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Research

My research interests are mostly in enumerative and algebraic combinatorics. Here are a few keywords from my papers: permutations, patterns, lattice paths, bijections, triangulations, generating functions. I have also done some work on algorithms, number theory, and applications of combinatorics to computational biology and dynamical systems.

Recent preprints

• Counting pattern-avoiding permutations by big descents (with Johnny Rivera Jr. and Y. Zhuang), preprint, arXiv:2408.15111.


• Penney's game for permutations (with Y. Lin), preprint, arXiv:2404.06585.


• Canon permutations and generalized descents of standard Young tableaux, preprint, arXiv:2403.14914.


• Statistics on clusters and r-Stirling permutations (with J. Troyka and Y. Zhuang), preprint, arXiv:2209.12436, to appear in J. Comb.

Journal publications

1. Triangular partitions: enumeration, structure, and generation (with A. Galván), Adv. in Appl. Math., to appear. arXiv:2312.16353. Code available here.


2. On individual leaf depths of trees, Discrete Appl. Math. 353 (2024), 151-180. arXiv:2302.05252.


3. Descents on nonnesting multipermutations, European J. Combin. (2023). arXiv:2204.00165.


4. Rowmotion on 321-avoiding permutations (with B. Adenbaum), Electron. J. Combin. 30(3) (2023), #P3.5, 26 pp.. arXiv:2212.11347.


5. Rowmotion on fences (with M. Plante, T. Roby and B. Sagan), Algebr. Comb. 6 (2023), no. 1, 17-36. arXiv:2108.12443.


6. Partitions with constrained ranks and lattice paths (with S. Corteel and C. Savage), ECA 3:3 (2023) Article #S2R18. arXiv:2211.08933.


7. On a dart game of Niedermaier, Adv. in Appl. Math. 145 (2023) 102483. arXiv:2209.03775.


8. Counting lattice paths by crossings and major index II: tracking descents via two-rowed arrays, Sém. Lothar. Combin. 87B (2023), Art. #7, 36 pp. arXiv:2112.05696.


9. Partial rank symmetry of distributive lattices for fences (with B. Sagan), Ann. Comb. 27 (2023), no. 2, 433-454. arXiv:2201.03044.


10. Counting lattice paths by crossings and major index I: the corner-flipping bijections, Combinatorial Theory 2 (2) (2022), #14. arXiv:2106.09878.


11. Cyclic pattern containment and avoidance (with R. Domagalski, J. Liang, Q. Minnich, B. Sagan, J. Schmidt and A. Sietsema), Adv. in Appl. Math. 135 (2022), 102320. [Download]


12. The degree of asymmetry of sequences (with E. Deutsch), ECA 2:1 (2022) Article S2R7. arXiv:2107.06188.


13. The degree of symmetry of lattice paths, Ann. Comb. 25 (2021), 877-911. arXiv:2002.12874.


14. Enumerating symmetric peaks in non-decreasing Dyck paths (with R. Florez and J.L. Ramirez), Ars Math. Contemp. 21 (2021) #P2.04.


15. Bijections for restricted inversion sequences and permutations with fixed points, Australas. J. Combin. 80 (2021), 106-115. arXiv:2006.13842.


16. Descents on quasi-Stirling permutations, J. Combin. Theory Ser. A, 180 (2021) 105429. arXiv:2002.00985.


17. Symmetric peaks and symmetric valleys in Dyck paths, Discrete Math. 344 (2021), 112364. arXiv:2008.05669.


18. Pervasive chromosomal instability and karyotype order during tumour evolution (with T. Watkins, E. Lim, M. Petkovic, ..., R. Schwarz, N. McGranahan, C. Swanton), Nature 587, 126-132 (2020).


19. Wilf equivalences between vincular patterns in inversion sequences (with J.S. Auli), Appl. Math. Comput. 388 (2020) 125514. arXiv:2003.11533.


20. A simple bijective proof of a familiar derangement recurrence, Fibonacci Quart. 59 (2021), 150--151. arXiv:2005.11312.


