About

My areas of research are algebraic & enumerative combinatorics, discrete mathematics, and their use in the analysis of complex systems.

I am a member of the Enumerative Combinatorics Group at UCD and serve as an Editorial Board member for the journal Enumerative Combinatorics and Applications.

Several of my more recent papers concern the combinatorics of the Abelian sandpile model, a topic that I am working on in conjunction with my PhD student Amal Alofi.

Degrees:
Ph.D. in Mathematics, Trinity College Dublin.
M.A. in Mathematical Sciences, University of Oxford.

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

Research

Preprints

  1. Amal Alofi and Mark Dukes.
    Parallelogram polyominoes and rectangular EW-tableaux: correspondences through the sandpile model.
    arXiv:2007.15932
  2. Mark Dukes and Anthony A. Casey.
    Combinatorial diversity metrics for the analysis of policy processes.
    arXiv:2008.10401       Data: Traces from Example 8 `Greek Gods'
  3. Mark Dukes.
    The sandpile model on the complete split graph, combinatorial necklaces, and tiered parking functions.
    arXiv:2006.08006

Publications

  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.
    Electronic Journal of Combinatorics 26 (2019), no. 3, P3.29.
  2. Mark Dukes, Thomas Selig, Jason P. Smith and Einar Steingrímsson.
    The Abelian sandpile model on Ferrers graphs – A classification of recurrent configurations.
    European Journal of Combinatorics 81 (2019) 221-241.
  3. Mark Dukes and Peter R.W. McNamara.
    Refining the bijections among ascent sequences, (2+2)-free posets, integer matrices and pattern-avoiding permutations.
    Journal of Combinatorial Theory Series A 167 (2019) 403-430.
  4. Mark Dukes and Thomas Selig.
    Decomposing recurrent states of the Abelian Sandpile Model.
    Séminaire Lotharingien de Combinatoire 77 (2018), B77g.
  5. 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).
  6. Mark Dukes and Chris D. White.
    Web matrices: structural properties and generating combinatorial identities.
    Electronic Journal of Combinatorics 23 (2016), no. 1, P1.45.
  7. 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.
  8. Mark Dukes.
    Generalized ballot sequences are ascent sequences.
    Australasian Journal of Combinatorics 64 (2016), no. 1, 61-63.
  9. 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.
  10. 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.
  11. 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.
  12. 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.
  13. 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.
  14. 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.
  15. 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.
  16. 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.    
  17. Anders Claesson, Mark Dukes and Martina Kubitzke.
    Partition and composition matrices.
    Journal of Combinatorial Theory Series A 118 (2011), no. 5, 1624-1637.
  18. 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.
  19. 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]
  20. 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.
  21. 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).
  22. Mark Dukes and Astrid Reifegerste.
    The area above the Dyck path of a permutation.
    Advances in Applied Mathematics 45 (2010), 15-23.
  23. Anders Claesson, Mark Dukes and Einar Steingrímsson.
    Permutations sortable by n – 4 passes through a stack.
    Annals of Combinatorics, 14 (2010) 45-51.
  24. François David, Mark Dukes, Thordur Jónsson, Sigurdur Örn Stefánsson.
    Random tree growth by vertex splitting.
    Journal of Statistical Mechanics (2009), no. 4, P04009.
  25. 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.
  26. 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.
  27. W. M. B. Dukes.
    Concerning the shape of a geometric lattice.
    Discrete Mathematics 308 (2008), 6632-6638.
  28. W. M. B. Dukes, T. Mansour and A. Reifegerste.
    Wilf classification of three and four letter signed patterns.
    Discrete Mathematics 308 (2008), 3125-3133.
  29. 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.
  30. W. M. B. Dukes and Toufik Mansour.
    Signed involutions avoiding 2-letter signed patterns.
    Annals of Combinatorics 11 (2007), 387-403.
  31. W. M. B. Dukes.
    Permutation statistics on involutions.
    European Journal of Combinatorics 28 (2007), 186-198.
  32. W. M. B. Dukes.
    On the number of matroids on a finite set.
    Séminaire Lotharingien de Combinatoire 51 (2004), B51g.
  33. 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.
  34. 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.
  35. W. M. B. Dukes.
    Bounds on the number of generalized partitions and some applications.
    Australasian Journal of Combinatorics 28 (2003), 257-262.
  36. W. M. B. Dukes.
    On a unimodality conjecture in matroid theory.
    Discrete Mathematics and Theoretical Computer Science 5 (2002), 181-190.
  37. 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 Peter R.W. McNamara.
    Refining the bijections among ascent sequences, (2+2)-free posets, integer matrices and pattern-avoiding permutations.
    FPSAC, Ljubljana, Slovenia, 2019.
    Sém. Lothar. Comb. 82B (2019) 12pp.
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. 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.
  7. 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.
  8. 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.
  9. 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

PhD Opportunities

If you are interested in doing a PhD 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 at mark.dukes@ucd.ie at the earilest opportunity. If you already have funding then the process is easier. If you do not have funding, then here is some information regarding potential sources:

  • 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 2020 if you are considering this for a September 2021 start.

    Irish Mathematical Olympiads Compendium

    All Irish Mathematical Olympiads from 1988 to present available in the IrMO Compendium.

    Teaching

    All material for current undergraduate courses can be found on UCD's latest e-learning system.

    2020–2021:

    • MATH20270: Theory of Games
    • MST30040: Differential Equations
    • MATH30290: Mathematical Logic

    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: 0000-0002-2779-2680
    MR Author ID: 696771
    Scopus Author ID: 25957999500
    Google Scholar: link

    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. Jason P. Smith, University of Aberdeen, UK.
    31. Sigurður Örn Stefánsson, Reykjavik, Iceland. www
    32. Einar Steingrímsson, Strathclyde, UK. www
    33. Vincent Vajnovszki, Dijon, France. www
    34. Chris D. White, Glasgow, UK. www
    35. Susan Yi Jun Wu, Tianjin, China.

    Service

    Editorial Board: Member of the editorial board of the journal Enumerative Combinatorics and Applications

    Journal Reviewing: Over the years I have regularly reviewed for the following journals:
    Advances in Applied Mathematics; Annals of Combinatorics; Discrete Mathematics; Electronic Journal of Combinatorics; European Journal of Combinatorics; Journal of Combinatorial Theory Series A.

    Other journals that I have reviewed for include: Australasian Journal of Combinatorics; Discrete Applied Mathematics; Graphs and Combinatorics; Information Processing Letters; Journal of Combinatorics; Journal of Integer Sequences; Journal of Statistical Planning and Inference; Journal of Statistical Physics; Proceedings of the American Mathematical Society; Pure Mathematics and Applications; Theoretical Computer Science.

    Funding Agencies: I have reviewed for the following research-funding agencies: EPSRC (United Kingdom); National Security Agency (USA); The Icelandic Research Fund (Iceland); Netherlands Organisation for Scientific Research (NWO) (Holland).