# Uniqueness and Additivity for n-Dimensional Binary Matrices with Respect to Their 1-Marginals

• E. Vallejo
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

## Abstract

In this chapter we deal with the question of when an n-dimensional binary matrix is uniquely determined by its 1-marginals and with the related notion of (0, 1)-additivity. We present a survey of known results; several of them have been considered before only in dimensions 2 and 3. Here, we show how to extend them to any dimension. The main results are characterizations of uniqueness and (0,1) additivity: one, of algebraic nature, involves matrices with integer entries; the other, of geometric nature, uses transportation polytopes and permutohedra.

## Keywords

Symmetric Group Binary Matrix Plane Partition Additive Matrice Dominance Order
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Avella-Alaminos, D., Vallejo, E.: Kronecker products and RSK-correspondences for 3-dimensional matrices. In preparation.Google Scholar
2. 2.
Billera, L.J., Sarangarajan, A.: The combinatorics of permutation polytopes. In: Billera, L.J., Greene, C., Simion, R., Stanley R.P. (eds.), Formal Power Series and Algebraic Combinatorics 1994, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 24, AMS, Providence, RI, pp. 1–23 (1996).Google Scholar
3. 3.
Brylawsky, T.: The lattice of integer partitions. Discr. Math., 6, 201–219 (1973).
4. 4.
Brualdi, R.A.: Matrices of zeros and ones with fixed row and column sum vectors. Lin. Algebra Appl., 33, 159–231 (1980).
5. 5.
Brualdi, R.A.: Minimal nonnegative integral matrices and uniquely determined (0,1)-matrices. Lin. Algebra Appl., 341, 351–356 (2002).
6. 6.
Brunetti, S., Del Lungo, A., Gerard, Y.: On the computational complexity of reconstructing three-dimensional lattice sets from their two-dimensional X-rays. Lin. Algebra Appl., 339, 59–73 (2001).
7. 7.
De Loera, J., Onn, S.: The complexity of three-way statistical tables. SIAM J. Comput., 33, 819–836 (2004).
8. 8.
Díaz-Leal, H., Mart’mez-Bernal, J., Romero, D.: Dimension of the fixed point set of a nilpotent endomorphism on the flag variety. Bol. Soc. Mat. Mexicana (3), 7, 23–33 (2001).
9. 9.
Fishburn, P.C., Lagarias, J.C., Reeds, J.A., Shepp, L.A.: Sets uniquely determined by projections on axes II. Discrete case. Discr. Math., 91, 149–159 (1991).
10. 10.
Fishburn, P.C., Shepp, L.A.: Sets of uniqueness and additivity in integer lattices. In: Herman, G.T., Kuba, A. (eds.), Discrete Tomography: Foundations, Algorithms, and Applications, Birkhäuser, Boston, MA, pp. 35–58 (1999).Google Scholar
11. 11.
Gale, D.: A theorem of flows in networks. Pacific J. of Math., 7, 1073–1082 (1957).
12. 12.
Gritzmann, P., de Vries, S.: On the algorithmic inversion of the discrete Radon transform. Theor. Comp. Science, 281, 455–469 (2001).
13. 13.
Gaiha, P., Gupta, S.K.: Adjacent vertices on a permutohedron. SIAM J. Appl. Math., 32, 323–327 (1977).
14. 14.
Grünbaum, B.: Convex Polytopes, 2nd edition. Springer, Berlin, Germany (2003).Google Scholar
15. 15.
Hardy, G., Littlewood, J.E., Polya, G.: Inequalities, 2nd edition Cambridge Univ. Press, Cambridge, UK (1952).
16. 16.
Klee, V., Witzgall, C.: Facets and vertices of transportation polytopes. In: Mathematics of Decision Sciences, Part I. Amer. Math. Soc., Providence, RI, pp. 257–282 (1968).Google Scholar
17. 17.
Kong, T.Y., Herman, G.T.: On which grids can tomographic equivalence of binary pictures be characterized in terms of elementary switching operations? Int. J. Imaging Syst. Technol., 9, 118–125 (1998).
18. 18.
Kuba, A., Herman, G.T.: Discrete tomography: A historical overview. In: Herman, G.T., Kuba, A. (eds.), Discrete Tomography. Foundations, Algorithms, and Applications. Birkhäuser, Boston, MA, pp. 3–34 (1999).Google Scholar
19. 19.
Jurkat, W.B., Ryser H.J.: Extremal configurations and decomposition theorems I. J. Algebra, 8, 194–222 (1968).
20. 20.
Marshall, A.W., Olkin, I.: Inequalities: Theory of Majorization and Its Applications. Academic Press, New York, NY (1979).
21. 21.
Minoux, M.: Solving integer minimum cost flows with separable convex cost objective polynomially. Math. Programm. Stud., 26, 237–239 (1986).
22. 22.
Muirhead, R.F.: Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters. Proc. Edinburgh Math. Soc., 21, 144–157 (1903).
23. 23.
Onn, S., Vallejo, E.: Permutohedra and minimal matrices. Lin. Algebra Appl., 412, 471–489 (2006).
24. 24.
Rado, R.: An inequality. J. London Math. Soc., 27, 1–6 (1952).
25. 25.
Ryser, H.J.: Combinatorial properties of matrices of zeros and ones. Canad. J. Math., 9, 371–377 (1957).
26. 26.
Ryser, H.: Combinatorial Mathematics, Mathematical Association of America, Washington, DC (1963).
27. 27.
Snapper, E.: Group characters and nonnegative integral matrices. J. Algebra, 19, 520–535 (1971).
28. 28.
Torres-Cházaro, A., Vallejo, E.: Sets of uniqueness and minimal matrices. J. Algebra, 208, 444–451 (1998).
29. 29.
Vallejo, E.: Reductions of additive sets, sets of uniqueness and pyramids. Discr. Math., 173, 257–267 (1997).
30. 30.
Vallejo, E.: Plane partitions and characters of the symmetric group. J. Algebraic Comb., 11, 79–88 (2000).
31. 31.
Vallejo, E.: The classification of minimal matrices of size 2×q. Lin. Algebra Appl., 340, 169–181 (2002).
32. 32.
Vallejo, E.: A characterization of additive sets. Discr. Math., 259, 201–210 (2002).
33. 33.
Vallejo, E.: Minimal matrices and discrete tomography. Electr. Notes Discr. Math., 20, 113–132 (2005).
34. 34.
Yemelichev, V.A., Kovalev, M.M., Kravtsov, M.K.: Polytopes, Graphs and Optimisation. Cambridge University Press, Cambridge, UK (1984).