21. On cyclic Schur-positive sets of permutations (with J. Bloom and Y. Roichman), Electron. J. Combin. 27(2) (2020), #P2.6. arXiv:1908.07920.


22. Consecutive patterns in inversion sequences II: avoiding patterns of relations (with J.S. Auli), J. Integer Seq. 22 (2019), Art. 19.7.5. arXiv:1906.07365.


23. Consecutive patterns in inversion sequences (with J.S. Auli), Discrete Math. Theor. Comput. Sci. 21:2 (2019), #6. arXiv:1904.02694.


24. Characterizations and enumerations of patterns of signed shifts (with K. Moore), Discrete Appl. Math. 277 (2020), 92-114. arXiv:1711.05213.


25. Exact and asymptotic enumeration of cyclic permutations according to descent set (with J. Troyka), J. Combin. Theory Ser. A 165 (2019), 360-391. arXiv:1710.05103.


26. Cyclic descents for near-hook and two-row shapes (with R. Adin and Y. Roichman), European J. Combin. 79 (2019), 152-178. arXiv:1801.00044.


27. A Markov chain for numerical chromosomal instability in clonally expanding populations (with A. Laughney and S. Bakhoum), PLoS Comput. Biol. 14(9): e1006447. [Here is the submitted version that includes supplementary figures and end-of-proof marks.]


28. Wilf equivalence relations for consecutive patterns (with T. Dwyer), Adv. in Appl. Math. 99 (2018), 134-157. arXiv:1801.08262.


29. A bijection between bargraphs and Dyck paths (with E. Deutsch), Discrete Appl. Math. 251 (2018), 340-344. arXiv:1705.05984.


30. Continued fractions for permutation statistics, Discrete Math. Theor. Comput. Sci. 19 (2018), #11. arXiv:1703.08742.


31. The probability of avoiding consecutive patterns in the Mallows distribution (with H. Crane and S. DeSalvo), Random Structures Algorithms 53 (2018), 417-447. arXiv:1609.01370.


32. On rotated Schur-positive sets (with Y. Roichman), J. Combin. Theory Ser. A 152 (2017), 121-137. arXiv:1609.07335.


33. Character formulas and descents for the hyperoctahedral group (with R. Adin, C. Athanasiadis and Y. Roichman), Adv. in Appl. Math. 87 (2017), 128-169. arXiv:1504.01283.


34. The structure of the consecutive pattern poset (with P. McNamara), Int. Math. Res. Not. IMRN 2018, no.7, 2099-2134. arXiv:1508.05963.


35. Statistics on bargraphs viewed as cornerless Motzkin paths (with E. Deutsch), Discrete Appl. Math. 221 (2017), 54-66. arXiv:1609.00088.


36. Two descent statistics over 321-avoiding centrosymmetric involutions (with M. Barnabei, F. Bonetti and M. Silimbani), Electron. J. Combin. 23 (2016), #P1.35. arXiv:1508.07515.


37. Schur-positive sets of permutations via products of grid classes (with Y. Roichman), J. Algebraic Combin. 45 (2017), 363-405. arXiv:1509.00045.


38. Symmetries of statistics on lattice paths between two boundaries (with M. Rubey), Adv. Math. 287 (2016), 347-388. arXiv:1305.2206.


39. Dynamics of tumor heterogeneity derived from clonal karyotypic evolution (with A. Laughney, S. Bakhoum and G. Genovese), Cell Rep. 12 (2015), 809-820.


40. Bijections for pairs of non-crossing lattice paths and walks in the plane, European J. Combin. 49 (2015), 25-41. arXiv:1406.1709.


41. Signed arc permutations (with Y. Roichman), J. Comb. 6 (2015), 205-234. arXiv:1402.0211.


42. A generating tree approach to k-nonnesting partitions and permutations (with M. Mishna, S. Burrill and L. Yen), Ann. Comb. 20 (2016), 453-485. arXiv:1108.5615.


43. Descent sets on 321-avoiding involutions and hook decompositions of partitions (with M. Barnabei, F. Bonetti , and M. Silimbani), J. Combin. Theory Ser. A 128 (2014), 132-148. arXiv:1401.3011.


