1. Introduction
In matrix theory, the permanent is a function of matrices that differs from the determinant primarily in its omission of permutation signs. While the determinant has a well-established strong geometric interpretation, the permanent has no such perspective. This distinction extends to their computational complexity; computing the determinant is polynomial time, while computing the permanent is computationally challenging, classified as a #P-complete problem. Despite their differences, both functions share a common combinatorial structure, being defined through summations over permutations. Let
be a matrix of size
m by
n with elements in a field
F and suppose that
. Then, the permanent and determinant of
A are defined as follows [
1]:
and
where the summations runs over on the set
, which includes all one-to-one functions defined from
to
. The only difference between these definitions is the inclusion of the permutation sign,
, in the determinant. However, this seemingly small distinction has major implications for their properties and computational feasibility. While significant progress has been made in the efficient computation of the determinants using various algebraic methods, the inherent complexity of the permanent has motivated researchers to explore alternative approaches. This has led to extensive research into new approaches for evaluating the permanent. A review of recent theoretical developments follows.
Björklund et al. [
2] approached the permanent evaluation problem using combinatorial and algebraic techniques. Their work emphasizes dynamic programming methods that compute intermediate results by constructing tables over subsets of column indices. Additionally, they extend Ryser’s permanent formula by presenting an optimized version for rectangular matrices, significantly improving its computational performance. These contributions provide theoretical insights that align with the subset-based combinatorial framework developed in this study.
Recent studies have also explored quantum-inspired approaches to permanent computation. For instance, Troyansky and Tishby [
3] constructed a quantum observable whose expectation value yields the permanent (for bosons) or determinant (for fermions) of a matrix, emphasizing the intrinsic variance in quantum measurements, which corresponds to the computational complexity gap between permanents and determinants. Similarly, Chabaud et al. [
4] employed quantum-inspired techniques to provide swift proofs of many important theorems about the permanent, such as the MacMahon Master Theorem.
Masschelein’s work [
5] extends Glynn’s algorithm to rectangular matrices and integrates it into a computational framework that dynamically selects the optimal algorithm—Naive, Ryser’s, or Glynn’s—based on the matrix’s properties. Similarly, Niu et al. [
6] introduced an algorithm for square matrices that constructs the permanent through submatrices. However, their work is limited to square matrices and emphasizes algorithmic efficiency over theoretical advancements.
In addition to algorithmic approaches, symbolic and algebraic methods have also been instrumental in advancing permanent computation. Chakraborty et al. [
7] employed symbolic techniques and dynamic programming to improve permanent computation for dense and structured matrices. Spies [
8] provided an algebraic reformulation of Ryser’s formula using polynomial expansions, offering new insights into the combinatorial structure of the permanent. Similarly, Callan [
9] explored Möbius inversion and trigonometric properties to derive closed-form solutions for specific matrix classes, contributing to the theoretical understanding of permanent computation.
Baykasoglu [
10] provided a graphical interpretation of permanent computation using directed graphs and subgraph enumeration. This alternative representation highlights the structural complexity of permanents and offers new insights into their combinatorial properties. The graphical approach is particularly useful for visualizing how terms in Ryser’s formula correspond to different structural patterns within the matrix.
The computation of determinants for rectangular matrices has been considered through various approaches. Cullis [
11] pioneered the study of determinant computation for non-square matrices using the determinants of sub-square matrices, a problem later explored by Radic [
12] and Joshi [
13]. Joshi’s work, in particular, introduced a combinatorial method for evaluating determinants of rectangular matrices, which serves as a conceptual framework relevant to this study. His determinant algorithm exhibits a combinatorial structure similar to the approach in this study. The method begins with a selection of elements from the matrix and proceeds by arranging columns into a specific order to determine the determinant recursively. This method will be described in detail in the next section.
