Permanent (mathematics)

In linear algebra, the permanent of a square matrix is a function of the matrix similar to the determinant. The permanent, as well as the determinant, is a polynomial in the entries of the matrix.^[1] Both are special cases of a more general function of a matrix called the immanant.

Definition

The permanent of an n×n matrix A = (a_i,j) is defined as

$${\displaystyle \operatorname {perm} (A)=\sum _{\sigma \in S_{n}}\prod _{i=1}^{n}a_{i,\sigma (i)}.}$$

 

The sum here extends over all elements σ of the symmetric group S_n; i.e. over all permutations of the numbers 1, 2, ..., n.

For example,

$${\displaystyle \operatorname {perm} {\begin{pmatrix}a&b\\c&d\end{pmatrix}}=ad+bc,}$$

 

and

$${\displaystyle \operatorname {perm} {\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix}}=aei+bfg+cdh+ceg+bdi+afh.}$$

 

The definition of the permanent of A differs from that of the determinant of A in that the signatures of the permutations are not taken into account.

The permanent of a matrix A is denoted per A, perm A, or Per A, sometimes with parentheses around the argument. Minc uses Per(A) for the permanent of rectangular matrices, and per(A) when A is a square matrix.^[2] Muir and Metzler use the notation ${\displaystyle {\overset {+}{|}}\quad {\overset {+}{|}}}$ .^[3]

The word, permanent, originated with Cauchy in 1812 as “fonctions symétriques permanentes” for a related type of function,^[4] and was used by Muir and Metzler^[5] in the modern, more specific, sense.^[6]

Properties

If one views the permanent as a map that takes n vectors as arguments, then it is a multilinear map and it is symmetric (meaning that any order of the vectors results in the same permanent). Furthermore, given a square matrix ${\displaystyle A=\left(a_{ij}\right)}$  of order n:^[7]

• perm(A) is invariant under arbitrary permutations of the rows and/or columns of A. This property may be written symbolically as perm(A) = perm(PAQ) for any appropriately sized permutation matrices P and Q,
• multiplying any single row or column of A by a scalar s changes perm(A) to s⋅perm(A),
• perm(A) is invariant under transposition, that is, perm(A) = perm(A^T).
• If ${\displaystyle A=\left(a_{ij}\right)}$  and ${\displaystyle B=\left(b_{ij}\right)}$  are square matrices of order n then,^[8]
  $${\displaystyle \operatorname {perm} \left(A+B\right)=\sum _{s,t}\operatorname {perm} \left(a_{ij}\right)_{i\in s,j\in t}\operatorname {perm} \left(b_{ij}\right)_{i\in {\bar {s}},j\in {\bar {t}}},}$$

   
  where s and t are subsets of the same size of {1,2,...,n} and ${\displaystyle {\bar {s}},{\bar {t}}}$  are their respective complements in that set.
• If ${\displaystyle A}$  is a triangular matrix, i.e. ${\displaystyle a_{ij}=0}$ , whenever ${\displaystyle i>j}$  or, alternatively, whenever ${\displaystyle i<j}$ , then its permanent (and determinant as well) equals the product of the diagonal entries:
  $${\displaystyle \operatorname {perm} \left(A\right)=a_{11}a_{22}\cdots a_{nn}=\prod _{i=1}^{n}a_{ii}.}$$

   

Relation to determinants

Laplace's expansion by minors for computing the determinant along a row, column or diagonal extends to the permanent by ignoring all signs.^[9]

For every ${\textstyle i}$ ,

$${\displaystyle \mathbb {perm} (B)=\sum _{j=1}^{n}B_{i,j}M_{i,j},}$$

 

where ${\displaystyle B_{i,j}}$  is the entry of the ith row and the jth column of B, and ${\textstyle M_{i,j}}$  is the permanent of the submatrix obtained by removing the ith row and the jth column of B.