44. Cyclic permutations realized by signed shifts (with K. Archer), J. Comb. 5 (2014), 1-30. arXiv:1304.7790.


45. Arc permutations (with Y. Roichman), J. Algebraic Combin. 39 (2014), 301-334. arXiv:1210.6056.


46. Inversion polynomials for 321-avoiding permutations (with S.-E. Cheng, A. Kasraoui and B. Sagan), Discrete Math. 313 (2013), 2552-2565. arXiv:1112.6014 (addendum: arXiv:1305.3845).


47. Total occurrence statistics on restricted permutations (with A. Burstein), Pure Math. Appl. (PU.M.A.) 24 (2013), 103-123. arXiv:1305.3177.


48. The most and the least avoided consecutive patterns, Proc. Lond. Math. Soc. 106 (2013), 957-979. arXiv:1203.1636.


49. Pattern avoidance in matchings and partitions (with J. Bloom), Electron. J. Combin. 20 (2013), #P5. arXiv:1211.3442.


50. Clusters, generating functions and asymptotics for consecutive patterns in permutations (with M. Noy), Adv. in Appl. Math. 49 (2012), 351-374. arXiv:1210.6061.


51. Fixed points and excedances in restricted permutations, Electron. J. Combin. 18 (2012), #P29.


52. Restricted simsun permutations (with E. Deutsch), Ann. Comb. 16 (2012), 253-269. arXiv:0912.1361.


53. Permutations and β-shifts, J. Combin. Theory Ser. A 118 (2011), 2474-2497. arXiv:1008.4167.


54. Descent sets of cyclic permutations, Adv. in Appl. Math. 47 (2011), 688-709. arXiv:0906.2795.


55. On basic forbidden patterns of functions (with Y. Liu), Discrete Appl. Math 159 (2011), 1207-1216. arXiv:0909.2277.


56. Cycle-up-down permutations (with E. Deutsch), Australas. J. Combin. 50 (2011), 187-199. arXiv:0909.5199.


57. The X-class and almost-increasing permutations, Ann. Comb. 15 (2011), 51-68. arXiv:0710.5168.


58. Improved bounds on the number of numerical semigroups of a given genus, J. Pure Appl. Algebra 214 (2010), 1862-1873. arXiv:0905.0489.


59. The largest and the smallest fixed points of permutations (with E. Deutsch), European J. Combin. 31 (2010), 1404-1409. arXiv:0904.2792.


60. The number of permutations realized by a shift, SIAM J. Discrete Math. 23 (2009), 765-786. arXiv:0909.2274.


61. Forbidden patterns and shift systems (with J.M. Amigó and M.B. Kennel), J. Combin. Theory Ser. A 115 (2008), 485-504. arXiv:0707.4628.


62. Generating trees for permutations avoiding generalized patterns, Ann. Comb. 11 (2007), 435-458. arXiv:0707.4633.


63. A bijection between 2-triangulations and pairs of non-crossing Dyck paths, J. Combin. Theory Ser. A 114/8 (2007), 1481-1503. arXiv:math.CO/0610235.


64. Bounds on the number of inference functions of a graphical model (with K. Woods), Statist. Sinica 17 (2007), 1395-1415. arXiv:math.CO/0610233.


65. The probability of choosing primitive sets (with K. Woods), J. Number Theory 125 (2007), 39-49. arXiv:math.NT/0607390.


66. Restricted Dumont permutations, Dyck paths, and noncrossing partitions (with A. Burstein and T. Mansour), Discrete Math. 306 (2006), 2851-2869. arXiv:math/0610234.


67. Combinatòria i biologia: funcions d'inferència i alineació de seqüències, Butl. Soc. Catalana Mat. 21 (2006), n. 1, 39-52. [Download]


68. Asymptotic enumeration of permutations avoiding generalized patterns, Adv. in Appl. Math. 36 (2006), 138-155. arXiv:math/0505254.


69. Old and young leaves on plane trees (with W.C. Chen and E. Deutsch), European J. Combin. 27 (2006), 414-427. arXiv:math/0410127.


