Journal of Applied Mathematics

Journal of Applied Mathematics / 2015 / Article

Research Article | Open Access

Volume 2015 |Article ID 879510 | 6 pages | https://doi.org/10.1155/2015/879510

On Distance -Fibonacci Numbers and Their Combinatorial and Graph Interpretations

Academic Editor: Ali R. Ashrafi
Received07 May 2015
Accepted02 Sep 2015
Published14 Sep 2015

Abstract

We introduce three new two-parameter generalizations of Fibonacci numbers. These generalizations are closely related to -distance Fibonacci numbers introduced recently. We give combinatorial and graph interpretations of distance -Fibonacci numbers. We also study some properties of these numbers.

1. Introduction

In general we use the standard terminology of the combinatorics and graph theory; see [1]. The well-known Fibonacci sequence is defined by the recurrence for with . The Fibonacci numbers have been generalized in many ways, some by preserving the initial conditions and others by preserving the recurrence relation. For example, in [2] -Fibonacci numbers were introduced and defined recurrently for any integer by for with , . In [3] the following generalization of the Fibonacci numbers was defined: for an integer such that and with and . Other interesting generalizations of Fibonacci numbers are presented in [4, 5]. In the literature there are different kinds of distance generalizations of . They have many graph interpretations closely related to the concept of -independent sets. We recall some of such generalizations:(1)Reference [6]. Consider for with for .(2)References [4, 7, 8]. Consider Fibonacci -numbers for any given and with and for .(3)Reference [9]. Consider for with for .(4)Reference [9]. Consider for with for , , , , for (5)Reference [9]. Consider for with for , , for , for (6)Reference [10]. Consider for with (7)Reference [11]. Consider for with(8)Reference [11]. Consider for with

In this paper we introduce three new two-parameter generalizations of distance Fibonacci numbers. They are closely related with the numbers , , presented in [10, 11]. We show their combinatorial and graph interpretations and we present some identities for them.

2. Distance -Fibonacci Numbers

Let , , and be integers. We define distance -Fibonacci numbers of the first kind by the recurrence relation with the following initial conditions:For we get . These numbers were introduced in [10].

If and , then gives the Fibonacci numbers . For and the numbers are the well-known Padovan numbers.

Table 1 includes the values of for special values of and .




Let , , and be integers. We define the distance -Fibonacci numbers of the second kind by the following recurrence relation: with initial conditions For we have then ; see [10]. Moreover, for and , .

In Table 2 a few first words of the distance -Fibonacci numbers of the second kind for special values of and are presented.




Let , , and be integers. We define distance -Fibonacci numbers of the third kind by the following recurrence relation: with initial conditions For we get . These numbers were introduced in [11]. For , , and we have . Moreover, for , , and .

Table 3 includes a few initial words of distance for special values of and .




By the definition of distance -Fibonacci numbers of three kinds we get for and the following relations:

3. Combinatorial and Graph Interpretations of Distance -Fibonacci Numbers

In this section we present some combinatorial and graph interpretations of distance -Fibonacci numbers. The classical Fibonacci numbers have many combinatorial interpretations. One of them is the interpretation related to set decomposition. We recall it. Let , , and be a family of disjoint subsets of such that(1),(2)if then contains two consecutive integers,(3).It is well known that the number of all families is equal to the classical Fibonacci numbers . We introduce analogous interpretation of distance -Fibonacci numbers.

Let and , , be the set of integers. Let . Assume that is a multifamily of two-element subsets of such that For fixed , by we denote a subfamily of such that . Analogously for fixed we define .

Let , , be a subfamily of such that and(a) for each , holds , for each , , holds ,(b) for each holds for and exactly one of the following conditions for and , respectively, is satisfied:(c1) or ,(c2),(c3) and if then either or .Assume that the condition (c1) is satisfied. Then the subfamily we will call a decomposition with repetitions of the set with the rest at the end.

Assume that the condition (c2) is satisfied. Then the subfamily we will call a perfect decomposition with repetitions of the set .

Assume that the condition (c3) is satisfied. Then the subfamily we will call a decomposition with repetitions of the set with the rest at the end or at the beginning.

Theorem 1. Let , , and be integers. Then the number of all decompositions with repetitions of the set with the rest at the end is equal to the number .

Proof (induction on ). Let , , and be integers. Let . Denote by the number of all decompositions with repetitions of with the rest at the end. Let . Then it is easily seen that there are exactly decompositions of . Thus we get . Let . Assume that equality holds for an arbitrary . We will show that .
Let and denote the number of all decompositions with repetitions of the set with the rest at the end such that and , respectively. It is easily seen that Moreover, we getBy the induction hypothesis and by recurrence (4) we obtainwhich ends the proof.

Analogously as Theorem 1 we can prove the following.

Theorem 2. Let , , and be integers. Then the number of all perfect decompositions with repetitions of the set is equal to the number .

Theorem 3. Let , , and be integers. Then the number of all decompositions with repetitions of the set with the rest at the end or at the beginning is equal to the number .

Distance -Fibonacci numbers of three kinds have a graph interpretation, too. It is connected with -distance -matchings in graphs. We recall the definition of a -distance -matching. Let and be any two graphs, let be an integer, and a -distance -matching of is a subgraph of such that all connected components of are isomorphic to and for each two components and from for each and holds . In case of and we obtain the definition of matching in classical sense. If covers the set (i.e., ), then we say that is a perfect matching of . For and the definition of -distance -matchings reduces to the definition of an independent set of a graph . In the literature the generalization of -matching of a graph is considered, too. For a given collection of graphs a -matching of is a family of subgraphs of such that each connected component of is isomorphic to some , . Moreover, the empty set is a -matching of , too. If for all , then we obtain the definition of -matching.

Among -matchings we consider such -matchings, where , , belong to the same class of graphs, namely, -vertex or -vertex paths ( and , resp.), .

Consider a multipath , where , , , and

Let , , and be integers. In the graph terminology the number is equal to the number of special -matchings of the multipath such that at most one vertex, namely, , does not belong to a -matching of the graph . We will call such matchings a quasi-perfect matching of . The number is equal to the number of such -matchings of that both vertex and vertex belong to some -matchings and , respectively, of the graph . In other words the number is equal to all perfect -matchings of the graph .

The number is equal to the number of special -matchings of the multipath such that at most one vertex either vertex or does not belong to a -matching of the graph .

Let be the number of all perfect -matchings of the graph .

Theorem 4. Let , , and be integers. Then .

Proof. Consider a multipath where vertices from are numbered in the natural fashion. Let and be the number of perfect -matchings of such that and , respectively. It is easily seen that .
Let be an arbitrary perfect -matching of , . Consider two cases:(1), where .
Then we can choose the edge on ways. Moreover, , where is an arbitrary -matching of the graph which is isomorphic to the multipath . Hence .(2), where .
Proving analogously as in case () we obtain .
Consequently Claim Proof. Assume now that the set corresponds to with the numbering in the natural fashion. Let be a multifamily of which gives a perfect decomposition of the set . Then every and correspond to subgraph and for , respectively, of . By Theorem 2 we get Moreover, by (6) we obtain , which ends the proof.

Analogously we can prove combinatorial interpretations of numbers and .

4. Identities for Distance -Fibonacci Numbers

In this section we give some identities and some relations between distance -Fibonacci numbers of three types.

Theorem 5. For , , and ,

Proof. We give the proof for distance -Fibonacci numbers of the first kind. By the definition of numbers , we havewhich ends the proof.

Corollary 6. For

Proof. For , , and by (19) we obtain Hence

Theorem 7. For , , and ,

Proof (induction on ). For we have