About

My area of research is discrete mathematics and its use in complex systems. Lately my research has focused on three particular themes:

The abelian sandpile model

The abelian sandpile model is a popular diffusion model which showcases the dynamical concept of self-organised criticality. Its dynamics on a general graph are governed by the graph's Laplacian and our work in this area has studied the model on the complete bipartite graph and its relation to q,t-Narayana polynomials, staircase polyominoes and bounce paths.

Web worlds, web diagrams and scattering amplitudes

These structures have their motivation in quantum chromodynamics where they are used in the calculation of scattering amplitudes. Our research has shown that the combinatorics of order-preserving maps on partially ordered sets lie at the root of being able to perform calculations related to these structures and amplitudes.

The combinatorics of permutations

Property classification theorems using forbidden sub-structure are widespread in mathematics. When one asks what connections emerge when this lens is pointed at permutations, a rich collection of results in enumerative combinatorics emerges. My research in this area includes the introduction of bivincular patterns and new connections to interval orders.

Degrees:
Ph.D. in Mathematics, Trinity College Dublin.
B.A.(Hons.) in Mathematical Sciences, University of Oxford.

Other:
P.G.Cert. in Academic Studies (Academic Practice).
Fellow of the HEA.

PhD Opportunities

If you are interested in doing a PhD with me in the areas of discrete mathematics, combinatorics, or discrete complex systems then I would be very happy to discuss this possibility with you, so please drop me an email. Here is some information regarding funding:

  • UCD Research Demonstrator positions, the deadline will be end-of-March 2019 this year for a September 2019 start. Two of my projects are listed on that page.
  • Government of Ireland Postgraduate Scholarships. The deadline for applying is usually the beginning of November the year before the position is due to start. Such applications require quite a non-trivial amount of work and I recommend coming to see me no later than 1st September 2018 if you are considering this for a September 2019 start.

    Research

    Preprints

    1. Mark Dukes, Thomas Selig, Jason P. Smith and Einar Steingrímsson
      Permutation graphs and the Abelian sandpile model, tiered trees and non-ambiguous binary trees.
      arXiv:1810.02437
    2. Mark Dukes, Thomas Selig, Jason P. Smith and Einar Steingrímsson
      The Abelian sandpile model on Ferrers graphs – A classification of recurrent configurations.
      arXiv:1809.07728
    3. Mark Dukes and Peter R.W. McNamara.
      Refining the bijections among ascent sequences, (2+2)-free posets, integer matrices and pattern-avoiding permutations.
      arXiv:1807.11505

