# Colin Defant

I am a 4th-year Ph.D. student in the Princeton Mathematics department. I graduated with a B.S. in mathematics from the University of Florida in 2017. During the summers of 2017, 2018, 2019, and 2020, I worked as an advisor at the University of Minnesota Duluth Mathematics REU Program

Contact

Papers

All of my papers are available on arXiv.org.

47. C. Defant and K. Zheng, Stack-sorting with consecutive-pattern-avoiding stacks. To appear in Adv. Appl. Math., (2021).

46. C. Defant, Fertility monotonicity and average complexity of the stack-sorting map. European J. Combin., 93 (2021), 17 pages.

https://arxiv.org/abs/2003.05935

45. C. Defant, Highly sorted permutations and Bell numbers. Enumer. Combin. Appl.1 (2021), 7 pages.

https://arxiv.org/abs/2012.03869

44. C. Defant, Enumeration of stack-sorting preimages via a decomposition lemma. To appear in Discrete Math. Theor. Comput. Sci., (2021).

https://arxiv.org/abs/1904.02829

43. A. Berger and C. Defant, On anti-powers in aperiodic recurrent words. Adv. Appl. Math., 121 (2020), 11 pages.

https://arxiv.org/abs/1902.01291

42. C. Defant and J. Propp, Quantifying noninvertibility in discrete dynamical systems. Electron. J. Combin., 2(2020), 22 pages.

https://arxiv.org/abs/2002.07144

41. C. Defant, Polyurethane toggles. Electron. J. Combin., 27 (2020), 16 pages.

https://arxiv.org/abs/1904.06283

40. A. Burcroff and C. Defant, Pattern-avoiding permutation powers. Discrete Math., 343 (2020), 11 pages.

https://arxiv.org/abs/1907.09451

39. C. Defant, N. Kravitz, and A. Sah, Supertrees. Electron. J. Combin., 27 (2020), 26 pages.

https://arxiv.org/abs/1908.03197

38. C. Defant, M. Engen, and J. A. Miller, Stack-sorting, set partitions, and Lassalle's sequence. J. Combin. Theory Ser. A, 175 (2020), 24 pages.

https://arxiv.org/abs/1809.01340

37. C. Defant, Stack-sorting preimages of permutation classes. Sém. Lothar. Combin., 82B (2020), 41 pages.

https://arxiv.org/abs/1809.03123

36. C. Defant, Catalan intervals and uniquely sorted permutations. J. Combin. Theory Ser. A, 174 (2020), 43 pages.

https://arxiv.org/abs/1904.02627

35. C. Defant and N. Kravitz, Stack-sorting for words. Australas. J. Combin., 77 (2020), 51-68

https://arxiv.org/abs/1809.09158

34. C. Defant, Counting 3-Stack-Sortable Permutations. J. Combin. Theory Ser. A, 172 (2020), 26 pages.

https://arxiv.org/abs/1903.09138

33. C. Defant, Fertility, strong fertility, and postorder Wilf equivalence. Australas. J. Combin., 76 (2020), 149-182.

https://arxiv.org/abs/1904.03115

32. N. Alon and C. Defant, Isoperimetry, stability, and irredundance in direct products. Discrete Math., 343 (2020), 19 pages.

https://arxiv.org/abs/1904.02595

31. C. Defant, Descents in t-sorted permutations. J. Comb., 11 (2020), 527-548.

https://arxiv.org/abs/1904.02613

30. C. Defant, Fertility numbers. J. Comb., 11 (2020), 511-526.

https://arxiv.org/abs/1809.04421

29. S. Cho, J. Cummings, C. Defant, and C. Sonneborn, An anti-Ramsey problem concerning complete bipartite graphs. AKCE Int. J. Graphs. Comb. (2019), In press.

https://arxiv.org/abs/1412.3066

28. C. Defant, Proofs of conjectures about pattern-avoiding linear extensions. Discrete Math. Theor. Comput. Sci., 21 (2019), #16, 9 pages.

https://arxiv.org/abs/1905.02309

27. C. Defant, Enumerating cliques in direct product graphs. J. Comb., 11 (2019), 351-358.

https://arxiv.org/abs/1707.05406

26. A. Adeniran, S. Butler, C. Defant, Y. Gao, P. E. Harris, C. Hettle, Q. Liang, H. Nam, and A. Volk, On the genus of a quotient of a numerical semigroup. Semigroup Forum, 98 (2019), 690-700.

https://arxiv.org/abs/1809.09360

25. C. Defant and S. Iyer, Domination and upper domination of direct product graphs. Discrete Math., 341 (2018), 2742-2752.

https://arxiv.org/abs/1708.01305

24. C. Defant, Flexible toggles and symmetric invertible asynchronous elementary cellular automata. Discrete Math., 341 (2018), 2367-2379.

https://arxiv.org/abs/1511.06966

23. C. Defant, Connected components of complex divisor functions. J. Number Theory, 190 (2018), 56-71.

https://arxiv.org/abs/1711.04244

22. C. Defant, Ranges of unitary divisor functions. Integers, 18 (2018), #A15, 10 pages.

http://math.colgate.edu/~integers/vol18.html

21.  K. Alladi and C. Defant, Revisiting the Riemann zeta function at positive even integers. Int. J. Number Theory14 (2018), 1849-1856.

