The mPath Cover Polynomial of a Graph and a model for General Coefficient Linear Recurrences

An m-path cover Γ = {P`1 , P`2, . . . , P`r} of a simple graph G is a set of vertex disjoint paths of G, each with `k ≤ m vertices, that span G. With every P` we associate a weight, ω(P`), and define the weight of Γ to be ω(Γ) = ∏r k=1 ω(P`k ). The m-path cover polynomial of G is then defined as Pm(G) = ∑ Γ ω(Γ), where the sum is taken over all m-path covers Γ of G. This polynomial is a specialization of the path-cover polynomial of Farrell. We consider the m-path cover polynomial of a weighted path P (m−1, n), and find the (m+1)-term recurrence that it satisfies. The matrix form of this recurrence yields a formula equating the trace of the recurrence matrix with the mpath cover polynomial of a suitably weighted cycle C(n). A directed graph, T (m), the edge-weighted m-trellis, is introduced and so a third way to generate the solutions to the above (m+ 1)-term recurrence is presented. We also give a model for general term linear recurrences and time dependent Markov chains. * Corresponding author.


Introduction, -Path Cover Polynomial, and Notation
Let be a graph with no loops or multiple edges, with vertex set ( ).
First we review some basic concepts to establish notation. A path ℓ in is a sequence of distinct vertices ℓ = [V 1 , V 2 , . . . , V ℓ ] where each pair (V , V +1 ) for 1 ≤ ≤ ℓ − 1 is an edge. The length of a path is the number of vertices in it. Thus a path of length 1 is a vertex and a path of length 2 is an edge, and ℓ has length ℓ. Path ℓ begins at vertex V 1 , its first vertex, and ends at vertex V ℓ , its last vertex. The path [V 1 , V 2 , . . . , V ℓ ] and its reverse [V ℓ , V ℓ−1 , . . . , V 1 ] are considered to be the same path. The set of vertices in ℓ is ( ℓ ) = {V 1 , V 2 , . . . , V ℓ }. Two paths ℓ and ℓ in are disjoint if ( ℓ ) ∩ ( ℓ ) = 0. The empty path has 0 vertices. Finally, recall that a subgraph of spans if it has the same vertex set as . Now we introduce the central concept of this paper. An -path ℓ has ℓ ≤ ; that is, it is a path of length at most for some fixed with 1 ≤ ≤ | ( )|.
With every -path ℓ we associate a weight, ( ℓ ), and then the weight of Γ is (Γ) = ∏ =1 ( ℓ ). Definition 1. The -path cover polynomial of , P ( ), is the sum of the weights of all -path covers of ; that is, where Γ is an -path cover of .
The path-cover polynomial (or path polynomial) of a graph is a specialization of the -cover polynomial of Farrell [1] where is restricted to be a path; see Farrell [2]. Thus our -path cover polynomial P ( ) is a further specialization to paths of length ℓ ≤ . See also Chow [3] and D' Antona and Munarini [4]. 2 International Journal of Combinatorics It seems that this research is the first direct consideration of the -path cover polynomial of a graph. See McSorley et al. [5] for specialization to the case = 2, where all classical orthogonal polynomials are generated as 2-path cover polynomials of suitably weighted paths. For related work see the theory of weighted linear species developed in Joyal [6] and Bergeron et al. [7]. In particular, Munarini [8] uses the -filtered linear partitions of a linearly ordered set to achieve some similar results; see especially our Sections 7 and 8.
In Section 2 we introduce a weighted path ( −1, ) and find the ( + 1)-term recurrence that its -path polynomial satisfies. In Section 3 the matrix form of this recurrence is presented and yields a trace formula that, in Section 4, gives the -path cover polynomial of a suitably weighted cycle ( ). Section 5 interprets our results in terms of a model for time-dependent Markov chains. In Section 6 a directed graph, ( ), the edge-weighted -trellis, is introduced and so a third way to generate the solutions to the above recurrence and trace is found. In Section 7 we model general constant coefficient linear recurrences, and we derive various relevant formulas with both algebraic and combinatorial proofs. Finally, in Section 8, we obtain a relevant new integer sequence and relate this sequence to known sequences in the literature.
, and so forth.
Vertices in ( − 1, ) (Section 2) and in subpaths of ( , ) will be labelled ; vertices in ( ) (Section 4) will be labelled V ; and vertices in ( ) (Section 6) will be labelled . For 1 ≤ ℓ ≤ we use indeterminate ℓ, as the weight of a path of length ℓ in . Throughout the paper ≥ 1 is fixed. In all the examples we set = 3, and many examples have = 4.
Example 7. For = 3 and = 4, (9) see Example 5. For a fixed with 0 ≤ ≤ − 1 we define the starting conditions, We then have the following recurrence; the proof is similar to the proof of Theorem 3, and setting = − 1 recovers Theorem 3.
We now work with the fundamental solutions to recurrence (2).
The first vertex 1 lies in every -path cover of ( − 1, 1) so, similar to the proof of Theorem 3, we have Thus, from above, using (12) again at the first line, the induction hypothesis at the second line and Theorem 8 at the last line. Hence the induction goes through and (15) is true for all ≥ 1.
(ii) the first fundamental solution to recurrence (2) is given by Corollary 12 is a useful technical result.
This completes study of the weighted path ( − 1, ).

