Determinants of generalized inary and matrices Dmitry Efimov arxiv:17005655v1 [mathra] 18 Fe 017 Department of Mathematics, Komi Science Centre UrD RAS, Syktyvkar, Russia Astract Under inary matrices we mean matrices whose entries take one of two values In this paper, explicit formulae for calculating the determinant of some type of inary Toeplitz matrices are otained Examples of the application of the determinant of inary Toeplitz matrices for the enumeration of even and odd permutations of different types are given Introduction Under inary matrices we will mean matrices whose elements can take only two values Matrices of this kind arise in different mathematical questions For example, this type includes such popular ojects in mathematics and its applications as matrices over the field GF) [1] The Hadamard s prolem of finding the maximal determinant of 1, 1)- matrices, ie matrices consisting from 1 and 1, is well known[] Further, 0, 1)-matrices are one of the favorite ojects of the enumerative cominatorics [3 6] Also one can use inary 1, x)-matrices for enumerative prolems [6] These examples can e extended One of the asic notion of the matrix theory is the notion of determinant There exist effective algorithms for the determinant calculation, for example, the modified method of Gaussian elimination, which run in polynomial time Nevertheless, in cases of some kinds of matrices it is possile to otain good explicit formulae for the determinant expression On the one hand, these formulae allow to draw certain conclusions aout properties of matrices On the other hand, they give even greater gain in the speed of the determinant calculation Such formulae are known only for very limited class of matrices Our paper is motivated to some extent y the recent work [7] in which the explicit formula for the determinant of some inary circulant matrices has een otained In the present paper we get explicit formulae for determinants of some kinds of inary Toeplitz matrices Considered matrices are close in their structure to and matrices, therefore we call them generalized and matrices For our purposes we use quite elementary methods Applying the Laplace expansion we otain recurrent formulae leading to required result Thepaperisorganizedasfollows Inthesecondsection wegiveandprovetheformulae, which allow to efficiently calculate determinants of generalized inary and matrices in an explicit form In the third section we give a few examples of the application of the determinant of such inary matrices for the enumeration of even and odd permutations of different types The study is supported y Program of UD RAS, project 15-16-1-3 email: defimov@dmkomiscru 1
1 The main part Let n,k,l e integers, 1 l k n Consider a inary Toeplitz matrix A = a ij ) of order n with the following elements: {, if l < j i < k; a ij = a, otherwise, where a and a ) elong to a commutative associative ring with a unit In the case a = 0 we get a so-called and matrix [8], p16) So we can say that we consider generalized inary and matrices Our purpose is to get explicit formulae for calculating the determinant of the matrix A We will divide this prolem into two cases Case 1 Let l = 1 In other words, let the first row of the matrix A have the form: k {}}{ [ } {{ a a} ], n and the i+1)-st row is otained from the first row y the removal of i elements on the right and addition of i elements a on the left: a A = 1) a Theorem 11 Let n p mod k), where 0 < p k Then the determinant of the matrix 1) is equal to: deta = a) n 1 + n p ) k a ) For the proof of Theorem 11 one can apply the method given in [7] for the proof of explicit formulae of the determinants of circulant matrices using the formula of the determinant of a lock matrix) But in this case another method will e more convenient First let us formulate some auxiliary statements Consider a square matrix of order n of the form: a a a a a 3) a a a a a a a a The given matrix is otained from the matrix 1) of order n 1) y addition of one more the last) row and one more the last) column, consisting entirely of elements a Let f n denote the determinant of such matrix Lemma 11 The determinant of the matrix 3) is equal to: f n = a) n 1 a 4)