70. Restricted Motzkin permutations, Motzkin paths, continued fractions, and Chebyshev polynomials (with T. Mansour), Discrete Math. 305 (2005), 170-189. arXiv:math/0610237.


71. Multiple pattern-avoidance with respect to fixed points and excedances, Electron. J. Combin. 11 (2004), #R51. arXiv:math/0311211.


72. Bijections for refined restricted permutations (with I. Pak), J. Combin. Theory Ser. A 105 (2004), 207-219. arXiv:math/0212328.


73. A simple and unusual bijection for Dyck paths and its consequences (with E. Deutsch), Ann. Comb. 7 (2003), 281-297. arXiv:math/0306125.


74. Consecutive patterns in permutations (with M. Noy), Adv. in Appl. Math. 30 (2003), 110-125. [Download]

Book Chapters

1. A survey of consecutive patterns in permutations, chapter of the book Recent Trends in Combinatorics (IMA Volume in Mathematics and its Applications) 601-618, Springer, 2016. arXiv:1504.07265.


2. Inference functions, chapter of the book Algebraic Statistics for Computational Biology, edited by L. Pachter and B. Sturmfels, Cambridge University Press, 2005.


3. Bounds for optimal sequence alignment (with F. Lam), chapter of the book Algebraic Statistics for Computational Biology, edited by L. Pachter and B. Sturmfels, Cambridge University Press, 2005.

Refereed Conference Proceedings

1. Combinatorial Properties of Triangular Partitions (with A. Galván), Proceedings of FPSAC 2024, Sém. Lothar. Combin. 91B.68 (2024), 12 pp.


2. The Distribution of Descents on Nonnesting Permutations, Proceedings of FPSAC 2023, Sém. Lothar. Combin. 89B.2 (2023), 12 pp.


3. Rowmotion on 321-Avoiding Permutations, Proceedings of FPSAC 2023, Sém. Lothar. Combin. 89B.16 (2023), 12 pp.


4. Walks in Simplices, Cylindric Tableaux, and Asymmetric Exclusion Processes, Proceedings of FPSAC 2022, Sém. Lothar. Combin. 86B.8 (2022), 12 pp.


5. Rowmotion on fences (with M. Plante, T. Roby and B. Sagan), Proceedings of FPSAC 2022, Sém. Lothar. Combin. 86B.40 (2022), 12 pp.


6. Enumerating Descents on Quasi-Stirling Permutations and Plane Trees, Extended Abstracts EuroComb 2021, Trends in Mathematics, vol 14. Birkhäuser, Cham., pp 32-37.


7. Measuring Symmetry in Lattice Paths and Partitions, Proceedings of FPSAC 2020, Sém. Lothar. Combin. 84B (2020), Article #26, 12 pp.


8. Inversion Sequences Avoiding Consecutive Patterns (with J.S. Auli), Proceedings of FPSAC 2020, Sém. Lothar. Combin. 84B (2020), Article #27, 12 pp.


9. The number of cycles with a given descent set (with J. Troyka), Proceedings of FPSAC 2018, Sém. Lothar. Combin. 80B (2018), Article #8, 12 pp.


10. Cyclic descent extensions and distributions (with R. Adin, V. Reiner and Y. Roichman), Proceedings of the GASCom 2018 Workshop, 32--42.


11. A necessary condition for c-Wilf equivalence (with T. Dwyer), Proceedings of FPSAC 2017, Sém. Lothar. Combin. 78B (2017), Article #69, 12 pp.


12. Patterns of negative shifts and signed shifts (with K. Archer and K. Moore), Proceedings of FPSAC 2017, Sém. Lothar. Combin. 78B (2017), Article #49, 12 pp.


13. Schur-positivity via products of grid classes (with Y. Roichman), Proceedings of FPSAC 2016, Discrete Math. Theor. Comput. Sci. proc. BC (2016), 443--454.


14. On intervals of the consecutive pattern poset (with P. McNamara), Proceedings of FPSAC 2016, Discrete Math. Theor. Comput. Sci. proc. BC (2016), 431--442.