    Publications

    1. Mark Dukes and Thomas Selig.
      Decomposing recurrent states of the Abelian Sandpile Model.
      Séminaire Lotharingien de Combinatoire 77 (2018), B77g.
    2. Tom S. Weber, Mark Dukes, Denise C. Miles, Stefan P. Glaser, Shalin H. Naik and Ken R. Duffy.
      Site-specific recombinatorics: in situ cellular barcoding with the Cre Lox system.
      BMC Systems Biology 10:43, (2016).
    3. Mark Dukes and Chris D. White.
      Web matrices: structural properties and generating combinatorial identities.
      Electronic Journal of Combinatorics 23 (2016), no. 1, P1.45.
    4. Jean-Christophe Aval, Michele D'Adderio, Mark Dukes, and Yvan Le Borgne.
      Two operators on sandpile configurations, the sandpile model on the complete bipartite graph, and a Cyclic Lemma.
      Advances in Applied Mathematics 73 (2016) 59-98.
    5. Mark Dukes.
      Generalized ballot sequences are ascent sequences.
      Australasian Journal of Combinatorics 64 (2016), no. 1, 61-63.
    6. Mark Dukes.
      Revstack sort, zigzag patterns, descent polynomials of t-revstack sortable permutations, and Steingrímsson's sorting conjecture.
      Electronic Journal of Combinatorics 21 (2014), no. 2, P2.2.
    7. Mark Dukes, Einan Gardi, Heather McAslan, Darren J. Scott, Chris D. White.
      Webs and posets.
      Journal of High Energy Physics 2014 (2014), no. 1, 1-43.
    8. Jean-Christophe Aval, Michele D'Adderio, Mark Dukes, Angela Hicks and Yvan Le Borgne.
      Statistics on parallelogram polyominoes and a q,t-analogue of the Narayana numbers.
      Journal of Combinatorial Theory Series A 123 (2014), no. 1, 271-286.
    9. Mark Dukes, Einan Gardi, Einar Steingrímsson, and Chris D. White.
      Web worlds, web-colouring matrices, and web-mixing matrices.
      Journal of Combinatorial Theory Series A 120 (2013), no. 5, 1012-1037.
    10. Mark Dukes and Yvan Le Borgne.
      Parallelogram polyominoes, the sandpile model on a complete bipartite graph, and a q,t-Narayana polynomial.
      Journal of Combinatorial Theory Series A 120 (2013), no. 4, 816-842.
    11. Michael H. Albert, M. D. Atkinson, Mathilde Bouvel, Anders Claesson and Mark Dukes.
      On the inverse image of pattern classes under bubble sort.
      Journal of Combinatorics 2 (2011), no. 2, 231-243.
    12. Mark Dukes, Sergey Kitaev, Jeffrey Remmel and Einar Steingrímsson.
      Enumerating (2+2)-free posets by indistinguishable elements.
      Journal of Combinatorics 2 (2011), no. 1, 139-163.
    13. Mark Dukes, Vit Jelinek and Martina Kubitzke.
      Composition matrices, (2+2)-free posets and their specializations.
      Electronic Journal of Combinatorics 18 (2011), no. 1, P44.    
    14. Anders Claesson, Mark Dukes and Martina Kubitzke.
      Partition and composition matrices.
      Journal of Combinatorial Theory Series A 118 (2011), no. 5, 1624-1637.
    15. Anders Claesson, Mark Dukes and Sergey Kitaev.
      A direct encoding of Stoimenow's matchings as ascent sequences.
      Australasian Journal of Combinatorics 49 (2011), 47-59.
    16. Fan Chung, Anders Claesson, Mark Dukes and Ronald Graham.
      Descent polynomials for permutations with bounded drop size.
      European Journal of Combinatorics 31 (2010), no. 7, 1853-1867.     [current-errata]
    17. Mireille Bousquet-Mélou, Anders Claesson, Mark Dukes and Sergey Kitaev.
      (2+2)-free posets, ascent sequences and pattern avoiding permutations.
      Journal of Combinatorial Theory Series A 117 (2010), no. 7, 884-909.
    18. Mark Dukes and Robert Parviainen.
      Ascent sequences and upper triangular matrices containing non-negative integers.
      Electronic Journal of Combinatorics 17 (2010), no. 1, #R53 (16pp).
    19. Mark Dukes and Astrid Reifegerste.
      The area above the Dyck path of a permutation.
      Advances in Applied Mathematics 45 (2010), 15-23.
    20. Anders Claesson, Mark Dukes and Einar Steingrímsson.
      Permutations sortable by n – 4 passes through a stack.
      Annals of Combinatorics, 14 (2010) 45-51.
    21. François David, Mark Dukes, Thordur Jónsson, Sigurdur Örn Stefánsson.
      Random tree growth by vertex splitting.
      Journal of Statistical Mechanics (2009), no. 4, P04009.
    22. Eva Y. P. Deng, W. M. B. Dukes, Toufik Mansour and Susan Y. J. Wu.
      Symmetric Schröder paths and restricted involutions.
      Discrete Mathematics 309 (2009), 4108-4115.
    23. W. M. B. Dukes, Vit Jelínek, Toufik Mansour and Astrid Reifegerste.
      New equivalences for pattern avoidance for involutions.
      Proceedings of the American Mathematical Society 137 (2009), 457-465.
    24. W. M. B. Dukes.
      Concerning the shape of a geometric lattice.
      Discrete Mathematics 308 (2008), 6632-6638.
    25. W. M. B. Dukes, T. Mansour and A. Reifegerste.
      Wilf classification of three and four letter signed patterns.
      Discrete Mathematics 308 (2008), 3125-3133.
    26. W. M. B. Dukes, Mark F. Flanagan, Toufik Mansour and V. Vajnovszki.
      Combinatorial Gray codes for classes of pattern avoiding permutations.
      Theoretical Computer Science 396 (2008), 35-49.
    27. W. M. B. Dukes and Toufik Mansour.
      Signed involutions avoiding 2-letter signed patterns.
      Annals of Combinatorics 11 (2007), 387-403.
    28. W. M. B. Dukes.
      Permutation statistics on involutions.
      European Journal of Combinatorics 28 (2007), 186-198.
    29. W. M. B. Dukes.
      On the number of matroids on a finite set.
      Séminaire Lotharingien de Combinatoire 51 (2004), B51g.
    30. T. C. Dorlas and W. M. B. Dukes.
      Fluctuations of the local magnetic field in frustrated mean-field Ising models.
      Markov Processes and Related Fields 10 (2004), 585-606.
    31. Ken Duffy and W. M. B. Dukes.
      On Knuth's generalization of Banach's matchbox problem.
      Mathematical Proceedings of the Royal Irish Academy 104 (2004), 107-118.
    32. W. M. B. Dukes.
      Bounds on the number of generalized partitions and some applications.
      Australasian Journal of Combinatorics 28 (2003), 257-262.
    33. W. M. B. Dukes.
      On a unimodality conjecture in matroid theory.
      Discrete Mathematics and Theoretical Computer Science 5 (2002), 181-190.
    34. T. C. Dorlas and W. M. B. Dukes.
      Large deviation approach to the generalised random energy model.
      Journal of Physics. A. Mathematical and General 35 (2002), 4385-4394.