Proof Sutracting the penultimate row from the last row and expanding the determinant along the last row, we get that f n = a)f n 1 Using the given formula consecutively to f n 1, f n and so on, we get 4) Suppose that k = n, ie consider the matrix of order n of the form: 5) a By g n denote the determinant of such matrix Lemma 1 One can calculate the determinant of the matrix 5) y the formula: Proof Similarly to the proof of the previous lemma g n = a) n 1 6) Consider the matrix 1) of order n with k < n Let d n denote the determinant of such matrix Lemma 13 The following equality holds: { a) d n = k d n k + a) n 1 a, k < n ; a) n 1 +a), k n 7) Proof Let k < n, ie the numer of elements in the first row of the matrix 1) is less than the numer of elements a Let us sutract the penultimate row of the matrix from the last row and expand the determinant along the last row Performing this procedure consecutively k times, we come to the equality: d n = a) k d n k + a) k f n k Sustituting here the formula 4), we get the first row of the equality 7) Let n k, ie the numer of elements in the first row of the matrix 1) is more or equal to the numer of elements a, ut less than n Executing the same k consecutive expansions along the last row as in the first case, we come to the equality: d n = a) k f n k +g n k ) Sustituting here the formulae 4) and 6), we get the second row of 7) Proof of the Theorem 11) Suppose that n p mod k), where 0 < p k, ie n = km+p, where m is a non-negative integer Assume that k < n Then m > 0 and, using the first row of the formula 7) recurrently m 1 times, we otain the equality: d n = a) n k p d k+p +m 1) a) n 1 a Since k+p k, then applying the second row of the formula 7) to d k+p and performing ovious transformations, we get ) If k = n, ie the first row of the matrix 1) entirely consists from elements, then p = n and the formula ) gives us d n = a) n 1, which corresponds to the statement of the Lemma 1 3
Case Let l > 1 In other words, let the i-th row of the matrix A, i = 1,,l, have the form: k+i 1 {}}{ [ } {{ a a} ], n and the l +j)-th row is otained from the l-th one y the removal of j elements on the right and addition of j elements a on the left: a A = 8) a The matrix A is not symmetric in general, ut it is persymmetric ie symmetric with respect of the secondary diagonal) like all Toeplitz matrices Lemma 14 In the matrix 8) exactly maxk +l n,0) rows consist entirely from elements Proof First note that each row of the matrix A contains no more than k+l 1 elements y definition Let k+l n 0 Hence k+l 1 < n, ie there is at least one element a in each row and there are no rows, consisting entirely from elements Let now k+l n > 0 or, alternatively, k+l 1 n If n = k+i 1, 1 i l, then the numer of rows, that consist entirely from elements, will e equal to l i+1 = k+l n Theorem 1 Let 1 < l k n and n p mod k +l 1), 0 p < k +l 1 Then the determinant of the matrix 8) is equal to: ) 1) k 1)l 1)n k+l 1 a) + n 1 n k l+1 k+l 1 a, p = 0, ) deta = 1) k 1)l 1)n 1) k+l 1 a) + n 1 n 1 k+l 1 a, p = 1, 0, otherwise Proof By the theorem condition n Let n p mod k +l 1), 0 p < k +l 1 This is equivalent to n = k +l 1)s+p, where s is a non-negative integer First consider the case when s = 0, ie when p = n and, respectively, n < k +l 1 By lemma 14 at least rows of the matrix A will consist entirely from elements in this case, therefore the determinant of the matrix A will e equal to 0, that corresponds to the formula 9) Now assume that s 1 The first row of the matrix A differs from the second one only y an element in the k + 1)-st column Let us sutract the first row from the second one and expand the determinant along the second row We will get that deta = 1) k 1 a)deta, where A is a matrix, whose first and second rows differ also only in the k+1)-st column Performing this procedure l 1 times, we otain that 9) 4
