The permanent, a new graph labeling, and a known integer sequence
Abstract.
Let be an by matrix. For any real number , we define the polynomial
as the permanent of , where is the number of inversions of the permutation in the symmetric group . In this note, motivated by this notion, we discuss a new graph labeling for trees whose matrices satisfy certain permanental identities. We relate the number of labelings of a path with a known integer sequence. Several examples are provided.
Key words and phrases:
permanent, permanent, determinant, permanent, graph, tree, path, graph labeling, integer sequence, Mathematica2000 Mathematics Subject Classification:
15A15, 05C50, 05C78, 05C30, 68R10, 11B831. Introduction
Given an matrix and a real number , we define the permanent of as the polynomial
(1.1) 
where is the number of inversions of the permutation in the symmetric group of degree , i.e., the number of interchanges of consecutive elements necessary to arrange in its natural order [13, p.1] or, equivalently,
For example, we have
and
The permanent of a square matrix is a natural extension of the determinant (setting ) and the permanent (setting ) which is in fact quite hard to compute [5, p.190]. In addition, making , we get the product of the main diagonal entries of the matrix.
This concept was introduced independently and almost simultaneously by different authors by a quarter of a century ago under different names in matrix theory and, surprisingly, in Grassmann algebras and quantum groups: permanent is just one of them [3, 4, 5, 9, 11, 12, 14, 15, 18, 19, 20]. Here we adopt one of the possible ways to call this matricial function (cf. also [6, 7, 8]).
We believe that this code will be particularly useful for further developments on the properties of the permanent.
It is clear that in general, under similarity, the permanent does not keep the same value, i.e., the polynomial is not necessarily the same as , for nonsingular. In particular, for permutation similarity. This means that interchanging rows and columns of the same indexes leads to possible different permanents. Since interchanging rows and columns does not change the underlying graph of the matrices involved, but the labeling of the vertices, we conclude that the permanent of a graph depends on its labeling.
After a first attempt to extend monotonic properties of the permanent of Jacobi positive definite matrices to more general acyclic matrices [8], it has recently been noticed a particular labeling for which the previous properties were indeed satisfied [6].
In this note we aim to discuss this new labeling for trees, counting them for paths. Incidentally, this process will lead to a new interpretation for a wellknown integer sequence.
2. A new graph labeling
Given a symmetric matrix , the graph of is defined by the zerononzero offmain diagonal pattern of . In general, the vertex labeling is not much discussed in matrix theory since most of the results involve the spectra of matrices, which do not change by such labelings. For example, the underlying graph of a tridiagonal matrix is a path with the vertices ordered successively and edges joining consecutive vertices and :
However, as we mentioned in the introduction, this is not the case for the permanent of a square matrix. For instance, we have
which is a polynomial of degree , and
which in turn has degree . The “graph” is the same but the labeling is not, i.e.,
and
respectively.
In order to establish several general results for the permanent, recently in [6], the second author introduced the following labeling:
Labeling 1.
Given two disjoint edges and , say , , and , then one of the following conditions must be fulfilled:


.
To the best of our knowledge, this labeling is new and will be referred to in the remaining of this paper as a labeling. For example, we have the following:
Interestingly, not all graphs allow such labeling: a complete graph of more than vertices is just an example. However, any tree allows labelings satisfying the conditions described. In what follows, we present an algorithm to construct one of such labeling:
Algorithm 1.
Let us consider a tree with a given number of vertices.

Choose any vertex from the tree and label it with , which will be the root.

Take the largest path attached to vertex .

Label the vertices of this path by , where is an edge, for .

Choose the vertex with largest label in the previous with degree greater than two, say .

Repeat step 2., replacing by .

Repeat step 3., labeling the vertices of the path by .

Once all vertices of degree more than two were considered, restart from 2., choosing the second largest path attached to the root and proceed until all vertices were considered.
We remark that, if there is more than one path attached to the root of the same largest size we choose arbitrarily one of them. Clearly, this is not the only way to construct such labeling.
As a simple example of the algorithm, we have
Returning to the permanent, as a consequence, for any matrix whose graph is a tree with the vertices labeled as described before, one always has
for any vertex , or
with , (cf. [8, 7, 6]). Here, is the matrix obtained from replacing the rows and columns indexed by , by zero, except the entries in the main diagonal, which are 1’s.
3. Counting labelings for paths
In this section we confine our study to paths. Our algorithm provides the following example for a path of vertices:
The following labeling is also a possibility
but
is not.
A Mathematica routine that computes all the possible labelings for an order path is given in Listing 2. For example, the distinct labelings for a path with for 4 vertices are
Table 3.1 presents the number of distinct labelings for paths of order up to , which were computed with the same routines.
order  #labelings 
2  1 
3  3 
4  8 
5  20 
6  48 
order  #labelings 

7  112 
8  256 
9  576 
10  1280 
11  2816 
Remarkably, this exhaustive enumeration leads us exactly to the integer sequence A001792 of the The OnLine Encyclopedia of Integer Sequences [17]. This sequence has many different interpretations. Originally, we will find it in [10] (cf. also [1, Table 22.3]) in the absolute value of the coefficients of for the Chebyshev polynomials of the first kind . The most simple formula is perhaps , for each positive integer . This sequence emerges also from the Bernoulli’s triangle rows sums [2, 16]. Nonetheless, we can recent find a vast number of interesting interpretations for this sequence. Namely, it is the determinant of the square matrix with 3’s on the diagonal and 1’s elsewhere, or the absolute value of the determinant of the Toeplitz matrix with first row containing the first integers [17].
Regarding other Mathematica routines, one can find several collected in [17] such as:
matrix[n_Integer /; n >= 1] := Table[Abs[p  q] + 1, q, n, p, n];
a[n_Integer /; n >= 1] := Abs[Det[matrix[n]]]
or
g[n_, m_, r_] := Binomial[n  1, r  1] Binomial[m + 1, r] r;
or
Table[1 + Sum[g[n, k  n, r], r, 1, k, n, 1, k  1], k, 1, 29]
or
LinearRecurrence[4, 4, 1, 3, 40]
or even
CoefficientList[Series[(1  x) / (1  2 x)^2, x, 0, 40], x]
One interesting open question is formally prove that this new labeling leads to the integer sequence A001792.
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