Skip to main content
SpringerLink
Log in

Chains and Antichains in the Bruhat Order for Classes of (0, 1)-Matrices

  • Conference paper
  • First Online:
Applied and Computational Matrix Analysis (MAT-TRIAD 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 192))

Included in the following conference series:

Abstract

Let \({\mathscr {A}}(R,S)\) denote the set of all matrices of zeros and ones with row sum vector R and column sum vector S. This set can be ordered by a generalization of the usual Bruhat order for permutations. Contrary to the classical Bruhat order on permutations, where permutations can be seen as permutation matrices, the Bruhat order on the class \({\mathscr {A}}(R,S)\) is not, in general, graded, and an interesting problem is the determination of bounds for the maximal length of chains and antichains in this poset. In this survey we aim to provide a self-contained account of the recent developments involving the determination of maximum lengths of chains and antichains in the Bruhat order on some classes of matrices in \({\mathscr {A}}(R,S)\).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Barvinok, A.: On the number of matrices and a random matrix with prescribed row and column sums and \(0-1\) entries. Adv. Math. 224(1), 316–339 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Borovnick, A.V., Gelfand, I.M., White, N.: Coxeter Matroids. Birkhäuser, Boston (2003)

    Google Scholar 

  3. Brualdi, R.A.: Matrices of zeros and ones with fixed row and column sum vectors. Linear Algebra Appl. 33, 159–231 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brualdi, R.A.: Short proofs of the Gale & Ryser and Ford & Fulkerson characterizations of the row and column sum vectors of \((0,1)\)-matrices. Math. Inequal. Appl. 4, 157–159 (2001)

    MathSciNet  MATH  Google Scholar 

  5. Brualdi, R.A.: Combinatorial matrix classes. Encyclopedia of Mathematics and its Applications, vol. 108. Cambridge University Press, Cambridge (2006)

    Google Scholar 

  6. Brualdi, R.A.: Algorithms for constructing \((0,1)\)-matrices with prescribed row and column sum vectors. Discrete Math. 306(3), 3054–3062 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brualdi, R.A., Deaett, L.: More on the Bruhat order for \((0,1)\)-matrices. Linear Algebra Appl. 421(2–3), 219–232 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brualdi, R.A., Hwang, S.-G.: A Bruhat order for the class of \((0,1)\)–matrices with row sum vector \(R\) and column sum vector \(S\). Electron. J. Linear Algebra 12, 6–16 (2004/05)

    Google Scholar 

  9. Brylawski, T.: The lattice of integer partitions. Discrete Math. 6, 201–219 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  10. Canfield, E.R., McKay, B.D.: Asymptotic enumeration of dense \(0-1\) matrices with equal row sums and equal column sums. Electron. J. Combin. 12, Research Paper 29, 31 pp. (2005)

    Google Scholar 

  11. Conflitti, A., da Fonseca, C.M., Mamede, R.: The maximal length of a chain in the Bruhat order for a class of binary matrices. Linear Algebra Appl. 436(3), 753–757 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Conflitti, A., da Fonseca, C.M., Mamede, R.: On the largest size of an antichain in the Bruhat order for \(A(2k, k)\). Order 30(1), 255–260 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dahl, G.: Network flows and combinatorial matrix theory, Lecture Notes (2010)

    Google Scholar 

  14. Dilworth, R.P.: A decomposition theorem for partially ordered sets. Ann. Math. 51(2), 161–166 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  15. Everett Jr., C.J., Stein, P.R.: The asymptotic number of integer stochastic matrices. Discrete Math. 1, 33–72 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fulkerson, D.R.: Zero-one matrices with zero trace. Pac. J. Math. 10, 831–836 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gale, D.: A theorem on flows in networks. Pac. J. Math. 7, 1073–1082 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gao, S., Matheis, K.: Closed formulas and integer sequences arising from the enumeration of (0, 1)-matrices with row sum two and some constant column sums. Congr. Numer. 202, 45–53 (2010)

    Google Scholar 

  19. Ghebleh, M.: On maximum chains in the Bruhat order of \(A(n, 2)\). Linear Algebra Appl. 446, 377–387 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ghebleh, M.: Antichains of \((0,1)\)-matrices through inversions. Linear Algebra Appl. 458, 503–511 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Goldstein, D., Stong, R.: On the number of possible row and column sums of \(0,1\)–matrices. Electron. J. Combin. 13(1), Note 8, 6 pp. (electronic) (2006)

    Google Scholar 

  22. Greenhill, C., McKay, B.D., Wang, X.: Asymptotic enumeration of sparse 0–1 matrices with irregular row and column sums. J. Combin. Theory Ser. A 113(2), 291–324 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Greenhill, C., McKay, B.D.: Asymptotic enumeration of sparse nonnegative integer matrices with specified row and column sums. Adv. Appl. Math. 41(4), 459–481 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Haber, R.M.: Term rank of \(0, l\) matrices. Rend. Sem. Mat. Univ. Padova 30, 24–51 (1960)

    MathSciNet  MATH  Google Scholar 

  25. Krause, M.: A simple proof of the Gale-Ryser theorem. Am. Math. Mon. 103, 335–337 (1996)

    Google Scholar 

  26. van Lint, J.H., Wilson, R.M.: A Course in Combinatorics. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  27. McKay, B.D., Wang, X.: Asymptotic enumeration of \(0-1\) matrices with equal row sums and equal column sums. Linear Algebra Appl. 373, 273–287 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mirsky, L.: A dual of Dilworth’s decomposition theorem. Am. Math. Mon. 78, 876–877 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  29. O’Neil, P.E.: Asymptotics and random matrices with row-sum and column-sum restrictions. Bull. Am. Math. Soc. 75, 1276–1282 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  30. Pérez-Salvador, B.R., de-los Cobos-Silva, S., Gutiérrez-Andrade, M.A., Torres-Chazaro, A.: A reduced formula for the precise number of \((0,1)\)–matrices in \({\fancyscript {A}}\) (R, S). Discrete Math. 256(1–2), 361–372 (2002)

    Google Scholar 

  31. Ryser, H.J.: Combinatorial properties of matrices of zeros and ones. Canad. J. Math. 9, 371–377 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ryser, H.J.: Combinatorial mathematics. The Carus Mathematical Monographs, vol. 14, (Wiley, New York, 1963)

    Google Scholar 

  33. Wang, B.-Y.: Precise number of \((0,1)\)-matrices in \(\mathfrak{A}(R, S)\). Sci. Sinica Ser. A 31(1), 1–6 (1988)

    MathSciNet  Google Scholar 

  34. Wang, B.-Y., Zhang, F.: On the precise number of \((0,1)\)-matrices in \(\mathfrak{A}(R, S)\). Discrete Math. 187(1–3), 211–220 (1998)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was partially supported by the Centre for Mathematics of the University of Coimbra – UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020.

Author information

Authors and Affiliations

  1. Department of Mathematics, CMUC, University of Coimbra, 3001-454, Coimbra, Portugal

    Ricardo Mamede

Authors
  1. Ricardo Mamede
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ricardo Mamede .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Coimbra, Coimbra, Portugal

    Natália Bebiano

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Mamede, R. (2017). Chains and Antichains in the Bruhat Order for Classes of (0, 1)-Matrices. In: Bebiano, N. (eds) Applied and Computational Matrix Analysis. MAT-TRIAD 2015. Springer Proceedings in Mathematics & Statistics, vol 192. Springer, Cham. https://doi.org/10.1007/978-3-319-49984-0_15

Download citation

Publish with us

Policies and ethics

Search

Navigation