deta = 1) k 1)l 1) a) l 1 deta, where a a a A = a a The part of the matrix A that is located aove the horizontal line consists from k rows, and in the first row there are exactly k elements In the lower right corner of the matrix A there is a sumatrix, which is formed y intersection of the last n k l+1 rows and columns of A The given sumatrix, in turn, also has the form 8) The algorithm can e repeated s times with an ovious shift at each step in the k rows down and in the k columns to the right Thus in the second step we sutract the k+1)-st row of the matrix A from the k+)-nd one and expand the determinant along the k +)-nd row, and repeat this procedure l 1 times and so on As a result we will get that deta = 1) k 1)l 1)s a) l 1)s deta, 10) where a a A = a a M The part of the matrix A that is located aove the horizontal line consists from ks rows, and in the first row there are exactly k elements In the lower right corner of the matrix A there is a sumatrix M, which is formed y intersection of the last n k+l 1)s = p rows and columns of the matrix A It is not hard to see that if 1 < p < k + l 1 then M contains two identical rows, hence the matrix A also contains two identical rows, and therefore its determinant and the determinant of the matrix A are equal to 0 If p = 0 then the matrix A is the matrix of the form 1) of order ks Then y Theorem 11 we get: deta = a) ks 1 + ks k ) a = a) ks 1 [+s 1)a] k Then, sustituting the given formula in 10) and taking into account that n = k+l 1)s, we finally otain: deta = 1) k 1)l 1)s a) k+l 1)s 1 [+s 1)a] = = 1) k 1)l 1)n k+l 1 a) n 1 + n k l+1 ) a k+l 1 If p = 1 then the matrix A will e the matrix of the form 1) of order ks+1 Then from Theorem 11 we get: deta = a) ks + ks+1 1 ) a = a) ks +sa) k 5
Then, sustituting the given formula in 10) and taking into account that n = k +l 1)s+1, we finally otain: deta = 1) k 1)l 1)s a) k+l 1)s +sa) = = 1) k 1)l 1)n 1) k+l 1 a) n 1 + n 1 ) k+l 1 a The application of determinants of inary matrices to the enumeration of permutations As mentioned in the Introduction, the inary matrices are one of the favorite ojects of the enumerative cominatorics In particular, they are applied for enumeration of permutations with restricted positions Following [6] let us descrie riefly this mechanism Let A = a ij ) e a 0,1)-matrix of order n Each of such matrices defines a class BA) of restricted permutations Namely a permutation p elongs to BA) if and only if the inequality M p A holds for its incidence matrix M p, ie each element of the matrix M p is not more than the corresponding element of the matrix A The matrix A is called the characteristic matrix of the class BA) It is not hard to see that the numer of permutations in the class BA) is equal to the permanent of the matrix A: BA) = pera Denote the numer of even and odd permutations from the class BA) y E A and O A, respectively It is ovious that E A +O A = pera It is easy to see also that E A O A = deta This implies the following formulae for calculating the total numer of even and odd permutations from the class BA): E A = pera+deta, O A = pera deta 11) As an example we calculate the numer of even and odd permutations π S n such that πi) i,i + 1 for i = 1,n 1 and πn) n The characteristic matrix of this class of permutations is the following 0,1)-matrix A n of order n: A n = 0 0 0 0 1 1 0 0 0 Such matrices arise in the variation of the famous ménage prolem, where not a round tale, ut one side of a rectangular tale is considered [5], ch 8) If we denote the permanent of such matrix y p n, then the sequence of permanents will satisfy the following recurrence relation: n 1)p n = n n 1)p n 1 +np n + 1) n+1, p 1 = p = 0 One can also calculate these permanents y the following explicit formula: n ) n k p n = n k)! 1) k k k=0 Let d n denote the determinant of the matrix A n Sustituting = 0, a = 1, k = to the formula ), we get d n = 1) n 1n p, n p mod ), 0 < p 6
One can also rewrite this equality in the following form: { n 1 n 1 d n = 1) n 1 =, if n odd;, if n even n Applying formulae 11), we otain sequences of the numer of even e n ) and odd o n ) permutations of the given type depending on the permutation order: n 1 3 4 5 6 7 8 9 10 p n 0 0 1 3 16 96 675 5413 48800 48859 d n 0 0 1 1 3 3 4 4 e n 0 0 1 1 9 47 339 705 440 4494 o n 0 0 0 7 49 336 708 4398 4498 As the second example, we will calculate the numer of even and odd permutations π S n such that πi) i > 1, i = 1,,n The characteristic matrix of the given class of permutations is the matrix of order n B n whose main diagonal and its neighoring diagonals are zero, and all other elements are equal to 1: B n = 