply for and for its cardinality.

A -interval of is a subset of the form having exactly 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 .

The cardinality of a linear species is the formal geometric series where and for .

A linear partition of a finite linear order is a family of nonempty disjoint intervals, called blocks, whose union is . For an integer , we will say that a block is an -block of when has cardinality .

Let and be two linear species, with . The composition is the species equivalent to giving a partition of , a structure of species on each block of , and a structure of species on .

The geometric species or uniform species is the species defined by (a singleton), for all finite linear orders .

The cardinality of is the geometric series The linear species is defined by for every . Its cardinality is the series For an integer , the -geometric species is defined as the singleton on every linear order of cardinality and as the empty set in all other cases: Thus the cardinality of is the series .

##### 4.1. The Linear Species of Adjacent Cycle Permutations

Let be the linear species of the linear partitions. Giving a linear partition of a finite linear order is equivalent to giving a partition of , a structure of a nonempty linear order on each block of , and a geometric structure on . Therefore we have the isomorphism

Then, passing to the cardinalities and setting , the relation (4.7) becomes

Expanding the series, we obtain

From the definition it follows that an adjacent cycle permutation of corresponds to a linear partition of the linear order in which the blocks are nonempty.

For example, the linear partitions of are The corresponding a.c. permutations of in cyclic form are Then the number of a.c. permutations coincides with the number of linear partitions of .

##### 4.2. The Linear Species of the Adjacent Cycle Permutations with Cycles

Let be the linear species of the linear partitions with blocks, and let be the cardinality of when .

Giving a structure of species on a finite linear order is equivalent to giving a partition of , a structure of linear order with at least element on each block of , and a structure of a linear order with elements on . Therefore we have

Passing to the cardinalities we have

For , we obtain

In the previous section we interpreted as the linear species of a.c. permutations; now we interpret as the linear species of a.c. permutations with cycles.

From Proposition 2.2 we have . This value coincides with the coefficient of in the previous generating function when .

##### 4.3. The Linear Species of Adjacent Cycle Derangements

An adjacent cycle derangement of is an a.c. permutation without fixed points.

The linear species of the a.c. derangements of a finite set corresponds to giving a finite linear order , a partition of , a structure of linear order of cardinality at least on each block of of , and a geometric structure on . Therefore Passing to the cardinalities we obtain As is the series of the Fibonacci numbers with initial conditions , , then

##### 4.4. The Linear Species of the Adjacent Cycle Derangements with Cycles

Let be the linear species of the a.c. derangements with cycles, and let be the cardinality of when .

In order to give a structure of species on a finite linear order one has to give a partition of , a structure of linear order with at least elements on each block of , and a structure of a linear order with elements on . Therefore we have

Passing to the cardinalities we have

For , we obtain which coincides with the result of Theorem 2.1.

In  the above-mentioned species were investigated in a different context.

#### 5. Adjacent Cycle Derangements of a Multiset

Let be a multiset, where , the multiplicity of the element , satisfies the condition . Recall, [5, page ], that the intercalation product of two permutations and of is obtained by the two-line notation of and , setting beside these two-line representations and sorting the columns into nondecreasing order of the top line.

It is known that every permutation of has a unique representation as the intercalation of cycles where , 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 in one-line form. Namely the one-line representation of is This procedure determines a function such that , which turns out to be a bijection.

In  it is proved that a permutation is -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 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 satisfy the condition that either or .

In particular an a.c. permutation without -cycles is an a.c. derangement.

In relation to the multiset , where , for , denote by and the sets of a.c. derangements of and a.c. derangements of having cycles, respectively, and and their cardinalities.

Theorem 5.1. Let be a multiset, where , for . Then where and ; it implies .

Proof. Let be an element of . Denote the elements of coinciding with . We have two cases to consider.(1) belongs to a cycle of with more than distinct elements. Then is the extension to of a permutation . (2) belongs to a permutation having 2 distinct elements. Then is product of a cycle and the cycle .
The sets and are clearly disjoint; then . Equating the cardinalities we obtain .

Note that if is the set , then , , and the initial conditions hold , ; thus we obtain the results of Theorem 2.4.

Proposition 5.2. Let be a multiset, where , for . Then

Proof. Clearly the sets form a partition of , and there is a bijection between and . Equating the cardinalities we obtain the relation (5.7).

In the case of the set we obtain Proposition 2.6.

In Proposition 2.2 it is proved that the number of a.c. permutations of is ; now we prove that this number coincides with the number of a.c. derangements of particular multisets.

Theorem 5.3. Let be an integer and a multiset, where for and . Then and .

Proof. Let be an a.c. permutation of the ordered set , having cycles; then is obtained by inserting dividers in the positions between an integer and its consecutive in the sequence . Let be the map that associates to the a.c. permutation of such that if in there is a divider between and then in a divider is inserted before last and last but one of the elements . Because the multiplicity of every element of , 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 be an a.c. derangement of having cycles. We may represent by inserting suitable dividers in the positions before last element of the sequence , having elements, . In correspondence of every possible divider before last we insert a divider between and in the sequence . Thus we obtain an a.c. permutation of the set having cycles. This implies that there is a bijection between and . 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 and possibly ).

For instance, for the a.c. permutations of are The corresponding a.c. derangements of are

Proposition 5.4. Let and be two multisets, where , for , , . Then .

Proof. By Theorem 5.3 the values of and do not depend on the multiplicity of .

As particular case we obtain the following result.

Proposition 5.5. Let be a multiset, where . Then and .

Proof. An a.c. derangement of the multiset , where , with cycles of length greater than coincides with the number of a.c. derangement of the set with cycles. Then and follow from Theorem 2.1 and Proposition 2.3.

For instance, let and . In order to obtain the a.c. derangements of we have to consider the three positions . Assign or to a position when there is or there is not a divider; we obtain that the possible sequences are , , , , and . The corresponding derangements are , , , , and .

Lemma 5.6. Let be a multiset, where .
Then .

Proof. The set of a.c. derangements of is partitioned into the sets , where , , is the set of the derangements which contain the cycle extended to the elements , and one of the derangements of the set . The case corresponds to the derangement . Thus the number of a.c. derangements is . Then the result follows.

Theorem 5.7. Let be a multiset, where for .
Then .

Proof. The proof is by induction on . For the result follows from Lemma 5.6.
Now assume the result holds for every integer lesser than .
Note that the elements of can be partitioned into two sets and , where consists on the set of the derangements which are the product of the cycle formed by the elements of the multiset and the derangements of the multiset , while is formed by the derangements obtained from the elements of , where the first cycle now contains the elements of .
It implies that and, by the induction assumption, .

Theorem 5.8. Let be a multiset where , , , and .
Then

Proof. The proof is by induction on .
Let . Then the set of derangements is partitioned into the set of derangements of with the addition of the cycle and the set of derangements obtained by extending last cycle of every derangement of to the elements .
Both the sets have cardinality by Proposition 5.5; then the number of derangements is .
Now we prove the result assuming it holds for every integer lesser than .
The set of derangements of is partitioned into the set of derangements of with the addition of the cycle and the set of derangements obtained from by the extension of last cycle of every derangement to the elements , . By the assumption hypothesis , and, since , we obtain the result.

#### Acknowledgment

The work was partially supported by MIUR (Ministero dell'Istruzione, dell'Università e della Ricerca).