Spectrum of Permanent’s Values and Its Extremal Magnitudes in and
Let denote the class of square matrices containing in each row and in each column exactly 1’s. The minimal value of , for which the behavior of the permanent in is not quite studied, is . We give a simple algorithm for calculation of upper magnitudes of permanent in and consider some extremal problems in a generalized class , the matrices of which contain in each row and in each column nonzero elements , , and and zeros.
The definition of permanent of a square matrix of order is , where sum is over all permutations of numbers . This definition is very combinatorial. For example, if is -matrix, then is the number of arrangements of nonattacking each other rooks on positions of 1’s of . Therefore, the most natural applications of the permanent are in combinatorics: enumerating the Latin rectangles and squares, permutations with restricted positions, and different problems in the theory of graphs. Many interesting examples of applications of the permanent one can find in , chapter 8 (and not only in Mathematics!). To applications of the permanent there promoted celebrate proofs of the best known van der Waerden’s and Minc’s conjectures by Egorychev , Falikman , and Brègman , respectively. For example, Alon and Friedland  excellently used the Minc-Bregman inequality for permanent of -matrices for finding the maximum number of perfect matchings in graphs with a given degree sequence.
Let denote the class of square matrices containing in each row and in each column exactly 1’s. If , then matrix is doubly stochastic, since all its row and column sums equal 1. Therefore, -matrices are also called doubly stochastic (0,1)-matrices. These matrices have especially simple and attractive structure and have many applications. It is important that especially class in permanent’s history was a nontrivial test area for different researches of the permanent. For example, one can mention a remarkable Merriell research  of the maximum of permanent on class . An author’s research  of the permanent on -matrices (see Section 9) has led to the disproof of an important Balasubramanian conjecture  which would yield a full proof of 1967 Ryser hypothesis  on transversals of Latin squares.
In the present paper we consider many other problems for . We study also the following natural generalization of . For given real or complex nonzero numbers , , and , denote by the class of square matrices containing every number from exactly one time in each row and in each column, such that the other elements are 0’s. It is clear that, if and , then the matrix is doubly stochastic.
Definition 1. One calls -spectrum in (denoting it the set of all the values which are taken by the permanent in .
Note that -spectrum in trivially is . It is known (cf. Tarakanov ) that
But, for , -spectrum of , generally speaking, is unknown. Greenstein (cf. , point 8.4, Problem 3) put the problem of describing the -spectrum in . In this paper we give an algorithm for calculation of upper values of -spectrum in . We also obtain several results for a generalized class with real nonzero numbers , , and . Some results of the present paper were announced by the author in .
2. What Is Known about ?
(1) Explicit formula for (cf. Stanley , chapter 1) is
(2) Asymptotic formula for (cf. O’Neil ) is where is arbitrarily small for sufficiently large .
In addition, note that with different , , and is, evidently, the number of 3-rowed Latin rectangles of length such that where is the number of reduced 3-rowed Latin rectangles with the first row . It is known (Riordan , pp. 204–210) that where is subfactorial: and is sequence of Lucas numbers of the Ménage problem which is defined by Cayley recursion (cf. , p. 201) (see , sequences A102761, A000186).
Denote, furthermore, by the set of matrices in with 1’s on the main diagonal. Note that
Indeed, it is well known that every -matrix has a diagonal of ones (i.e., a set of 1’s not two in the same row or column). Let be such a diagonal. There exists a permutation of rows and columns such that will be the main diagonal of . Nevertheless, and (8) follows.
(4) Asymptotic formula for (Shevelev ) is where and is arbitrarily small for sufficiently large .
(5) Denote by the set of symmetric matrices in . -spectrum on is given by the following theorem (Shevelev ).
Theorem 2. Let denote the set of all partitions of with parts more than or equal to 3. To every partition , , put in a correspondence the number where sequence is defined by the recursion Then one has
(6) The maximal value of permanent in was found by Merriell .
Theorem 3. If , , then
Note that the case of (16) easily follows from the Minc-Bregman inequality for permanent of -matrices.
(7) Put . In case of , Bolshakov  showed that the second maximal of permanent in (such that interval is free from values of permanent in ) equals
Note that both and are attained in .
(8) Denote by the minimal value of permanent in . In 1979, Voorhoeve  obtained a beautiful lower estimate for :
This estimate remains the best even after proof by Egorychev  and Falikman , the famous van der Waerden conjectural lower estimate for every doubly stochastic matrix . Indeed, this estimate yields only , such that (18) is stronger for .
(9) Bolshakov  found -spectrum in in cases . Namely, he added to the evident -spectrums and also the following -spectrums:
3. A Generalization of Theorem 2 on Matrices of Class with Symmetric Positions of Elements
Denote by the set of matrices in with ’s on the main diagonal. It is clear that, together with (4),
Note that, as for sets , , we have
Denote, furthermore, by the set of matrices in with symmetric positions of elements: if and only if .
-spectrum on is given by the following theorem.
