Mark Dukes

Preprints

54. Mark Dukes and Andrew Mullins.
    Knuth's big-chooser matchbox process: the case of many matchboxes.
    Submitted for publication, 2025.
53. 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

Publications

52. Mark Dukes and Anton Sohn.
    Characterizing traces of processes defined by precedence and response constraints: an order theory approach.
    Discrete Applied Mathematics 368 (2025), 112-125.   DOI
51. Michele D'Adderio, Mark Dukes, Alessandro Iraci, Alexander Lazar, Yvan Le Borgne, and Anna Vanden Wyngaerd.
    Shuffle theorems and sandpiles.
    Communications in Mathematical Physics 406 (2025), Article 83.   DOI
50. Mark Dukes and Andrew Mullins.
    A bump statistic on permutations resulting from the Robinson-Schensted correspondence.
    Annals of Combinatorics, 29 (2025), no. 2, 375-394.   DOI
49. Amal Alofi and Mark Dukes.
    A note on the lacking polynomial of the complete bipartite graph.
    Discrete Mathematics 348 (2025), no. 2, 114323.   DOI
48. Mark Dukes and Bruce Sagan.
    Difference ascent sequences.
    Advances in Applied Mathematics 159 (2024), 102736.   DOI
47. 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
46. 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
45. Beáta Bényi, Anders Claesson, and Mark Dukes.
    Weak ascent sequences and related combinatorial structures.
    European Journal of Combinatorics 108 (2023), 103633.   DOI
44. 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].
43. 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
42. 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]
41. 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
40. Mark Dukes.
    Fagan's Construction, Strange Roots, and Tchoukaillon Solitaire.
    Journal of Integer Sequences 24 (2021), Article 21.7.1.
39. Mark Dukes and Toufik Mansour.
    A maxdrop statistic for standard Young tableaux.
    Discrete Mathematics, Algorithms and Applications 14 (2022), no. 2, 2150105.   DOI
38. 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
37. 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
36. 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
35. 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
34. Mark Dukes and Thomas Selig.
    Decomposing recurrent states of the Abelian Sandpile Model.
    Séminaire Lotharingien de Combinatoire 77 (2018), B77g.
33. 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
32. 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
31. 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
30. Mark Dukes.
    Generalized ballot sequences are ascent sequences.
    Australasian Journal of Combinatorics 64 (2016), no. 1, 61-63.
29. 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.
28. 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.
27. 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.
26. 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.
25. 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.
24. 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.
23. 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.
22. 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.    
21. Anders Claesson, Mark Dukes and Martina Kubitzke.
    Partition and composition matrices.
    Journal of Combinatorial Theory Series A 118 (2011), no. 5, 1624-1637.
20. 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]
18. 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).
16. Mark Dukes and Astrid Reifegerste.
    The area above the Dyck path of a permutation.
    Advances in Applied Mathematics 45 (2010), 15-23.
15. Anders Claesson, Mark Dukes and Einar Steingrímsson.
    Permutations sortable by n – 4 passes through a stack.
    Annals of Combinatorics 14 (2010) 45-51.
14. François David, Mark Dukes, Thordur Jónsson, Sigurdur Örn Stefánsson.
    Random tree growth by vertex splitting.
    Journal of Statistical Mechanics (2009), no. 4, P04009.
13. 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.
12. 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.
11. W. M. B. Dukes.
    Concerning the shape of a geometric lattice.
    Discrete Mathematics 308 (2008), 6632-6638.
10. W. M. B. Dukes, T. Mansour and A. Reifegerste.
    Wilf classification of three and four letter signed patterns.
    Discrete Mathematics 308 (2008), 3125-3133.
9. 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.
8. W. M. B. Dukes and Toufik Mansour.
   Signed involutions avoiding 2-letter signed patterns.
   Annals of Combinatorics 11 (2007), 387-403.
7. W. M. B. Dukes.
   Permutation statistics on involutions.
   European Journal of Combinatorics 28 (2007), 186-198.
6. W. M. B. Dukes.
   On the number of matroids on a finite set.
   Séminaire Lotharingien de Combinatoire 51 (2004), B51g.
5. 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.
4. 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.
3. W. M. B. Dukes.
   Bounds on the number of generalized partitions and some applications.
   Australasian Journal of Combinatorics 28 (2003), 257-262.
2. W. M. B. Dukes.
   On a unimodality conjecture in matroid theory.
   Discrete Mathematics and Theoretical Computer Science 5 (2002), 181-190.
1. 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

13. 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.
12. Anders Claesson, Mark Dukes, Atli Fannar Franklín, and Sigurdur Örn Stefánsson.
    Counting tournament score sequences.
    EUROCOMB 2023, Prague, Czech Republic.
11. Anders Claesson, Mark Dukes, and Beata Benyi.
    Weak ascent sequences, permutations, matrices, and posets.
    Permutation Patterns 2023, Dijon, France.
10. 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.
9. 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.
8. 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.
6. 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.
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.
3. 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.
2. 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.
1. 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

• Surveys in Combinatorics 2017.
  Cambridge University Press, 2017 (with A. Claesson, S. Kitaev, D. Manlove, and K. Meeks).
  
• The Irish Mathematical Olympiads Compendium 1988 - 2024.
  E-book, 2024.

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.

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

1. Michael H. Albert, Otago, New Zealand.
2. Amal Alofi, UCD, Ireland.
3. Mike D. Atkinson, Otago, New Zealand.
4. Jean-Christophe Aval, Bordeaux, France.
5. Beáta Bényi, Baja, Hungary.
6. Mireille Bousquet-Mélou, Bordeaux, France.
7. Mathilde Bouvel, Nancy, France.
8. Anthony A. Casey, Copenhagen, Denmark.
9. Fan Chung, San Diego, USA.
10. Anders Claesson, Reykjavík, Iceland.
11. Michele D'Adderio, Pisa, Italy.
12. François David, Paris, France.
13. Eva Y. P. Deng, Dalian, China.
14. Tony Dorlas, DIAS, Ireland.
15. Ken Duffy, Northeastern University, USA.
16. Mark Flanagan, UCD, Ireland.
17. Einan Gardi, Edinburgh, UK.
18. Ronald Graham, San Diego, USA.
19. Angela Hicks, Lehigh University, USA.
20. Alessandro Iraci, Pisa, Italy.
21. Vit Jelínek, Prague, Czech Republic.
22. Þórður Jónsson, Reykjavík, Iceland.
23. Sergey Kitaev, Strathclyde, UK.
24. Martina Kubitzke, Osnabruck, Germany.
25. Alexander Lazar, Brussels, Belgium.
26. Yvan Le Borgne, Bordeaux, France.
27. Toufik Mansour, Haifa, Israel.
28. Heather McAslan, Sussex, UK.
29. Andrew Mullins, UCD, Ireland.
30. Peter R.W. McNamara, Bucknell, USA.
31. Robert Parviainen, USA.
32. Astrid Reifegerste, Magdeburg, Germany.
33. Jeffrey Remmel, San Diego, USA.
34. Bruce Sagan, Michigan, USA.
35. Darren J. Scott, Durham, UK.
36. Anton Sohn, UCD, Ireland.
37. Thomas Selig, Suzhou, China.
38. Jason P. Smith, Nottingham, UK.
39. Sigurður Örn Stefánsson, Reykjavík, Iceland.
40. Einar Steingrímsson, Strathclyde, UK.
41. Vincent Vajnovszki, Dijon, France.
42. Anna Vanden Wyngaerd, Brussels, Belgium.
43. Chris D. White, QMUL, UK.
44. Susan Yi Jun Wu, Tianjin, China.