https://arxiv.org/abs/1707.04379

20. C. Defant, Poset pattern-avoidance problems posed by Yakoubov. J. Comb., 9 (2017), 233-257.

https://arxiv.org/abs/1608.03951

19. C. Defant, Anti-power prefixes of the Thue-Morse word. Electron. J. Combin., 24 (2017), Paper #P1.32, 17 pages.

http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i1p32

18. C. Defant, Preimages under the stack-sorting algorithm. Graphs Combin., 33 (2017), 103-122.

17. C. Defant, Postorder preimages. Discrete Math. Theor. Comput. Sci., 19 (2017), #3, 15 pages.

https://dmtcs.episciences.org/3116

16. C. Defant, On Schemmel nontotient numbers. J. Comb. Number Theory, 8 (2016), 233-239.

https://www.novapublishers.com/catalog/product_info.php?products_id=61147

15. C. Defant, Upper bounds for Stern's diatomic sequence and related sequences. Electron. J. Combin., 23 (2016) , Paper #P4.8, 47 pages.

http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p8

14. C. Defant, Unitary Cayley graphs of Dedekind domain quotients. AKCE Int. J. Graphs Comb., 13 (2016), 65-75.

http://www.sciencedirect.com/science/article/pii/S0972860016300275

13. C. Defant, Complex divisor functions. Analysis, Geometry and Number Theory, 1 (2016), 22-48.

https://arxiv.org/abs/1502.03417

12. C. Defant, Binary codes and period-2 orbits of sequential dynamical systems. Discrete Math. Theor. Comput. Sci., 19 (2015), #10, 12 pages.

https://arxiv.org/abs/1509.03907

11. C. Defant, On 2-powerfully perfect numbers in three quadratic rings. Notes on Number Theory and Discrete Mathematics, 23 (2017), 1-11.

http://nntdm.net/volume-23-2017/number-2/

10. C. Defant, A note about iterated arithmetic functions. Miskolc Math. Notes16 (2015), 763-767.

http://mat76.mat.uni-miskolc.hu/mnotes/article/1448

9. C. Defant, On arithmetic functions related to iterates of the Schemmel totient functions. J. Integer Seq. 18 (2015), Article 15.2.1.

https://cs.uwaterloo.ca/journals/JIS/VOL18/Defant/defant5.pdf

8. C. Defant, On the density of ranges of generalized divisor functions with restricted domains. Unif. Distrib. Theory, 10 (2015), 19-33.

https://math.boku.ac.at/udt/vol10/no1/02Defant.pdf

7. C. Defant, Unitary multiperfect numbers in certain quadratic rings. Int. J. Math. Comput. Sci., 10 (2015), 127-138.

http://ijmcs.future-in-tech.net/10.2/R-DefantUnitary.pdf

6. C. Defant, On the density of ranges of generalized divisor functions. Notes on Number Theory and Discrete Mathematics, 21 (2015), 80-87.

http://nntdm.net/volume-21-2015/number-3/80-87/

5. C. Defant, On ranges of variants of the divisor functions that are dense. Int. J. Number Theory, 11 (2015), 1905-1912.

https://arxiv.org/abs/1507.01128

4. C. Defant, On sparsely Schemmel totient numbers. Integers, 15 (2015), #A18, 13 pages.

http://www.integers-ejcnt.org/vol15.html

3. C. Defant, An arithmetic function arising from the Dedekind $\psi$ function. Math. J. Okayama Univ., 59 (2017), 81-92.

http://www.math.okayama-u.ac.jp/mjou/mjou59/_06_Defant.pdf

2. C. Defant, Multiperfect numbers in certain quadratic rings. Int. J. Math Comput. Sci., 9 (2014), 49-61.

http://ijmcs.future-in-tech.net/9.2/R-MultiperfectDefant.pdf

1. C. Defant, An extension of the abundancy index to certain quadratic rings. Int. J. Math Comput. Sci., 9 (2014), 63-82.

Submitted Papers

8. C. Defant. Fertilitopes. Submitted.

https://arxiv.org/abs/2102.11836

7. C. Defant and S. Hopkins, Symmetry of Narayana numbers and rowvacuation of root posets. Submitted.

https://arxiv.org/abs/2101.02329

6. C. Defant, A. Elvey Price, and A. J. Guttmann, Asymptotics of 3-stack-sortable permutations. Submitted.

https://arxiv.org/abs/2009.10439

5. N. Alon, C. Defant, and N. Kravitz, Typical and extremal aspects of friends-and-strangers graphs. Submitted.

https://arxiv.org/abs/2009.07840

4. C. Defant and N. Kravitz, Friends and strangers walking on graphs. Submitted.

https://arxiv.org/abs/2009.0504

3. C. Defant and Noah Kravitz, Promotion sorting. Submitted.

https://arxiv.org/abs/2005.07187

2. C. Defant, Troupes, cumulants, and stack-sorting. Submitted.

1. C. Defant, Motzkin intervals and valid hook configurations. Submitted.

https://arxiv.org/abs/1904.10451

My interests are primarily in enumerative and algebraic combinatorics. A large portion of my work in combinatorics concerns a specific function called the stack-sorting map. In order to understand this function, I've developed a theory of new combinatorial objects that I call valid hook configurations. I've also explored these new objects in their own right. I especially enjoy finding other parts of combinatorics that are connected with stack-sorting and valid hook configurations. For example, I have found that valid hook configurations show up in a very simple yet surprising combinatorial formula in free probability theory; this has allowed me to use free probability in order to understand the stack-sorting map and to use a generalization of stack-sorting to derive new combinatorial facts about free and classical cumulants.

Some Talks

Songs

These are some simple short songs that I made in 2014 for a computer game that a team of undergraduates at Cornell made.

Song 1 - Unknown Artist
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Song 2 - Colin Defant
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Song 3 - Colin Defant
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Song 4 - Colin Defant
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Song 5 - Colin Defant
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