Theorem 4. If to every partition , , , corresponds the number where sequence is defined by the recursion then one has
Proof. Let be the symmetric permutation group of elements . Two positions , are called independent if , . We will say that in the matrix , a weight is appropriated to the position . Let have no cycle of length less than . Consider a map
appropriating to the position the weight .
Now we need several lemmas.
Lemma A. (1) The map is bijective; (2) if is a set of pairwise independent positions, then is also a set of pairwise independent positions.
Proof. (a) Consider two distinct positions such that, at least, one of the two inequalities holds Let such that, say, . Show that . Indeed, if we suppose that ; then ; that is, has a cycle of length in spite of the condition. Conversely, if , then , since has not any cycle of length less than as well.
(b) Let positions (26) be independent. Both inequalities (27) hold and, as in (a), we have , ; that is, the positions , are independent as well.
Lemma B. Let have no cycle of length less than . Then -matrix having 1’s on only positionsis an incidence matrix of .
Proof. Since has no cycles of length less than , then is a permutation of numbers . Thus the set of positions of 1’s of matrix coincides with the set of 1’s of the incidence matrix of .
Let be -matrix with 1’s on positions only.
Lemma C. Let have not got any cycle of length less than . If and are the incidence matrices of and , then one hasProof. Both formulas follow from Lemma B.
Noting that , where is the identity matrix, we conclude that Moreover, since, by the bijection , to every diagonal (i.e., to every set of pairwise independent positions) of the matrix corresponds one and only one diagonal of the matrix with the same products of weights, then we have Note that from the definition it follows that, for every matrix , we have a representation where is the incidence matrix of a substitution . In case when has not got any cycle of length less than , the matrix is a completely indecomposable matrix in . Thus, by (31), all completely indecomposable matrices of have the same permanent, equal to .
In general, a substitution with the incidence matrix in (32) cannot have cycles of length less than 3. Indeed, if, for some , we have either or , then in both cases which means coincidence of positions 1’s of the matrices and in the th row.
Let be an arbitrary substitution with cycles of length more than 2. Let where , , and , be the decomposition of in a product of cycles. Then the matrix is a direct sum of the matrices such that, by (31), and we have It is left to notice that Minc  found a recursion (23) for and, as well known, the multiplication of an matrix by does not change its permanent. Therefore, .
Example 5. Let us find .
We have the following partitions of 11 with the parts not less than 3: According to (23), for , we have , and for
Using induction, we find
In the following examples we calculate -spectrum for arbitrary .
Example 6. Let us find .
By induction, for , we have
Further, again using induction, one can find that, if is even, then and, if is odd, then
Example 7. Analogously, in case of , for , we have
It is interesting that, in case of multiple of 3, the permanent omits the value .
4. Merriell-Type Theorems in a Subclass of
Note that in class the Minc-Bregman inequality and the Merriell theorem, generally speaking, do not hold even for positive , , and . Nevertheless, some restrictions on , , and allow proving some analogs of the Merriell theorem. Recall that (16) is attained in . Denote by the maximal value of permanent in .
Theorem 8. Consider a class with the numbers , , and satisfying “triangle inequalities” and the following additional conditions: where sequence is defined by recursion (23). Then, for multiple of 3, one has
Proof. Note that conditions (44)-(45) are satisfied, for example, in case . Using induction, let us prove that
Indeed, for , this inequality is trivial, while, for , it follows from the first condition (45). Let it hold for . Then, according to (23), we have
Note that, according to Theorem 4, the equality in (47) holds in a direct sum of -matrices of which corresponds to the partition . Let now . By Theorem 4, there exists a partition of with the parts not less than 3, , such that and, in view of (47), we have This proves (46).
Example 9. Consider case .
Thus the condition means that or And it is easy to verify that the second condition in (45) is satisfied as well. As a corollary, we obtain the following result.
Theorem 10. If (52) holds, then, for multiple of 3, one has
Lemma 11. If , then sequence which is defined by recursion (23) has the form
Theorem 12. If (57) holds, then, for multiple of 4, one has
Proof. From (51) and (57), we conclude that
Let us show that, for ,
For , inequality (60) is trivial. For , we have and thus, using Lemma 11, we have
Let now . By Theorem 4, there exists a partition of with the parts not less than 3, , such that and, in view of (60), we have with the equality in a direct sum of -matrices of which corresponds to the partition .
Note that, if , then in Theorem 10 we have only maximizing matrix (up to a permutation of the rows and columns) which corresponds to the partition ; in Theorem 12 we also have only maximizing matrix (up to a permutation of the rows and columns) which corresponds to the partition . It is interesting that, only in case of the equality , where , Theorems 10–12 are true for every multiple of 12 with the equality of the maximums . Thus, up to a positive factor , the class possesses an interesting extremal property: it contains maximizing matrices (up to a permutation of the rows and columns), instead of only maximizing matrix, if .
Indeed, the number of the maximizing matrices (up to a permutation of the rows and columns) is defined by the number of the following partitions of (mod 12):
5. Estimate of Cardinality of -Spectrum on Circulants in and
Denote by the set of the circulants in . Note that a circulant has a form , , where is -matrix with 1’s on positions only. Multiplying by we obtain circulant of the form with . Since is defined by a choice of two different values , then trivially
Now we prove essentially more exact and practically unimprovable estimate.