For example, expanding along the first column,

$${\displaystyle {\begin{aligned}\operatorname {perm} \left({\begin{matrix}1&1&1&1\\2&1&0&0\\3&0&1&0\\4&0&0&1\end{matrix}}\right)={}&1\cdot \operatorname {perm} \left({\begin{matrix}1&0&0\\0&1&0\\0&0&1\end{matrix}}\right)+2\cdot \operatorname {perm} \left({\begin{matrix}1&1&1\\0&1&0\\0&0&1\end{matrix}}\right)\\&{}+\ 3\cdot \operatorname {perm} \left({\begin{matrix}1&1&1\\1&0&0\\0&0&1\end{matrix}}\right)+4\cdot \operatorname {perm} \left({\begin{matrix}1&1&1\\1&0&0\\0&1&0\end{matrix}}\right)\\={}&1(1)+2(1)+3(1)+4(1)=10,\end{aligned}}}$$

 

while expanding along the last row gives,

$${\displaystyle {\begin{aligned}\operatorname {perm} \left({\begin{matrix}1&1&1&1\\2&1&0&0\\3&0&1&0\\4&0&0&1\end{matrix}}\right)={}&4\cdot \operatorname {perm} \left({\begin{matrix}1&1&1\\1&0&0\\0&1&0\end{matrix}}\right)+0\cdot \operatorname {perm} \left({\begin{matrix}1&1&1\\2&0&0\\3&1&0\end{matrix}}\right)\\&{}+\ 0\cdot \operatorname {perm} \left({\begin{matrix}1&1&1\\2&1&0\\3&0&0\end{matrix}}\right)+1\cdot \operatorname {perm} \left({\begin{matrix}1&1&1\\2&1&0\\3&0&1\end{matrix}}\right)\\={}&4(1)+0+0+1(6)=10.\end{aligned}}}$$

 

On the other hand, the basic multiplicative property of determinants is not valid for permanents.^[10] A simple example shows that this is so.

$${\displaystyle {\begin{aligned}4&=\operatorname {perm} \left({\begin{matrix}1&1\\1&1\end{matrix}}\right)\operatorname {perm} \left({\begin{matrix}1&1\\1&1\end{matrix}}\right)\\&\neq \operatorname {perm} \left(\left({\begin{matrix}1&1\\1&1\end{matrix}}\right)\left({\begin{matrix}1&1\\1&1\end{matrix}}\right)\right)=\operatorname {perm} \left({\begin{matrix}2&2\\2&2\end{matrix}}\right)=8.\end{aligned}}}$$

 

Unlike the determinant, the permanent has no easy geometrical interpretation; it is mainly used in combinatorics, in treating boson Green's functions in quantum field theory, and in determining state probabilities of boson sampling systems.^[11] However, it has two graph-theoretic interpretations: as the sum of weights of cycle covers of a directed graph, and as the sum of weights of perfect matchings in a bipartite graph.

Applications

Symmetric tensors

The permanent arises naturally in the study of the symmetric tensor power of Hilbert spaces.^[12] In particular, for a Hilbert space ${\displaystyle H}$ , let ${\displaystyle \vee ^{k}H}$  denote the ${\displaystyle k}$ th symmetric tensor power of ${\displaystyle H}$ , which is the space of symmetric tensors. Note in particular that ${\displaystyle \vee ^{k}H}$  is spanned by the symmetric products of elements in ${\displaystyle H}$ . For ${\displaystyle x_{1},x_{2},\dots ,x_{k}\in H}$ , we define the symmetric product of these elements by

$${\displaystyle x_{1}\vee x_{2}\vee \cdots \vee x_{k}=(k!)^{-1/2}\sum _{\sigma \in S_{k}}x_{\sigma (1)}\otimes x_{\sigma (2)}\otimes \cdots \otimes x_{\sigma (k)}}$$

 

If we consider ${\displaystyle \vee ^{k}H}$  (as a subspace of ${\displaystyle \otimes ^{k}H}$ , the kth tensor power of ${\displaystyle H}$ ) and define the inner product on ${\displaystyle \vee ^{k}H}$  accordingly, we find that for ${\displaystyle x_{j},y_{j}\in H}$ 

$${\displaystyle \langle x_{1}\vee x_{2}\vee \cdots \vee x_{k},y_{1}\vee y_{2}\vee \cdots \vee y_{k}\rangle =\operatorname {perm} \left[\langle x_{i},y_{j}\rangle \right]_{i,j=1}^{k}}$$

 

Applying the Cauchy–Schwarz inequality, we find that ${\displaystyle \operatorname {perm} \left[\langle x_{i},x_{j}\rangle \right]_{i,j=1}^{k}\geq 0}$ , and that

$${\displaystyle \left|\operatorname {perm} \left[\langle x_{i},y_{j}\rangle \right]_{i,j=1}^{k}\right|^{2}\leq \operatorname {perm} \left[\langle x_{i},x_{j}\rangle \right]_{i,j=1}^{k}\cdot \operatorname {perm} \left[\langle y_{i},y_{j}\rangle \right]_{i,j=1}^{k}}$$