Our study aims to provide evidence that the evaluation of the permanent of a rectangular matrix can be performed using its square submatrices. While adapting existing determinant methods for non-square matrices may seem natural, such approaches are devoid of a direct theoretical basis. To fill this gap, we introduce a rigorous combinatorial framework specifically designed for the permanence of rectangular matrices. This framework bridges combinatorics and matrix analysis while linking the evaluation of the permanent to a subset-sum problem, providing a novel perspective.
2. Background and Notation
The problem of evaluating the permanent of a rectangular matrix has been addressed using different approaches, ranging from classical algebraic formulations to algorithmic improvements. In this section, we provide an overview of the existing methodologies and introduce the basic notation used in our analysis.
Binet [
14], who is credited with introducing the definition of the permanent, initially restricted it to matrices of orders
, defining it as the sum of all possible products of
m elements, with no two elements from the same row or column. Minc [
15] later generalized Binet’s definition to matrices of arbitrary finite orders
m and
n, formalizing it in Equation (
1).
The study of permanents for rectangular matrices has gained interest, particularly following Ryser’s work [
16], which introduced a formula based on the Inclusion–Exclusion principle. This formula, known as Ryser’s theorem, is as follows:
Let
A be a matrix of size
with
. Let
denote a matrix obtained from
A by replacing
r columns of
A by zeros. Let
denote the product of the row sums of
, and let
denote the sums of the
over all of the choices for
. Then,
Minc [
15] observed and claimed that, although the Inclusion–Exclusion principle is formally established by Ryser, the foundational structure of permanent evaluation, as initially introduced by Binet, was inherently compatible with this idea. Binet formulated the permanents of
matrices under the restriction
. Building on this, Minc further refined and simplified Ryser’s formula, leading to the following expression in [
17]:
where
and
.
In [
17], Minc presents another formula for computing the permanent of an
matrix,
, expressed in terms of the
R functions, which is formally stated as the following theorem:
Let
A be an
matrix,
. Then,
where
In this formula, denotes the set of increasing sequences of positive integers such that . For the R-function is the symmetrized sum of all distinct products of the so that in each product the sequences , partition the set .
In [
18], Bebiano noted that the permanent of a rectangular
matrix
, can be obtained by evaluating the permanent of the
matrix
where
C is the
matrix all of whose entries are 1. Then,
As mentioned in the Introduction, we will also consider approaches for rectangular determinants. In this context, we will focus on Joshi’s method for rectangular determinant evaluation. Joshi’s rectangular determinant algorithm exhibits a combinatorial structure, similar to the approach in this study. The method begins with a selection of elements from the matrix and proceeds by arranging columns into a specific order to determine the determinant recursively. Joshi has provided the following formula for evaluating the determinant in [
13].
For an
matrix
A with real entries, let
be defined as above; then,
The sub-square matrices used in Joshi’s algorithm are denoted by . To form the matrices , m columns from n columns should be selected . The column with the smallest column number among these m columns is labeled as the dth column, where . Then, the other columns are determined by choosing columns among the ones right of the dth column regarding their order. Thus, each new column is located to the right of the previous column. According to Joshi’s definition, for example, in computing the determinant of a matrix A of order , the submatrices of order used are given by the set . Here, the index i ranges from 1 to . The set consists of all 3 by 3 submatrices, the first column of which is the first column of matrix A. Similarly, includes all 3 by 3 submatrices, the first column of which is the second column of matrix A. Following this pattern, represents the submatrices, the first column of which is the th column of matrix A. Therefore, the set represents the submatrix formed by the last three columns of matrix A. On the other hand, the columns used to construct the matrices are determined by the sets . Note that is the maximum value of p, indicating the number of elements in . m-tuples which are the elements of sets are denoted as which is constructed as where The components other than the first component of these m-tuples are determined from the set . Therefore, sets are formed by ’s. There emerge different ’s for set, different ’s for set, and finally different ’s for set. Therefore, , , …, , …, .