    Conference Proceedings

    1. Mark Dukes and Thomas Selig.
      Decomposing recurrent states of the abelian sandpile model.
      Discrete Mathematics Days, Barcelona, 2016.
      Electronic Notes in Discrete Mathematics 54 (2016) 97-102.
    2. Jean-Christophe Aval, Michele d'Adderio, Mark Dukes, Angela Hicks and Yvan Le Borgne.
      A q,t-analogue of Narayana numbers.
      25th International Conference on Formal Power Series & Algebraic Combinatorics, Paris, 2013.
      Discrete Math. Theor. Comput. Sci. Proc. AS (2013) 623-634.
    3. Mark Dukes and Yvan Le Borgne.
      The sandpile model on a bipartite graph, parallelogram polyominoes, and a q,t-Narayana polynomial.
      24th International Conference on Formal Power Series & Algebraic Combinatorics, Nagoya, 2012.
      Discrete Math. Theor. Comput. Sci. Proc. AR (2012) 337-348.
    4. Anders Claesson, Mark Dukes and Martina Kubitzke.
      Partition and composition matrices: two matrix analogues of set partitions.
      23rd International Conference on Formal Power Series & Algebraic Combinatorics, Reykjavik, 2011.
      Discrete Math. Theor. Comput. Sci. Proc. AO (2011) 221-232.
    5. Fan Chung, Anders Claesson, Mark Dukes and Ronald Graham.
      Descent polynomials for permutations with bounded drop size.
      22nd International Conference on Formal Power Series & Algebraic Combinatorics, San Francisco, 2010.
      Discrete Math. Theor. Comput. Sci. Proc. AN (2010) 247-258.
    6. Mireille Bousquet-Mélou, Anders Claesson, Mark Dukes and Sergey Kitaev.
      Unlabeled (2+2)-free posets, ascent sequences and pattern avoiding permutations.
      21st International Conference on Formal Power Series & Algebraic Combinatorics, Austria, 2009.
      Discrete Math. Theor. Comput. Sci. Proc. AK (2009) 216-228.
    7. Mark Dukes, Vit Jelínek, Toufik Mansour and Astrid Reifegerste.
      Equivalences for pattern avoiding involutions and classification.
      20th International Conference on Formal Power Series & Algebraic Combinatorics, Chile, 2008.
      Discrete Math. Theor. Comput. Sci. Proc. AJ (2008) 181-188.
    8. Mark Dukes and Toufik Mansour.
      Involutions avoiding the class of permutations in Sk with prefix 12.
      19th International Conference on Formal Power Series & Algebraic Combinatorics, China, 2007.
      PDF format.