 

Cycle covers

Main article: Vertex cycle cover

Any square matrix ${\displaystyle A=(a_{ij})_{i,j=1}^{n}}$  can be viewed as the adjacency matrix of a weighted directed graph on vertex set ${\displaystyle V=\{1,2,\dots ,n\}}$ , with ${\displaystyle a_{ij}}$  representing the weight of the arc from vertex i to vertex j. A cycle cover of a weighted directed graph is a collection of vertex-disjoint directed cycles in the digraph that covers all vertices in the graph. Thus, each vertex i in the digraph has a unique "successor" ${\displaystyle \sigma (i)}$  in the cycle cover, and so ${\displaystyle \sigma }$  represents a permutation on V. Conversely, any permutation ${\displaystyle \sigma }$  on V corresponds to a cycle cover with arcs from each vertex i to vertex ${\displaystyle \sigma (i)}$ .

If the weight of a cycle-cover is defined to be the product of the weights of the arcs in each cycle, then

$${\displaystyle \operatorname {weight} (\sigma )=\prod _{i=1}^{n}a_{i,\sigma (i)},}$$

 

implying that

$${\displaystyle \operatorname {perm} (A)=\sum _{\sigma }\operatorname {weight} (\sigma ).}$$

 

Thus the permanent of A is equal to the sum of the weights of all cycle-covers of the digraph.

Perfect matchings

A square matrix ${\displaystyle A=(a_{ij})}$  can also be viewed as the adjacency matrix of a bipartite graph which has vertices ${\displaystyle x_{1},x_{2},\dots ,x_{n}}$  on one side and ${\displaystyle y_{1},y_{2},\dots ,y_{n}}$  on the other side, with ${\displaystyle a_{ij}}$  representing the weight of the edge from vertex ${\displaystyle x_{i}}$  to vertex ${\displaystyle y_{j}}$ . If the weight of a perfect matching ${\displaystyle \sigma }$  that matches ${\displaystyle x_{i}}$  to ${\displaystyle y_{\sigma (i)}}$  is defined to be the product of the weights of the edges in the matching, then

$${\displaystyle \operatorname {weight} (\sigma )=\prod _{i=1}^{n}a_{i,\sigma (i)}.}$$

 

Thus the permanent of A is equal to the sum of the weights of all perfect matchings of the graph.

Permanents of (0, 1) matrices

Enumeration

The answers to many counting questions can be computed as permanents of matrices that only have 0 and 1 as entries.

Let Ω(n,k) be the class of all (0, 1)-matrices of order n with each row and column sum equal to k. Every matrix A in this class has perm(A) > 0.^[13] The incidence matrices of projective planes are in the class Ω(n² + n + 1, n + 1) for n an integer > 1. The permanents corresponding to the smallest projective planes have been calculated. For n = 2, 3, and 4 the values are 24, 3852 and 18,534,400 respectively.^[13] Let Z be the incidence matrix of the projective plane with n = 2, the Fano plane. Remarkably, perm(Z) = 24 = |det (Z)|, the absolute value of the determinant of Z. This is a consequence of Z being a circulant matrix and the theorem:^[14]

If A is a circulant matrix in the class Ω(n,k) then if k > 3, perm(A) > |det (A)| and if k = 3, perm(A) = |det (A)|. Furthermore, when k = 3, by permuting rows and columns, A can be put into the form of a direct sum of e copies of the matrix Z and consequently, n = 7e and perm(A) = 24^e.

Permanents can also be used to calculate the number of permutations with restricted (prohibited) positions. For the standard n-set {1, 2, ..., n}, let ${\displaystyle A=(a_{ij})}$  be the (0, 1)-matrix where a_ij = 1 if i → j is allowed in a permutation and a_ij = 0 otherwise. Then perm(A) is equal to the number of permutations of the n-set that satisfy all the restrictions.^[9] Two well known special cases of this are the solution of the derangement problem and the ménage problem: the number of permutations of an n-set with no fixed points (derangements) is given by

$${\displaystyle \operatorname {perm} (J-I)=\operatorname {perm} \left({\begin{matrix}0&1&1&\dots &1\\1&0&1&\dots &1\\1&1&0&\dots &1\\\vdots &\vdots &\vdots &\ddots &\vdots \\1&1&1&\dots &0\end{matrix}}\right)=n!\sum _{i=0}^{n}{\frac {(-1)^{i}}{i!}},}$$