Matrix Formulation and Trace
We set up our ( + 1)-term recurrence (2) in matrix form.
(28) With tr denoting trace, we have the following.
Lemma 13. For ≥ 1, We now apply these results to the weighted cycle ( ).

Weighted Cycle ( ) and Trace
We introduce the weighted cycle ( ) for ≥ 1 shown in Figure 1. It has vertices labelled {V 1 , V 2 , . . . , V } and edges. It is weighted as follows: for 1 ≤ ℓ ≤ , let ℓ be a path of length ℓ that traverses ( ) clockwise and ends at vertex V . We define ( ℓ ) = ℓ, .
Lemma 14. For 1 ≤ ≤ ≤ the following -path cover polynomials, the first which comes from ( ) and the second from ( − 1, ), are equal: Proof. Except for vertex labels, the weighted paths Hence the result is obtained.
Definition 15. For ≥ 1 let C ( ) be the -path cover polynomial of the weighted ( ).
In the following, when necessary, we reduce subscripts on , V, and the second subscript on , all modulo . We write Proof. Consider the weighted ( ). Vertex V 1 lies in everypath cover of ( ). Suppose, in such an -path cover, it is covered by a path ℓ of length ℓ that begins at V − and ends at V − −1+ℓ , for some ∈ {−1, 0, 1, . . . , ℓ − 2}. Now 1 ≤ ℓ ≤ ; that is, +2 ≤ ℓ ≤ . The sum of the weights of all such paths is then letting = + 2 at the second line, and using subscript reduction modulo and Lemma 14 at the third line, then Corollary 12 at the fourth line, and Lemma 13 at the last line.

Markov Chain Interpretation
In this section we consider an interesting special case, where in the matrix formulation of the recurrence we have stochastic matrices. A matrix of Form (26) can be considered a transition matrix for a Markov chain with states under the conditions Because the probabilities , vary with , these are the transition matrices for a nonhomogeneous Markov chain. Note also that, as transition matrices are multiplied from left to right, the process is effectively time reversed. In fact, This process is often referred to as a ladder process. From any state , with < , the process jumps with certainty to + 1, then to +2, and so forth, up the ladder, till it reaches state . At that point it jumps randomly back down the ladder to one of the intermediates states , 1 ≤ < , and the procedure repeats. Because all of the matrices are stochastic, the row sums of matrices such as , , see (28), will all equal 1. Recall from Section 3 that Thus, we have the following.
Furthermore, under the assumption > 0, forall , it is immediate that the chain is irreducible and aperiodic, hence ergodic. That is, exists and has equal rows, and each row proportional to the left-invariant vector indicated above normalized to row sum 1.
Thus, for large , if we randomly choose an -path cover of ( − 1, ) then the probability that it belongs to thefundamental solution is 2( − + 1)/ ( + 1). In particular, the first fundamental solution satisfies So the -path cover polynomial model provides a combinatorial model for nonhomogeneous Markov chains. A closely related model, the trellis, is discussed in detail below.

