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We provide a method to determine if a q-ary multidimensional matrix is lonesum or not by using properties of line sums of lonesum multidimensional matrices. In particular, we establish a graphic method that uses edge-colored graphs to determine if a binary multidimensional matrix is lonesum or not. We also provide two methods to determine if a q-ary multidimensional matrix is lonestructure or not. The first one uses properties of line structures of lonestructure multidimensional matrices and the second one uses edge-colored directed multigraphs.

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A lonesum matrix is a matrix that can be uniquely reconstructed from its
row and column sums. Kaneko defined the poly-Bernoulli numbers
$B_m^{(n)}$ by a generating function, and Brewbaker computed the number
of binary lonesum $m\times n$-matrices and showed that this number
coincides with the poly-Bernoulli number $B_m^{(-n)}$. We compute the
number of $q$-ary lonesum $m\times n$-matrices, and then provide
generalized Kaneko's formulas by using the generating function for the
number of $q$-ary lonesum $m\times n$-matrices. In addition, we define
two types of $q$-ary lonesum matrices that are composed of strong and
weak lonesum matrices, and suggest further researches on lonesum
matrices. \

Consider the class AP(R, S) of (0, 1)-matrices with row sum vector R, column sum vector S, and zeros in all positions outside a certain set P. It is assumed that P satisfies a certain monotonicity property. We show the existence of a canonical matrix in this matrix
class and give a simple algorithm for finding this matrix. Moreover, a classical interchange result of Ryser is generalized
to the class AP(R, S) and the uniqueness question for the class AP(R, S) is discussed.

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We show that the number of (0,1)-matrices with n rows and k columns uniquely reconstructable from their row and column sums is the poly-Bernoulli number B n (-k) . Combinatorial proofs for both the sieve and closed formulas are presented. In addition, we prove an analogue of Fermat’s Little Theorem: For a positive integer n and prime number p we have B n (-p) ≡2 n (modp)· Also, an analogue to Fermat’s Last Theorem is presented: For all positive integers {x,y,z} and n>1 there exist no solution to the equation B x (-n) +B y (-n) =B z (-n) ·

This paper is concerned with a matrix A of m rows and n columns, all of whose entries are 0’s and 1’s. Let the sum of row i of A be denoted by r
i
(i = 1, ... , m) and let the sum of column i of A be denoted by S
i
(i = 1, ... ,n).

For q∈C transcendental over Q, we give an algorithmic construction of an order-isomorphism between the set of H-primes of Oq(Mn(C)) and the sub-poset S of the (reverse) Bruhat order of the symmetric group S2n consisting of those permutations that move any integer by no more than n positions. Further, we describe the permutations that correspond via this bijection to rank tH-primes. More precisely, we establish the following result. Imagine that there is a barrier between positions n and n+1. Then a 2n-permutation σ∈S corresponds to a rank tH-invariant prime ideal of Oq(Mn(C)) if and only if the number of integers that are moved by σ from the right to the left of this barrier is exactly n−t. The existence of such an order-isomorphism was conjectured by Goodearl and Lenagan.

This thesis shows that the number of (0,1)-matrices with n rows and k columns uniquely reconstructible from their row and column sums are the poly-Bernoulli numbers of negative index, B[subscript n superscript (-k)] . Two proofs of this main theorem are presented giving a combinatorial bijection between two poly-Bernoulli formula found in the literature. Next, some connections to Fermat are proved showing that for a positive integer n and prime number p B[subscript n superscript (-p) congruent 2 superscript n (mod p),] and that for all positive integers (x, y, z, n) greater than two there exist no solutions to the equation: B[subscript x superscript (-n)] + B[subscript y superscript (-n)] = B[subscript z superscript (-n)]. In addition directed graphs with sum reconstructible adjacency matrices are surveyed, and enumerations of similar (0,1)-matrix sets are given as an appendix. Typescript (photocopy) Thesis (M.S.)--Iowa State University, 2005. Includes bibliography.

Let 𝕂 be a (commutative) field and consider a nonzero element q in 𝕂 that is not a root of unity. Goodearl and Lenagan (20028.
Goodearl , K. R. ,
Lenagan , T. H. ( 2002 ). Prime ideals invariant under winding automorphisms in quantum matrices . Internat. J. Math 13 : 497 – 532 . [CROSSREF] View all references) have shown that the number of ℋ-primes in R = O q (ℳ n (𝕂)) that contain all (t + 1) × (t + 1) quantum minors but not all t × t quantum minors is a perfect square. The aim of this paper is to make precise their result: we prove that this number is equal to (t!) 2 S(n + 1, t + 1)2, where S(n + 1, t + 1) denotes the Stirling number of the second kind associated to n + 1 and t + 1. This result was conjectured by Goodearl, Lenagan, and McCammond. The proof involves some closed formulas for the poly-Bernoulli numbers that were established by Kaneko (199710.
Kaneko , M. ( 1997 ). Poly-Bernoulli numbers . J. Théorie Nombres Bordeaux 9 : 221 – 228 . View all references) and Arakawa and Kaneko (19991.
Arakawa , T. ,
Kaneko , M. ( 1999 ). On poly-Bernoulli numbers . Comment Math. Univ. St. Paul 48 ( 2 ): 159 – 167 . View all references).

Constructing (0, 1)-matrices with given line sums and certain fixed zeros, Advances in Discrete Tomography and Its Applications

- R A Brualdi
- G Dahl

R.A. Brualdi, G. Dahl, Constructing (0, 1)-matrices with given line sums and certain fixed zeros, Advances in Discrete Tomography and Its Applications, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2007, pp. 113–123.