 

where J is the n×n all 1's matrix and I is the identity matrix, and the ménage numbers are given by

$${\displaystyle {\begin{aligned}\operatorname {perm} (J-I-I')&=\operatorname {perm} \left({\begin{matrix}0&0&1&\dots &1\\1&0&0&\dots &1\\1&1&0&\dots &1\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&1&1&\dots &0\end{matrix}}\right)\\&=\sum _{k=0}^{n}(-1)^{k}{\frac {2n}{2n-k}}{2n-k \choose k}(n-k)!,\end{aligned}}}$$

 

where I' is the (0, 1)-matrix with nonzero entries in positions (i, i + 1) and (n, 1).

Bounds

The Bregman–Minc inequality, conjectured by H. Minc in 1963^[15] and proved by L. M. Brégman in 1973,^[16] gives an upper bound for the permanent of an n × n (0, 1)-matrix. If A has r_i ones in row i for each 1 ≤ i ≤ n, the inequality states that

$${\displaystyle \operatorname {perm} A\leq \prod _{i=1}^{n}(r_{i})!^{1/r_{i}}.}$$

 

Van der Waerden's conjecture

In 1926, Van der Waerden conjectured that the minimum permanent among all n × n doubly stochastic matrices is n!/nⁿ, achieved by the matrix for which all entries are equal to 1/n.^[17] Proofs of this conjecture were published in 1980 by B. Gyires^[18] and in 1981 by G. P. Egorychev^[19] and D. I. Falikman;^[20] Egorychev's proof is an application of the Alexandrov–Fenchel inequality.^[21] For this work, Egorychev and Falikman won the Fulkerson Prize in 1982.^[22]

Computation

Main articles: Computing the permanent and Sharp-P-completeness of 01-permanent

The naïve approach, using the definition, of computing permanents is computationally infeasible even for relatively small matrices. One of the fastest known algorithms is due to H. J. Ryser.^[23] Ryser's method is based on an inclusion–exclusion formula that can be given^[24] as follows: Let ${\displaystyle A_{k}}$  be obtained from A by deleting k columns, let ${\displaystyle P(A_{k})}$  be the product of the row-sums of ${\displaystyle A_{k}}$ , and let ${\displaystyle \Sigma _{k}}$  be the sum of the values of ${\displaystyle P(A_{k})}$  over all possible ${\displaystyle A_{k}}$ . Then

$${\displaystyle \operatorname {perm} (A)=\sum _{k=0}^{n-1}(-1)^{k}\Sigma _{k}.}$$

 

It may be rewritten in terms of the matrix entries as follows:

$${\displaystyle \operatorname {perm} (A)=(-1)^{n}\sum _{S\subseteq \{1,\dots ,n\}}(-1)^{|S|}\prod _{i=1}^{n}\sum _{j\in S}a_{ij}.}$$

 

The permanent is believed to be more difficult to compute than the determinant. While the determinant can be computed in polynomial time by Gaussian elimination, Gaussian elimination cannot be used to compute the permanent. Moreover, computing the permanent of a (0,1)-matrix is #P-complete. Thus, if the permanent can be computed in polynomial time by any method, then FP = #P, which is an even stronger statement than P = NP. When the entries of A are nonnegative, however, the permanent can be computed approximately in probabilistic polynomial time, up to an error of ${\displaystyle \varepsilon M}$ , where ${\displaystyle M}$  is the value of the permanent and ${\displaystyle \varepsilon >0}$  is arbitrary.^[25] The permanent of a certain set of positive semidefinite matrices can also be approximated in probabilistic polynomial time: the best achievable error of this approximation is ${\displaystyle \varepsilon {\sqrt {M}}}$  (${\displaystyle M}$  is again the value of the permanent).^[26]

MacMahon's master theorem

Main article: MacMahon's master theorem

Another way to view permanents is via multivariate generating functions. Let ${\displaystyle A=(a_{ij})}$  be a square matrix of order n. Consider the multivariate generating function:

$${\displaystyle {\begin{aligned}F(x_{1},x_{2},\dots ,x_{n})&=\prod _{i=1}^{n}\left(\sum _{j=1}^{n}a_{ij}x_{j}\right)\\&=\left(\sum _{j=1}^{n}a_{1j}x_{j}\right)\left(\sum _{j=1}^{n}a_{2j}x_{j}\right)\cdots \left(\sum _{j=1}^{n}a_{nj}x_{j}\right).\end{aligned}}}$$