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 This example is linked with another variation of the ménage prolem, where a rectangular tale is considered and additional restriction on the placement of men is imposed The explicit formula of the total numer of such permutations of order n or, alternatively, of the value of the permanent of B n was found y VS Shevelev see the review [6]) Here we will not give it, ut indicate only that the sequence {p n} of such numers has the id-numer A001883 in [9] Let d n denote the determinant of the matrix B n Sustituting = 0, a = 1, k = l = to the formula 9), we get 3 n d n = 3, if p = 0; n 1 3, if p = 1; 0, if p = ; wheren p mod 3) Applyingformulae 11), we otain sequences of the numer of even e n) and odd o n) permutations of the given type depending on the permutation order: n 1 3 4 5 6 7 8 9 10 p n 0 0 0 1 4 9 06 1708 1570 159737 d n 0 0 0 1 0 1 0 3 e n 0 0 0 1 14 104 854 7850 79870 o n 0 0 0 0 15 10 854 785 79867 Another version of the application of inary matrices to enumeration of permutations has een descried also in [6] It consists in the following Let us consider a inary matrix A of order n in which some elements areequal to the variale, and the other elements are equal to 1 It is easy to see that the coefficient on k in the permanent pera will e equal to the numer of permutations π S n whose incidence matrices have exactly k units in the positions, in which there are elements in the matrix A Respectively, the coefficient on k in the determinant deta will e equal to the difference etween the numer of even and odd permutations of such type Then calculating in expressions 1 deta±pera) the coefficient on k, we get the numer of even and odd permutations of such kind 7
Let us give an example of the application of this scheme Let π e a permutation Recall that a numer i for which πi) i is called a weak excedance of π [10], p 40) Let us find the numer of even and odd permutations of order n with exactly k weak excedances Consider a inary matrix of order n C n whose elements on the main diagonal and aove it are equal to, and other elements are equal to 1: C n = 1 It is well known [10], p 39 40) that the permanent of this matrix is equal to the Eulerian polynomial of order n, and, respectively, the coefficient on k is equal to the Eulerian numer Tn,k) A0089 in [9]) By the formula ) we get that detc n = 1) n 1 It follows that the coefficient on k in detc n is equal to: ) n 1 cn,k) = 1) n k k 1 Hencethenumerofevenandoddpermutationsofordernwithexactly k weakexcedances is equal to: e n,k = 1 )) n 1 Tn,k)+ 1) n k, o n,k = 1 )) n 1 Tn,k) 1) n k k 1 k 1 Let, for example, k = Then e n, = 1 Tn,)+ 1)n n 1)), o n, = 1 Tn,) 1)n n 1)) and we get the following tale: n 1 3 4 5 6 7 8 9 10 Tn,) 0 1 4 11 6 57 10 47 50 1013 cn,) 0 1 3 4 5 6 7 8 9 e n, 0 1 1 7 11 31 57 17 47 511 o n, 0 0 3 4 15 6 63 10 55 50 Let us write out all permutations of order 4 with weak excedances: 143, 143, 413, 314, 314, 341, 341, 413, 413, 431, 431 total 7 even and 4 odd permutations weak excedances are in old, even permutations are underlined) Acknowledgment The author is grateful to VS Shevelev for useful pointers to the literature and comments Referenses [1] Boston N Spaces of constant rank matrices over GF) // Electronic Journal of Linear Algera 010 Vol 0 P 1 5 [] Seerry J, Xia T, Koukouvinos C, Mitrouli M The maximal determinant and sudeterminants of ±1 matrices // Linear Algera and its Applications 003 Vol 373 P 97 310 [3] Brualdi R A, Ryser H J Cominatorial matrix theory Camridge University Press, 1991 [4] Minc H Permanents Reading, MA: Addison-Wesley, 1978 8
[5] Ryser H J Cominatorial Mathematics Mathematical Association of America, 1963 [6] Shevelev V S Some prolems of the theory of enumerating the permutations with restricted position // Journal of Soviet Mathematics 199 Vol 614) P 7 317 [7] Kravvaritis Ch Determinant evaluations for inary circulant matrices // Special Matrices 014 Vol P 187 199 [8] Golu J H, Van Loan C F Matrix computations The Johns Hopkins University Press, 1996 [9] Sloane N J A The On-Line Encyclopedia of Integer Sequences Availale at http://oeisorg/ [10] Stanley R P Enumerative Cominatorics Volume 1 Camridge University Press, nd edition, 011 64 p 9