Theorem 13. One has
Proof. Let us return back to the general form
Note that is defined by a choice of a vector , but its rotation, that is, a passage to a vector of the form , does not change the magnitude of . Indeed it corresponds to the multiplication by , and our statement follows from the equality . Besides, its reflection relative to some diameter of the imaginary circumference of the rotation, by the symmetry, keeps magnitude of the permanent. Since geometrically three points on the imaginary circumference define a triangle, then our problem reduces to a triangle case of the following general problem, posed by Professor Richard H. Reis (South-East University of Massachusetts, USA) in a private communication to Hansraj Gupta in 1978:
Let a circumference is split by the same parts. It is required to find the number of the incongruent convex -gons, which could be obtained by connection of some from dividing points. Two -gons are considered congruent if they are coincided at the rotation of one relatively other along the circumference and (or) by reflection of one of the -gons relatively some diameter.
In 1979, Gupta  gave a solution of the Reis problem in the form
If we denote by the set of the circulants in , then from our arguments it follows that In case , from (71)-(72) we find and (69) follows.
Example 14. In case we have only two incongruent triangles corresponding to circulants and .
Nevertheless, the calculations give . Thus , and .
Example 15. In case we have three incongruent triangles corresponding to circulants , , and .
The calculations give and , while . Thus , and .
Note that a respectively large magnitude of is explained by its decomposability in a direct product of circulants , such that .
It is clear that, in case of circulants in , the upper estimate (69) yields either the same estimate, if , or , if (and in the symmetric cases), or , if , , and are distinct numbers.
Add that, using a bijection, one can apply formula (71) to enumerate the two-color bracelets of beads, of which are black and are white (see, e.g., the author’s explicit formulas for sequences A032279-A032282, A005513-A005516 in ).
6. Algorithm of Calculations of Upper Magnitudes of the Permanent in
Lemma 16. For , one has
Proof. By the usual way, from (14) we find
where is the golden ratio.
Denote . Since , then , and, consequently, if , then . Therefore, we have
Note that, actually, the difference between the hand sides of (76) is more than .
Let now , , 1, and 2, and . Let denote the set of all partitions of with parts more than or equal to . For us an important role play cases , 4. To , put in a correspondence the sets
In case , when , let us agree that is a singleton .
Consider now the set :
Theorem 17 (algorithm of calculation of upper magnitudes of the permanent in ). If , then the set taken in decreasing order of its elements gives upper magnitudes of the permanent in .
Proof. Note that the proof is the same for every value of . Therefore, let us consider, say, . From (74) it follows that, if contains parts 3 and , then, for , we have
This means that, for the formation of the list of all upper magnitudes of the permanent in in the condition , which are bounded from below by , it is sufficient to consider only a part of the spectrum containing numbers , where with the opposite condition . From the equality with the condition , we have
Since does not depend on , there is only a finite assembly of such partition for arbitrary . This ensures a possibility of the realization of the algorithm.
For the considered , for , we have , where , and has the form , . Thus we should choose only , and this yields
In order to use Theorem 17 for calculating the upper magnitudes of the permanent in , in case, say, ,(1)we write a list of partition of numbers , , with the parts not less than 4;(2)we compare the corresponding values of with and keep only ;(3)after that, we take in decreasing order the numbers .
Below we give the first 10 upper magnitudes of the permanent in , for , via numbers (14). Consider
Formula (82) shows that is attained in . Moreover
7. Main Conjectural Inequality for Maximum of Permanent in Completely Indecomposable -Matrices
Denote by the set of completely indecomposable matrices in , that is, the set of -matrices containing no -submatrices. Let denote the maximum of permanent in . Our very plausible conjecture which we call “main conjectural inequality (MCI)” is the following.
Conjecture 18 (cf. Shevelev , pp. 165-166). For , one has
In Lemma 16 we essentially proved that in subclass the MCI is valid.
Besides, in all known cases MCI holds. Moreover, as we shall see, our algorithm of calculation of the consecutive upper magnitudes of permanent in , which is based on MCI, reproduces all Merriell’s and Bolshakov’s results for and . Note also that, for sufficiently large , the number of consecutive upper magnitudes of permanent in grows very quickly with every step of extension of the list of known -spectrums for small . For example, using the found by Bolshakov , , we obtain, for sufficiently large , 4, 7, and 11 upper values of in cases , , and respectively. Further, after calculating , the number of upper values of, for example, increases by more than thrice.
8. Algorithm of Calculations of Upper Magnitudes of the Permanent in Based on MCI
Let , , 1, 2, and . Let denote the set of all partitions of with parts more than or equal to . For us an important role play cases . To , put in a correspondence sets where runs through all values of permanent in set of completely indecomposable matrices in (in case , when , let us agree that is a singleton ).
Consider now the set :
Theorem 19 (algorithm of calculation of upper magnitudes of the permanent in ). If , then the set taken in decreasing order of its elements gives upper magnitudes of the permanent in .
Proof. We need three lemmas.
Lemma 20. For , one has <