 

The coefficient of ${\displaystyle x_{1}x_{2}\dots x_{n}}$  in ${\displaystyle F(x_{1},x_{2},\dots ,x_{n})}$  is perm(A).^[27]

As a generalization, for any sequence of n non-negative integers, ${\displaystyle s_{1},s_{2},\dots ,s_{n}}$  define:

$${\displaystyle \operatorname {perm} ^{(s_{1},s_{2},\dots ,s_{n})}(A)}$$

 

as the coefficient of ${\displaystyle x_{1}^{s_{1}}x_{2}^{s_{2}}\cdots x_{n}^{s_{n}}}$  in${\displaystyle \left(\sum _{j=1}^{n}a_{1j}x_{j}\right)^{s_{1}}\left(\sum _{j=1}^{n}a_{2j}x_{j}\right)^{s_{2}}\cdots \left(\sum _{j=1}^{n}a_{nj}x_{j}\right)^{s_{n}}.}$ 

MacMahon's master theorem relating permanents and determinants is:^[28]

$${\displaystyle \operatorname {perm} ^{(s_{1},s_{2},\dots ,s_{n})}(A)={\text{ coefficient of }}x_{1}^{s_{1}}x_{2}^{s_{2}}\cdots x_{n}^{s_{n}}{\text{ in }}{\frac {1}{\det(I-XA)}},}$$

 

where I is the order n identity matrix and X is the diagonal matrix with diagonal ${\displaystyle [x_{1},x_{2},\dots ,x_{n}].}$ 

Rectangular matrices

The permanent function can be generalized to apply to non-square matrices. Indeed, several authors make this the definition of a permanent and consider the restriction to square matrices a special case.^[29] Specifically, for an m × n matrix ${\displaystyle A=(a_{ij})}$  with m ≤ n, define

$${\displaystyle \operatorname {perm} (A)=\sum _{\sigma \in \operatorname {P} (n,m)}a_{1\sigma (1)}a_{2\sigma (2)}\ldots a_{m\sigma (m)}}$$

 

where P(n,m) is the set of all m-permutations of the n-set {1,2,...,n}.^[30]

Ryser's computational result for permanents also generalizes. If A is an m × n matrix with m ≤ n, let ${\displaystyle A_{k}}$  be obtained from A by deleting k columns, let ${\displaystyle P(A_{k})}$  be the product of the row-sums of ${\displaystyle A_{k}}$ , and let ${\displaystyle \sigma _{k}}$  be the sum of the values of ${\displaystyle P(A_{k})}$  over all possible ${\displaystyle A_{k}}$ . Then^[10]

$${\displaystyle \operatorname {perm} (A)=\sum _{k=0}^{m-1}(-1)^{k}{\binom {n-m+k}{k}}\sigma _{n-m+k}.}$$

 

Systems of distinct representatives

The generalization of the definition of a permanent to non-square matrices allows the concept to be used in a more natural way in some applications. For instance:

Let S₁, S₂, ..., S_m be subsets (not necessarily distinct) of an n-set with m ≤ n. The incidence matrix of this collection of subsets is an m × n (0,1)-matrix A. The number of systems of distinct representatives (SDR's) of this collection is perm(A).^[31]

See also

• Computing the permanent
• Bapat–Beg theorem, an application of permanents in order statistics
• Slater determinant, an application of permanents in quantum mechanics
• Hafnian