Edge-Weighted -Trellis ( )
In this section we deal with the edge-weighted -trellis, ( ), shown in Figure 3, and give another method of generating ( ) , and C ( ). The vertices of ( ) are labelled { 1 , 2 , . . . , }. All edges in ( ) are directed, with arrows as shown. All circuits in ( ) are directed and are traversed in the direction of the arrows. We use to denote a directed circuit in ( ), which we simply call a circuit. A circuit is based at vertex if it begins and ends at vertex . A circuit may pass through the same vertex more than once. The length of a circuit is the number of edges in it.
The weights on the edges of ( ) are taken from {1, 1, , . . . , , } where ≥ 1, as shown. The weight of circuit , ( ), is the product of the weights of all the edges in . If the edge with weight , is traversed as the th edge in , then , is a factor in ( ); thus the meaning of , here is different from that in Sections 2 and 4. We allow empty circuits with length 0.
Definition 20. Let T ( , 0) = 1 and, for ≥ 1, let T ( , ) be the sum of the weights of all circuits in ( ) that are based at vertex with length .
putting ℓ = + at the second line, then using Corollary 22 with = ℓ + 1 − and = − ℓ at the third line, and finally using Corollary 12 at the last line.

Homogeneous Case, ℓ, → ℓ
In this section, we consider the case of constant coefficients, that is, where the indeterminates ℓ, are independent of .
Notation. We use * to modify a path or expression or matrix in which weights or indeterminates ℓ, are replaced with ℓ . First we review some known properties of -path polynomials using standard techniques. Then we show how our model recovers these results combinatorially.

Constant Coefficient
Recurrences. This subsection mainly establishes notation and recalls basic results of interest.
Consider the recurrence We begin with the first fundamental solution. The following is standard and readily derived via geometric series and multinomial expansion.
The matrix takes the form, confor Section 5.1, International Journal of Combinatorics so that det( − ) = 1 − ∑ =1 . Define the ( + 1)st fundamental solution to recurrence (55) to be the one with initial conditions and denote this fundamental solution by ℎ ( +1) , with ℎ = ℎ (1) . Then the entries in the bottom row of ( ) are exactly the values In general, The fundamental solutions for > 0 can be expressed in terms of the first fundamental solution as follows.
Proposition 28. The ( + 1)st fundamental solution to the recurrence (55) is given by where ℎ denotes the first fundamental solution.
Proof. We will illustrate for ≤ 2 that shows how the general case works. We have For = 1, we obtain 0 for nonpositive , except for = −1, as required. Similarly, for = 2, for nonpositive we obtain 1 precisely for = −2; otherwise we get 0. Note that the subtractions are necessary to cancel off terms when 0 ≥ > − . Since the coefficients are independent of , these are indeed solutions to the recurrence. Thus the result is obtained. Now for the trace, we have the following.
Proposition 29. The trace of ( ) is given by Proof. From (60), we have, using Proposition 28, (next, interchanging the order of summation) Remark 30. These are a variation on Newton's Identities relating power sum symmetric functions and elementary symmetric functions. Here, the homogeneous symmetric functions, ℎ , play a role as well.

Combinatorial Proofs.
We now show how these formulas may be derived combinatorially by our model with the specialization ℓ, → ℓ . The weighted path * (2, 3) looks like Notation. Consistent with the above, we use ℎ or H to represent expressions in which we have replaced ℓ, with ℓ . Thus H ( , ) = P * ( , ), for 0 ≤ ≤ −1; see Definition 6 of weighted path ( , ).

