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Multidimensional matrices uniquely recovered by their lines

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Abstract

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. \
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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.