Notes

1. Marcus, Marvin; Minc, Henryk (1965). "Permanents". Amer. Math. Monthly. 72 (6): 577–591. doi:10.2307/2313846. JSTOR 2313846.
2. Minc (1978)
3. Muir & Metzler (1960)
4. Cauchy, A. L. (1815), "Mémoire sur les fonctions qui ne peuvent obtenir que deux valeurs égales et de signes contraires par suite des transpositions opérées entre les variables qu'elles renferment.", Journal de l'École Polytechnique, 10: 91–169
5. Muir & Metzler (1960)
6. van Lint & Wilson 2001, p. 108
7. Ryser 1963, pp. 25 – 26
8. Percus 1971, p. 2
9. Percus 1971, p. 12
10. Ryser 1963, p. 26
11. Aaronson, Scott (14 Nov 2010). "The Computational Complexity of Linear Optics". arXiv:1011.3245 [quant-ph].
12. Bhatia, Rajendra (1997). Matrix Analysis. New York: Springer-Verlag. pp. 16–19. ISBN 978-0-387-94846-1.
13. Ryser 1963, p. 124
14. Ryser 1963, p. 125
15. Minc, Henryk (1963), "Upper bounds for permanents of (0,1)-matrices", Bulletin of the American Mathematical Society, 69 (6): 789–791, doi:10.1090/s0002-9904-1963-11031-9
16. van Lint & Wilson 2001, p. 101
17. van der Waerden, B. L. (1926), "Aufgabe 45", Jber. Deutsch. Math.-Verein., 35: 117.
18. Gyires, B. (1980), "The common source of several inequalities concerning doubly stochastic matrices", Publicationes Mathematicae Institutum Mathematicum Universitatis Debreceniensis, 27 (3–4): 291–304, MR 0604006.
19. Egoryčev, G. P. (1980), Reshenie problemy van-der-Vardena dlya permanentov (in Russian), Krasnoyarsk: Akad. Nauk SSSR Sibirsk. Otdel. Inst. Fiz., p. 12, MR 0602332. Egorychev, G. P. (1981), "Proof of the van der Waerden conjecture for permanents", Akademiya Nauk SSSR (in Russian), 22 (6): 65–71, 225, MR 0638007. Egorychev, G. P. (1981), "The solution of van der Waerden's problem for permanents", Advances in Mathematics, 42 (3): 299–305, doi:10.1016/0001-8708(81)90044-X, MR 0642395.
20. Falikman, D. I. (1981), "Proof of the van der Waerden conjecture on the permanent of a doubly stochastic matrix", Akademiya Nauk Soyuza SSR (in Russian), 29 (6): 931–938, 957, MR 0625097.
21. Brualdi (2006) p.487
22. Fulkerson Prize, Mathematical Optimization Society, retrieved 2012-08-19.
23. Ryser (1963, p. 27)
24. van Lint & Wilson (2001) p. 99
25. Jerrum, M.; Sinclair, A.; Vigoda, E. (2004), "A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries", Journal of the ACM, 51 (4): 671–697, CiteSeerX 10.1.1.18.9466, doi:10.1145/1008731.1008738, S2CID 47361920
26. Chakhmakhchyan, Levon; Cerf, Nicolas; Garcia-Patron, Raul (2017). "A quantum-inspired algorithm for estimating the permanent of positive semidefinite matrices". Phys. Rev. A. 96 (2): 022329. arXiv:1609.02416. Bibcode:2017PhRvA..96b2329C. doi:10.1103/PhysRevA.96.022329. S2CID 54194194.
27. Percus 1971, p. 14
28. Percus 1971, p. 17
29. In particular, Minc (1978) and Ryser (1963) do this.
30. Ryser 1963, p. 25
31. Ryser 1963, p. 54

References

• Brualdi, Richard A. (2006). Combinatorial matrix classes. Encyclopedia of Mathematics and Its Applications. Vol. 108. Cambridge: Cambridge University Press. ISBN 978-0-521-86565-4. Zbl 1106.05001.
• Minc, Henryk (1978). Permanents. Encyclopedia of Mathematics and its Applications. Vol. 6. With a foreword by Marvin Marcus. Reading, MA: Addison–Wesley. ISSN 0953-4806. OCLC 3980645. Zbl 0401.15005.
• Muir, Thomas; Metzler, William H. (1960) [1882]. A Treatise on the Theory of Determinants. New York: Dover. OCLC 535903.
• Percus, J.K. (1971), Combinatorial Methods, Applied Mathematical Sciences #4, New York: Springer-Verlag, ISBN 978-0-387-90027-8
• Ryser, Herbert John (1963), Combinatorial Mathematics, The Carus Mathematical Monographs #14, The Mathematical Association of America
• van Lint, J.H.; Wilson, R.M. (2001), A Course in Combinatorics, Cambridge University Press, ISBN 978-0521422604

Further reading

• Hall Jr., Marshall (1986), Combinatorial Theory (2nd ed.), New York: John Wiley & Sons, pp. 56–72, ISBN 978-0-471-09138-7 Contains a proof of the Van der Waerden conjecture.
• Marcus, M.; Minc, H. (1965), "Permanents", The American Mathematical Monthly, 72 (6): 577–591, doi:10.2307/2313846, JSTOR 2313846

External links

• Permanent at PlanetMath.
• Van der Waerden's permanent conjecture at PlanetMath.