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. (Commuted to an M.A.)

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

Research

Preprints

  1. Mark Dukes and Thomas Selig.
    Decomposing recurrent states of the Abelian Sandpile Model.
    Submitted for publication.

Publications

  1. 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, to appear.
  2. Mark Dukes and Chris D. White.
    Web matrices: structural properties and generating combinatorial identities.
    Electronic Journal of Combinatorics 23 (2016), no. 1, P1.45.
  3. 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.
  4. Mark Dukes.
    Generalized ballot sequences are ascent sequences.
    Australasian Journal of Combinatorics 64 (2016), no. 1, 61-63.
  5. 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.
  6. 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.
  7. 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.
  8. 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.
  9. 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.
  10. 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.
  11. 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.
  12. 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.    
  13. Anders Claesson, Mark Dukes and Martina Kubitzke.
    Partition and composition matrices.
    Journal of Combinatorial Theory Series A 118 (2011), no. 5, 1624-1637.
  14. 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.
  15. 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]
  16. 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.
  17. 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).
  18. Mark Dukes and Astrid Reifegerste.
    The area above the Dyck path of a permutation.
    Advances in Applied Mathematics 45 (2010), 15-23.
  19. Anders Claesson, Mark Dukes and Einar Steingrímsson.
    Permutations sortable by n – 4 passes through a stack.
    Annals of Combinatorics, 14 (2010) 45-51.
  20. François David, Mark Dukes, Thordur Jónsson, Sigurdur Örn Stefánsson.
    Random tree growth by vertex splitting.
    Journal of Statistical Mechanics (2009), no. 4, P04009.
  21. 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.
  22. 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.
  23. W. M. B. Dukes.
    Concerning the shape of a geometric lattice.
    Discrete Mathematics 308 (2008), 6632-6638.
  24. W. M. B. Dukes, T. Mansour and A. Reifegerste.
    Wilf classification of three and four letter signed patterns.
    Discrete Mathematics 308 (2008), 3125-3133.
  25. 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.
  26. W. M. B. Dukes and Toufik Mansour.
    Signed involutions avoiding 2-letter signed patterns.
    Annals of Combinatorics 11 (2007), 387-403.
  27. W. M. B. Dukes.
    Permutation statistics on involutions.
    European Journal of Combinatorics 28 (2007), 186-198.
  28. W. M. B. Dukes.
    On the number of matroids on a finite set.
    Séminaire Lotharingien de Combinatoire 51 (2004), B51g.
  29. 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.
  30. 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.
  31. W. M. B. Dukes.
    Bounds on the number of generalized partitions and some applications.
    Australasian Journal of Combinatorics 28 (2003), 257-262.
  32. W. M. B. Dukes.
    On a unimodality conjecture in matroid theory.
    Discrete Mathematics and Theoretical Computer Science 5 (2002), 181-190.
  33. 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. 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.
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. 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.
  7. 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.
    To appear in Graph Polynomials, eds. M. Dehmer, I. Gutman, X. Li, and Y. Shi. CRC Press.

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 Starter Grant
    Amount: £11,000
    Dates: Jan 2012 - Dec 2013

Lecturing

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

2015–2016: (University of Strathclyde)

  • CS208: Logic and Algorithms
  • CS408: Individual Project
  • CS411: Theory of Computation
  • CS412: Information Access and Mining

2014–2015: (University of Strathclyde)

  • CS208: Logic and Algorithms
  • CS408: Individual Project
  • CS411: Theory of Computation
  • CS412: Information Access and Mining

2013–2014: (University of Strathclyde)

  • CS208: Logic and Algorithms
  • CS408: Individual Project
  • CS411: Theory of Computation
  • CS412: Information Access and Mining

2012–2013: (University of Strathclyde)

  • CS101: Topics in Computing I: Discrete Mathematics
  • CS208: Logic and Algorithms
  • CS408: Individual Project
  • CS412: Information Access and Mining

2011–2012: (University of Strathclyde)

  • CS101: Topics in Computing I: Discrete Mathematics
  • CS208: Logic and Algorithms
  • CS408: Individual Project
  • CS412: Information Access and Mining

2008–2009:

  • E724: Discrete Mathematics (Reykjavík University)

2007–2008:

  • E401: Probability Theory (Reykjavík University)
  • 09.10.74: Combinatorics (University of Iceland)

2006–2007: (University College Dublin)

  • MATH10120: Linear Algebra with Applications to Economics
  • MATH10140: Advanced Calculus
  • MST30030: Financial Mathematics
  • MATH20050: Vector Spaces and Coding

2005–2006: (University College Dublin)

  • MATH10120: Linear Algebra with Applications to Economics
  • MATH10140: Advanced Calculus
  • MATH2204: Probability and Statistics
  • MATH2104: Functions of Several Variables
  • MST2001: Calculus of Several Variables

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

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. Stefan Glaser, Melbourne, Australia.
  16. Ronald Graham, San Diego, USA. www
  17. Angela Hicks, Stanford, USA. www
  18. Vit Jelínek, Prague, Czech Republic. www
  19. Þórður Jónsson, Reykjavík, Iceland. www
  20. Sergey Kitaev, Strathclyde, UK. www
  21. Martina Kubitzke, Frankfurt, Germany. www
  22. Yvan Le Borgne, Bordeaux, France. www
  23. Toufik Mansour, Haifa, Israel. www
  24. Heather McAslan, Sussex, UK.
  25. Denise Miles, Melbourne, Australia.
  26. Shalin Naik, Melbourne, Australia.
  27. Robert Parviainen, WGS Gaming, USA.
  28. Astrid Reifegerste, Magdeburg, Germany. www
  29. Jeffrey Remmel, San Diego, USA. www
  30. Darren J. Scott, Durham, UK.
  31. Sigurður Örn Stefánsson, Reykjavik, Iceland. www
  32. Einar Steingrímsson, Strathclyde, UK. www
  33. Vincent Vajnovszki, Dijon, France. www
  34. Tom Weber, Maynooth, Ireland.
  35. Chris D. White, Glasgow, UK. www
  36. Susan Yi Jun Wu, Tianjin, China.