    Book Chapters

    1. Mark Dukes and Yvan Le Borgne.
      New aspects of the abelian sandpile model on graphs and their polynomials.
      Graph Polynomials, eds. M. Dehmer, I. Gutman, X. Li, and Y. Shi. Chapman and Hall/CRC Press. 2016.

    Books Edited

    1. Anders Claesson, Mark Dukes, Sergey Kitaev, David Manlove, Kitty Meeks.
      Surveys in Combinatorics 2017.
      Cambridge University Press, 2017.

    Research Funding

    Principal Investigator
        EPSRC grant: New combinatorial perspectives on the abelian sandpile model
        Amount: £354,282
        Grant reference: EP/M015874/1
        Dates: May 2015 - May 2018.
    (Ownership tranferred to E. Steingrímsson on 23/9/2016 on departing the UK, and my role changed to Co-Investigator.)

    Co-Investigator
        Icelandic Research Fund Excellence grant: Combinatorics on permutations and words
        Amount: ISK 67,609,000
        Grant reference: 90038011, 90038012, 90038013
        Dates: January 2009 - December 2011.

    University of Strathclyde Faculty of Science Grant
        Amount: £11,000
        Dates: Jan 2012 - Dec 2013

    Lecturing

    All material for current courses can be found on UCD's Blackboard.

    2018–2019:

    • MATH20300: Linear Algebra 2
    • MATH20270: Theory of Games
    • MST30040: Differential Equations

    Contact

      Postal address:

       Dr Mark Dukes
       UCD School of Mathematics and Statistics
       University College Dublin
       Belfield, Dublin 4, Ireland

    E-mail: mark.dukes@ccc.oxon.org, mark.dukes@ucd.ie

    ORCID: 0000000227792680
    MR Author ID: 696771

    Co-authors

    1. Michael H. Albert, Otago, New Zealand. www
    2. Mike D. Atkinson, Otago, New Zealand. www
    3. Jean-Christophe Aval, Bordeaux, France. www
    4. Mireille Bousquet-Mélou, Bordeaux, France. www
    5. Mathilde Bouvel, Zurich, Switzerland. www
    6. Fan Chung, San Diego, USA. www
    7. Anders Claesson, Strathclyde, UK. www
    8. Michele D'Adderio, Brussels, Belgium. www
    9. François David, Paris, France. www
    10. Eva Y. P. Deng, Dalian, China.
    11. Tony Dorlas, Dublin, Ireland. www
    12. Ken Duffy, Maynooth, Ireland. www
    13. Mark Flanagan, Dublin, Ireland. www
    14. Einan Gardi, Edinburgh, UK. www
    15. Ronald Graham, San Diego, USA. www
    16. Angela Hicks, Stanford, USA. www
    17. Vit Jelínek, Prague, Czech Republic. www
    18. Þórður Jónsson, Reykjavík, Iceland. www
    19. Sergey Kitaev, Strathclyde, UK. www
    20. Martina Kubitzke, Frankfurt, Germany. www
    21. Yvan Le Borgne, Bordeaux, France. www
    22. Toufik Mansour, Haifa, Israel. www
    23. Heather McAslan, Sussex, UK.
    24. Peter R.W. McNamara, Bucknell, USA. www
    25. Robert Parviainen, WGS Gaming, USA.
    26. Astrid Reifegerste, Magdeburg, Germany. www
    27. Jeffrey Remmel, San Diego, USA. www
    28. Darren J. Scott, Durham, UK.
    29. Thomas Selig, Strathclyde, UK. www
    30. Sigurður Örn Stefánsson, Reykjavik, Iceland. www
    31. Einar Steingrímsson, Strathclyde, UK. www
    32. Vincent Vajnovszki, Dijon, France. www
    33. Chris D. White, Glasgow, UK. www
    34. Susan Yi Jun Wu, Tianjin, China.