15. The frequency of pattern occurrence in random walks (with M. Martinez), Proceedings of FPSAC 2015, Discrete Math. Theor. Comput. Sci. proc. FPSAC'15 (2015), 217-228.


16. Periodic patterns of signed shifts (with K. Archer), Proceedings of FPSAC 2013, Discrete Math. Theor. Comput. Sci. proc. AS (2013), 873-884.


17. Patterns in matchings and rook placements (with J. Bloom), Proceedings of FPSAC 2013, Discrete Math. Theor. Comput. Sci. proc. AS (2013), 909-920.


18. Bijections for lattice paths between two boundaries (with M. Rubey), Proceedings of FPSAC 2012, Discrete Math. Theor. Comput. Sci. proc. AR (2012), 827-838.


19. Arc Permutations (extended abstract) (with Y. Roichman), Proceedings of FPSAC 2012, Discrete Math. Theor. Comput. Sci. proc. AR (2012), 259-270.


20. Consecutive patterns in permutations: clusters and generating functions (with M. Noy), Proceedings of FPSAC 2012, Discrete Math. Theor. Comput. Sci. proc. AR (2012), 247-258.


21. Generating trees for partitions and permutations with no k-nestings (with M. Mishna, S. Burrill and L. Yen), Proceedings of FPSAC 2012, Discrete Math. Theor. Comput. Sci. proc. AR (2012), 409-420.


22. Allowed patterns of beta-shifts, Proceedings of FPSAC 2011, Discrete Math. Theor. Comput. Sci. proc. AO (2011), 293-304.


23. Sorting by Placement and Shift (with P. Winkler), Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2009. arXiv:0809.2957.


24. Permutations realized by shifts, Proceedings of FPSAC 2009, Discrete Math. Theor. Comput. Sci. proc. AK (2009), 361-372.


25. Pattern avoidance in dynamical systems (with J.M. Amigó and M.B. Kennel), Proceedings of FPSAC 2008, Discrete Math. Theor. Comput. Sci. proc. AJ (2008), 71-82.


