About

My areas of research are enumerative combinatorics, combinatorial physics, and the use of discrete mathematics in the analysis of complex systems. A prominant theme in my research is the algorithmic construction and analysis of bijections between discrete objects. Computer algebra plays a considerable role in this discovery process with the Pythonesque SageMath being a favorite tool. Currently my main university administration role is as Head of Subject (Mathematics). I serve as both an Editorial Board member and a Managing Editor for the journal Enumerative Combinatorics and Applications. I'm also a member of the Enumerative Combinatorics Group at UCD.

Researcher profile links: MathReviews; Google Scholar; ORCID; Scopus.

My Erdős number is 2 via Dukes ↔ Chung/Graham ↔ Erdős. Some academic genealogy fun.

Research

Preprints

  1. Henri Derycke, Mark Dukes, and Yvan Le Borgne.
    The sandpile model on the complete split graph: q,t-Schröder polynomials, sawtooth polyominoes, and a cyclic lemma.
    arXiv:2402.15372
  2. Michele D'Adderio, Mark Dukes, Alessandro Iraci, Alexander Lazar, Yvan Le Borgne, and Anna Vanden Wyngaerd.
    Shuffle theorems and sandpiles.
    arXiv:2401.06488
  3. Mark Dukes and Anton Sohn.
    Characterizing traces of processes defined by precedence and response constraints: an order theory approach.
    arXiv:2311.12218