First Fundamental Solution.
Proposition 27 is readily seen from the weighting of path * ( − 1, ). For the first International Journal of Combinatorics 11 fundamental solution, there are no vertices with weight 1, and no edges weighted 0. The first vertex has weight 1 , and so on.
In an -path cover the exponent ℓ is the number of paths of length ℓ, for each 1 ≤ ℓ ≤ , and the multinomial coefficient counts the number of -path covers obtained from any fixed set of -paths. So this model gives a visual interpretation to the analytic formula.

Higher Fundamental
where, at the second line, we note that in every -path cover of the weighted path * (0, + ) vertex + must lie on a path ℓ of length ℓ and weight ℓ where 1 ≤ ℓ ≤ , and at the last line we use (68). This gives the result.

Trace Formula.
We now give a combinatorial derivation of the trace formula in Proposition 29.
First let T ( ) be the sum of the weights of all circuits of length in * ( ) and the -trellis with edge-weights ℓ, replaced by ℓ ; that is, T ( ) = ∑ =1 T * ( , ); see Section 6.
Theorem 33. For any ≥ 1, Proof. We recall that the indeterminates in any term of T ( ) are initially ordered according to the edges traversed in the corresponding circuit; see Example 26. Let X = ℓ 1 ℓ 2 ⋅ ⋅ ⋅ ℓ be a typical ordered term in T ( ) with all 1's removed and with first indeterminate . We first show that term X occurs times in T ( ).
Now consider an occurrence of X in which = , namely, Let Then the sequence of edges traversed in * ( ) corresponding to Z begins at 1 and ends at 1 , and so is a circuit based at 1 , with length −1−( −1) = − . Thus Z ∈ T * ( 1 , − ). Conversely given any Z ∈ T * ( 1 , − ) then Z 1 −1 is an occurrence of term X starting with 1 0 and ending with 1 −1 . Thus (∑ = X)/ = T * ( 1 , − ) and ∑ = X = T * ( 1 , − ). Now we can partition the weighted circuits of * ( ) of length by their first indeterminate (ignoring the edges of weight 1 preceding this first indeterminate). That is, we can partition the terms of T ( ) by their first indeterminate . So, using the above arguments, we have Furthermore, T * ( 1 , − ) = P * (0, − ) = (1) * , − = ℎ − ; the first equality is Theorem 21, the second is Corollary 11(ii), and the third is by definition of ℎ . So finally, Example 34. See Examples 17 and 26. Here = 3 and = 4: where, at line 2, we have rearranged the terms according to their first indeterminate , using Example 26, and combined like terms.

Sequences, ℓ, → 1
In Section 7 we specialized by replacing weights ℓ, with ℓ . In this section we specialize further by replacing all weights ℓ, with 1. We denote this operation by #. We then use these # matrices to count -path covers of the path and cycle.
Consider the triangle, in bold, where ( ) is the ( , ) entry for all ≥ 1 and 1 ≤ ≤ ; it counts the number of -path covers of a cycle with vertices. We have entered the sequence obtained from reading this triangle row-by-row to the Online Encyclopedia of Integer Sequences [9]; it is sequence A185722.
Each of the 10 columns of the square array (see Table 1) appears as a sequence in [9]; for example, the second column ( = 2) gives sequence A000204 and the third column ( = 3) gives A001644. Thus we have a new combinatorial interpretation for each of these sequences and a connection between them.
A closely related sequence is A126198 (replace " " by " " in its description). Let ( , ) be the ( , ) entry of the triangle corresponding to A126198, then ( , ) counts the number of compositions of integer into parts of size ≤ . Now consider vertices arranged in a path. A composition of into parts of size ≤ corresponds naturally to an -path cover of this path with vertices by identifying a part of size ℓ in the composition with a path of length ℓ in the corresponding -path cover. This correspondence can also be reversed. Thus in our terminology, ( , ) is the number of -path covers of a path with vertices, and, from Corollary 11(ii) and our operation #, we have ( , ) = (1)# , = P # (0, ). The ( , ) main diagonal entry in this triangle is ( , ) = 2 −1 (as is well known, there are