The approach presented in this study differs significantly from those in previous studies, particularly related to the evaluation of permanents and determinants. Unlike prior work, which often relies on algorithmic or recursive formulations, our work introduces an entirely different framework. This framework provides a combinatorial interpretation of the rectangular permanent problem, linking its evaluation to the solution of a structured combinatorial problem. Furthermore, our perspective is designed as a proof-driven approach, aiming to establish a theoretical foundation rather than optimize the permanent computation. The following sections will develop this theoretical basis and demonstrate its consistency through numerical examples.
3. Evaluation of Permanent for Rectangular Matrices
In this section, we provide two proofs demonstrating that the permanent of a rectangular matrix can be computed using the permanents of its largest square submatrices. Our goal is to develop a theoretical analysis of this problem. To illustrate the problem more concretely, we introduce the statement given in the following theorem. Although this statement is not a complete and explicit direct computation formula, it contributes to a better understanding and analysis of the problem under consideration.
Theorem 1. Let be a rectangular matrix of order with . The permanent of A is given bywhere denotes the matrix whose columns are those indexed by the entries of the tuples satisfyingwhereand the index i ranges over as many numbers as needed to account for all possible m-tuples satisfying the condition (4). Proof. To compute
, we expand it using the permanent definition given by Equation (
1). The resulting expansion consists of
terms. The structure of each term is of the form
. These terms are grouped according to the column indices of their constituent factors (i.e., elements of the matrix). Specifically, within each group, the sum of the column indices of the matrix elements forming the terms is identical and satisfies the condition
where
. Using this grouping process, let us express
where each
represents a sum of
terms of the form
. For the
sums, we provide the following explanations.
is associated with
. Consequently, it consists of terms indexed by the tuple
, i.e.,
Here, the set denoted as represents the set of all one-to-one functions defined from to . Accordingly, corresponds to the permanent of the m by m submatrix consisting of the first m columns of matrix A.
is associated with
. It consists of terms indexed by the tuple
, i.e.,
This indicates that corresponds to the permanent of the m by m submatrix consisting of the first columns and the th column of A.
We note that some
values can yield multiple
tuples when
. In such cases, multiple
summations arise. For instance, the equation
gives two distinct
m-tuples, under this condition:
As a result, two different summations,
and
are presented. So,
corresponds to the permanent of the
m by
m submatrix consisting of the first
columns and the
th column of matrix
A. Similarly,
corresponds to the permanent of the
m by
m submatrix formed by the first
columns, along with the
mth and
th columns of matrix
A.
As exemplified for some values of
p above, each
corresponds to the permanent of a distinct
m by
m submatrix. The selection of column indices via the grouping procedure determines a unique submatrix
, whose permanent contributes to the summation in Equation (
7). This approach ensures that all possible submatrices of order
within
A are systematically considered, confirming the completeness of the expansion given by Equation (
3).
Note that the summation symbol iterating over the index
i arises from the fact that multiple
m-tuples are obtained from certain
. The index
i is not explicitly defined in Equation (
3), though its iteration count can be easily calculated. The index
i serves to illustrate the possible multiple
m-tuples formed by
and to make them more noticeable. For a clearer understanding of the
i-loop, see
Section 4 and
Section 5. □
The proof given above utilizes a special grouping process to evaluate the permanent of m by n matrix A, thereby forming the permanents of m by m submatrices. On the other hand, an alternative proof will be presented, in which we employ a generating function approach to systematically count the tuples corresponding to the values of , generating the sub-square permanents.
An Alternative Proof of Theorem 1
Proof. Let
denote the number of distinct
tuples associated with each
. Additionally, consider the array
, which contains the values
and consists of
elements. This array represents a structured mapping of the submatrix selection process based on column index sums. The main objective of this proof is to demonstrate that the equality
holds.
A detailed examination of the distribution of
values reveals a significant symmetry arising from the grouping process defined in the theorem. This symmetry is confirmed by the equality of the sequence sizes:
For example, the sets corresponding to and have the same count, as do those for and . To analyze this distribution further, we apply the generating function approach to count the sets .