Publications

  1. Amal Alofi and Mark Dukes.
    A note on the lacking polynomial of the complete bipartite graph.
    Discrete Mathematics 348 (2025), no. 2, 114323.   DOI
  2. Mark Dukes and Bruce Sagan.
    Difference ascent sequences.
    Advances in Applied Mathematics 159 (2024), 102736.   DOI
  3. Mark Dukes and Andrew Mullins.
    A bump statistic on permutations resulting from the Robinson-Schensted correspondence.
    Annals of Combinatorics, to appear.   DOI
  4. Mark Dukes.
    An Ising model having permutation spin motivated by a permutation complexity measure.
    Physica A: Statistical Mechanics and its Applications 626 (2023), 129090.   DOI
  5. Anders Claesson, Mark Dukes, Atli Fannar Franklín, and Sigurdur Örn Stefánsson.
    Counting tournament score sequences.
    Proceedings of the American Mathematical Society 151 (2023), 3691-3704.   DOI
  6. Beáta Bényi, Anders Claesson, and Mark Dukes.
    Weak ascent sequences and related combinatorial structures.
    European Journal of Combinatorics 108 (2023), 103633.   DOI
  7. Mark Dukes.
    A simple declarative model of the Federal Disaster Assistance Policy - modelling and measuring transparency.
    EURO Journal on Decision Processes 11 (2023), 100035.   DOI   [github-repo].
  8. Mark Dukes.
    Stakeholder utility measures for declarative processes and their use in process comparisons.
    IEEE Transactions on Computational Social Systems 9 (2022), no. 2, 546-558.   DOI
  9. Mark Dukes and Anthony A. Casey.
    Combinatorial diversity metrics for declarative processes: an application to policy process analysis.
    International Journal of General Systems 50 (2021), no. 4, 367-387.   DOI   [Example 8: Greek Gods]
  10. Mark Dukes.
    The sandpile model on the complete split graph, Motzkin paths, and tiered parking functions.
    Journal of Combinatorial Theory Series A 180 (2021) Article 105418.   DOI
  11. Mark Dukes.
    Fagan's Construction, Strange Roots, and Tchoukaillon Solitaire.
    Journal of Integer Sequences 24 (2021), Article 21.7.1.
  12. Mark Dukes and Toufik Mansour.
    A maxdrop statistic for standard Young tableaux.
    Discrete Mathematics, Algorithms and Applications 14 (2022), no. 2, 2150105.   DOI
  13. Amal Alofi and Mark Dukes.
    Parallelogram polyominoes and rectangular EW-tableaux: correspondences through the sandpile model.
    Enumerative Combinatorics and Applications 1 (2021), no. 1, Article S2R8.   DOI
  14. 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.   DOI
  15. 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.   DOI
  16. 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.   DOI
  17. Mark Dukes and Thomas Selig.
    Decomposing recurrent states of the Abelian Sandpile Model.
    Séminaire Lotharingien de Combinatoire 77 (2018), B77g.
  18. 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).   DOI
  19. Mark Dukes and Chris D. White.
    Web matrices: structural properties and generating combinatorial identities.
    Electronic Journal of Combinatorics 23 (2016), no. 1, P1.45.   DOI
  20. 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.   DOI
  21. Mark Dukes.
    Generalized ballot sequences are ascent sequences.
    Australasian Journal of Combinatorics 64 (2016), no. 1, 61-63.
  22. 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.
  23. 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.
  24. 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.
  25. 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.
  26. 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.
  27. 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.
  28. 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.
  29. 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.    
  30. Anders Claesson, Mark Dukes and Martina Kubitzke.
    Partition and composition matrices.
    Journal of Combinatorial Theory Series A 118 (2011), no. 5, 1624-1637.
  31. 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.
  32. 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]
  33. 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.
  34. 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).
  35. Mark Dukes and Astrid Reifegerste.
    The area above the Dyck path of a permutation.
    Advances in Applied Mathematics 45 (2010), 15-23.
  36. Anders Claesson, Mark Dukes and Einar Steingrímsson.
    Permutations sortable by n – 4 passes through a stack.
    Annals of Combinatorics 14 (2010) 45-51.
  37. François David, Mark Dukes, Thordur Jónsson, Sigurdur Örn Stefánsson.
    Random tree growth by vertex splitting.
    Journal of Statistical Mechanics (2009), no. 4, P04009.
  38. 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.
  39. 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.
  40. W. M. B. Dukes.
    Concerning the shape of a geometric lattice.
    Discrete Mathematics 308 (2008), 6632-6638.
  41. W. M. B. Dukes, T. Mansour and A. Reifegerste.
    Wilf classification of three and four letter signed patterns.
    Discrete Mathematics 308 (2008), 3125-3133.
  42. 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.
  43. W. M. B. Dukes and Toufik Mansour.
    Signed involutions avoiding 2-letter signed patterns.
    Annals of Combinatorics 11 (2007), 387-403.
  44. W. M. B. Dukes.
    Permutation statistics on involutions.
    European Journal of Combinatorics 28 (2007), 186-198.
  45. W. M. B. Dukes.
    On the number of matroids on a finite set.
    Séminaire Lotharingien de Combinatoire 51 (2004), B51g.
  46. 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.
  47. 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.
  48. W. M. B. Dukes.
    Bounds on the number of generalized partitions and some applications.
    Australasian Journal of Combinatorics 28 (2003), 257-262.
  49. W. M. B. Dukes.
    On a unimodality conjecture in matroid theory.
    Discrete Mathematics and Theoretical Computer Science 5 (2002), 181-190.
  50. 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. Michele D'Adderio, Mark Dukes, Alessandro Iraci, Alexander Lazar, Yvan Le Borgne, and Anna Vanden Wyngaerd.
    Shuffle theorems and sandpiles.
    FPSAC 2024, Bochum, Germany.
    Sém. Lothar. Comb. 91B (2019), Paper #70,12pp.
  2. Anders Claesson, Mark Dukes, Atli Fannar Franklín, and Sigurdur Örn Stefánsson.
    Counting tournament score sequences.
    EUROCOMB 2023, Prague, Czech Republic.
  3. Anders Claesson, Mark Dukes, and Beata Benyi.
    Weak ascent sequences, permutations, matrices, and posets.
    Permutation Patterns 2023, Dijon, France.
  4. Mark Dukes.
    Chip-firing on the complete split graph: Motzkin words and tiered parking functions.
    EuroComb 2021.
    In: Nešetřil, J., Perarnau, G., Rué, J., Serra, O. (eds) Extended Abstracts EuroComb 2021. Trends in Mathematics, vol 14. Birkhäuser, Cham.
  5. 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.
  6. 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.
  7. 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.
  8. 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.
  9. 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.
  10. 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.
  11. 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.
  12. 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.
  13. 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.

Books Edited

Book Chapters

  • 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.

Notes/Other

  • Jean-Christophe Aval, Mark Dukes and Yvan Le Borgne.
    A note on a bijection of Aldred, Atkinson and McCaughan.
    PDF, 2015.

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

Irish Mathematical Olympiads Compendium

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

Students/Teaching

Current PhD students:

  • Andrew Mullins
  • Anton Sohn

Former PhD students and Postdocs:

  • Amal Alofi (2018-2022, UCD). PhD Thesis:"Some results concerning tableaux and sandpile models on the complete bipartite graph".
  • Dr Thomas Selig (2015-2016, Univ. Strathclyde). Postdoctoral Researcher funded by my EPSRC grant.


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

Contact

  Postal address:

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

Service

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

Managing Editor: A managing editor for 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).