26. A bijection between 2-triangulations and pairs of non-crossing Dyck paths, Proceedings of FPSAC 2007.


27. Bounds on the number of inference functions of a graphical model (with K. Woods), Proceedings of FPSAC 2006.


28. Old and young leaves on plane and binary trees (with E. Deutsch), Proceedings of FPSAC 2005.


29. Restricted Motzkin permutations (with T. Mansour), Proceedings of FPSAC 2004.


30. Fixed points and excedances in restricted permutations, Proceedings of FPSAC 2003. arxiv:math.CO/0212221.


31. Forbidden subwords in permutations (with M. Noy), Proceedings of FPSAC 2001.

Slides and videos of selected presentations

• Penney's game for permutations, [AMS Fall Eastern Sectional Meeting, Albany, NY, October 2024].
• Combinatorial properties of triangular partitions, [Michigan Tech University, Seminar in Partition Theory, q-Series and Related Topics, January 2024]. Video recording.
• Descents on noncrossing and nonnesting permutations, [Queen Mary, University of London, February 2023]. Here is the video recording of a similar talk at the NY Combinatorics Seminar.
• Counting lattice paths by the number of crossings, major index, and descents, [Combinatorial and Algebraic Enumeration, Waterloo, Canada, May 2022].
• Partial rank symmetry of distributive lattices for fences, [University of Minnesota Combinatorics Seminar, April 2022].
• Rowmotion on 321-avoiding permutations, [BIRS Dynamical Algebraic Combinatorics workshop, November 2021]. Here is the video recording of the talk.
• Counting lattice paths by the number of crossings and major index, [Lattice Paths, Combinatorics and Interactions, June 2021].
• Bijections for derangements and pattern-avoiding inversion sequences, [Permutation Patterns 2021 Virtual Workshop, June 2021].
• Descents on quasi-Stirling permutations, [UMass Amherst Discrete Math Seminar, March 2020], [UVM Combinatorics Seminar, September 2021].
• The probability of avoiding consecutive patterns in the Mallows distribution, [Joint Mathematics Meetings, Denver, CO, January 2020].
• Consecutive patterns in permutations and inversion sequences, [Virtual Combinatorics Colloquium, December 2019]. Here is the video of my virtual talk. Check out the VCC webpage.
• Consecutive patterns in inversion sequences, [AMS Fall Southeastern Sectional Meeting, Gainesville, FL, November 2019].
• A Markov-chain model of chromosomal instability, [Dagstuhl Seminar on Genomics, Pattern Avoidance, and Statistical Mechanics, November 2018].
• Schur-positive grid classes and cyclic descents of SYT, [Oberwolfach Research Institute for Mathematics (MFO), Workshop on Enumerative Combinatorics, May 2018].
• Cyclic descents of standard Young tableaux, [Workshop on Algorithmic and Enumerative Combinatorics, ESI, Vienna, October 2017].
• The structure of the consecutive pattern poset, [MIT Combinatorics Seminar, March 2017].
• Schur-positive grid classes, [Keynote talk, Graduate Student Combinatorics Conference, Clemson University, April 2016].
• A Markov chain to probe chromosomal instability in tumor evolution and drug resistance [Sloan-Kettering Institute, Computational Biology Program, August 2015].
• Some unsolved problems in mathematics and computation [Johns Hopkins Center for Talented Youth, October 2014].
• Two EC tidbits [Richard Stanley's 70th Birthday Conference, June 2014].
• Consecutive patterns in permutations [Plenary talk at Permutation Patterns 2013; Video of talk at IMA, November 2014].
• Bijections for lattice paths between two boundaries [SIAM DM'12, CanaDAM 2013].
• Consecutive patterns in permutations: clusters, generating functions, and asymptotics [Slides February 2012].
• Allowed patterns of β-shifts [FPSAC 2011 Slides]. Permutations and β-shifts [Slides October 2010].
• The number of numerical semigroups of a given genus [Slides June 2010]/
• Descent sets of cyclic permutations [Slides Oct 2009, Slides April 2010].
• Sorting by placement and shift [Slides Jan 2009]. A greedy sorting algorithm [Slides February 2010].
• Discrete Mathematics Day, Middlebury College, September 2007, Forbidden patterns in telling random from deterministic time series. [Slides]
• MIT Combinatorics Seminar, September 2006, A bijection between 2-triangulations and pairs of non-crossing Dyck paths. [Slides]
• International Conference on Permutation Patterns, PP'06, Reykjavik, Iceland, June 2006, Generating trees for permutations avoiding generalized patterns. [Slides]
• Institut Mittag-Leffler Seminar, Djursholm, Sweden, May 2005, Combinatorics from biology: inference functions and sequence alignment. [Slides]
• International Conference on Permutation Patterns, PP'05, Gainesville, Florida, March 2005, Asymptotic enumeration of permutations avoiding generalized patterns. [Slides]
• MIT Combinatorics Seminar, February 2005. Inference functions and sequence alignment. [Slides]
• Stanford University Representation Theory Seminar, December 2004, Refined enumeration of pattern-avoiding permutations. [Slides]
• International Conference on Permutation Patterns, PP'04, Nanaimo, Canada, July 2004, Simultaneous pattern-avoidance with respect to fixed points and excedances. [Slides]
• Thesis defense, MIT, April 2004, Statistics on Pattern-avoiding Permutations . [Slides]
• UC Berkeley Combinatorics Seminar, February 2004, Refined enumeration of pattern-avoiding permutations. [Slides]
• MIT Combinatorics Seminar, November 2003. Pattern-avoiding permutations: old results and new developments. [Slides]
• FPSAC 2003, Vadstena, Sweden, June 2003. Fixed points and excedances in restricted permutations. [Slides]
• MIT Simple Person's Applied Math Seminar, November 2002. What pattern avoidance shouldn't have avoided. [Slides]

Unpublished material

• Consecutive patterns in circular permutations (with B. Sagan), arXiv:2107.04717. (These results have been merged into paper 64 above.)
• An involution on lattice paths between two boundaries. [Download] (A version of this note appears as part of paper 37 above.)
• Statistics on Pattern-avoiding Permutations, Ph.D. thesis, MIT, 2004. [Download pdf, ps]