To achieve this, we make use of the function defined in [
19], given by
where
j represents the columns of matrix
A. Expanding Equation (
15) with respect to
y, we obtain
where
is a generating function of the form
Then, it follows that the equality
corresponds to (
14), and this can be proved by induction on
n.
For the base case
, Equation (
15) simplifies to
while Equation (
16) gives
, thus verifying Equation (
18) for
.
Assuming Equation (
18) holds for
, the function
for
can be written as
Thus, the coefficient of the term with
in the expansion is
Given that the equality
holds for
, the corresponding relation
is therefore satisfied. Substituting these equalities into Equation (
21) after taking the limit enables the continuation of the proof.
Applying the limit of Equation (
21) and substituting the previously established equalities for
and
, we obtain
Utilizing Pascal’s identity, this expression simplifies to , thereby completing the proof. □
4. Combinatorial Symmetries in Grid Shading Problem and Rectangular Matrix Permanents
The problem of identifying subsets within a set of integers whose sum equals a specified value k is a well-known form of the subset-sum problem. A specific case of this problem concerns finding subsets that not only satisfy the sum condition but also contain an equal number of elements. This variant introduces additional combinatorial complexity, as it requires balancing the subset cardinality alongside the sum constraint.
The grid shading problem, as introduced in [
20], presents a combinatorial approach to subset selection by determining subsets of grid columns based on specific sum conditions. We can use this approach to interpret the combinatorial structure in our study.
In solving the grid shading problem, the author of [
20] considered the number of iso-square blocks in an
rectangular grid, where cells are shaded according to a defined rule. Each distinct shading pattern, referred to as a proper shading, corresponds to a unique configuration derived from these sum-based constraints.
Notably, the number of possible proper shadings exhibits an unimodal sequence, which aligns with the structured sequences that emerge in our method when analyzing submatrices of a given order. For a
grid, the resulting sequence of proper shadings forms the following unimodal distribution:
A visualization of this sequence is shown in [
20], using a
grid. Additionally, the following unimodal sequence is formed by the number of proper shadings in the
grid:
We provide the original visualization of this sequence, corresponding to the
grid, in
Figure 1. Similarly, another unimodal sequence
is formed by the numbers of the proper shadings that can be obtained by the
grid.
Our procedure for determining submatrices based on the sum of column indices provides a strong connection with Proctor’s grid shading problem [
20]. The sequence
, where
, corresponds precisely to the sequence obtained through
proper shadings in an
grid. This direct correspondence establishes a meaningful combinatorial link between the two problems, reinforcing the foundation of our approach.
For instance, for a matrix A, the sequence , where , corresponds to the sequence obtained by shading a grid in Proctor’s problem.
5. Illustrative Example
Consider a matrix
A of order
. Based on the condition given by Equation (
5), the values of
r range among
, leading to the identification of the corresponding submatrices. Then, we obtain the data as in
Table 1.
For example, if
, then
. This implies the condition
which gives two possible tuples
and
.
Let us define the following selection principle:
The corresponding products of the form are then generated according to this principle.
For , all possible quartic products are determined as follows:
If , then . Choosing indicates , yielding and . For , yields and . For , yields and .
If , then . Choosing indicates , yielding and . For , yields and . For , yields and .
If , then . Choosing indicates , yielding the terms and . For , yields and . For , yields and .
If , then . Choosing indicates , yielding the terms and . For , yields and . For , yields and .
The sum of the 24 quartic terms obtained via the set
is denoted by
. According to the permanent definition in Equation (
1), the sum
corresponds to the permanent of the submatrix
which consists of the first, second, fourth, and fifth columns of matrix
A.
From
Table 1, we obtain the sequence
and so
Indeed, the number of different submatrices of order
for a matrix of order
is
. Note that the sequence
overlaps with the sequence obtained by the Proctor’s grid shadings of order
. As a visual representation of these results,
Figure 1 showing the
grid shading patterns is given above.