Lecture Notes

  1. S. Chern, 32 Lectures on the Theory of Numbers. Version 1.01 (May 2023)

Preprints

  1. G. E. Andrews and S. Chern, Linked partition ideals and a family of quadruple summations, submitted. Available at arXiv:2301.11137. download
  2. S. Chern, S. Fu, and Z. Lin, Burstein's permutation conjecture, Hong and Li's inversion sequence conjecture, and restricted Eulerian distributions, submitted. Available at arXiv:2209.12137. download
  3. S. Chern, Z. Li, D. Stanton, T. Xue, and A. J. Yee, The Ariki–Koike algebras and Rogers–Ramanujan type partitions, submitted. Available at arXiv:2209.07713. download
  4. S. Chern, $q$-Log-concavity and $q$-unimodality of Gaussian polynomials and a problem of Andrews and Newman, submitted. download
  5. S. Chern and D. Tang, General coefficient-vanishing results associated with theta series, submitted. Available at arXiv:2208.09882. download
  6. S. Chern, Linked partition ideals and Euclidean billiard partitions, submitted. Available at arXiv:2012.14485. download

Accepted Papers

  1. J. M. Campbell and S. Chern, Nearly self-conjugate integer partitions, New Zealand J. Math., accepted. download
  2. S. Chern, Asymmetric Rogers–Ramanujan type identities. I. The Andrews–Uncu Conjecture, Proc. Amer. Math. Soc., accepted. doi: 10.1090/proc/16332.
  3. S. Chern, On 0012-avoiding inversion sequences and a conjecture of Lin and Ma, Quaest. Math., accepted. doi: 10.2989/16073606.2022.2039973.
  4. S. Chern, Weighted partition rank and crank moments. I. Andrews–Beck type congruences, Proceedings of the Conference in Honor of Bruce Berndt, accepted. download

Publications

  1. S. Chern, Nonmodular infinite products and a conjecture of Seo and Yee, Adv. Math. 417 (2023), Paper No. 108932, 40 pp. MR4551879. doi: 10.1016/j.aim.2023.108932.
  2. S. Chern, Linked partition ideals and a Schur-type identity of Andrews, Combinatorial and Additive Number Theory V, 107–117, Springer Proc. Math. Stat., 395, Springer, Cham, 2022. doi: 10.1007/978-3-031-10796-2_5.
  3. S. Chern, Linked partition ideals and Andrews–Gordon type series for Alladi and Gordon's extension of Schur's identity, Rocky Mountain J. Math. 52 (2022), no. 6, 2009–2026. MR4527006. doi: 10.1216/rmj.2022.52.2009.
  4. G. E. Andrews, S. Chern, and Z. Li, On the $k$-measure of partitions and distinct partitions, Algebr. Comb. 5 (2022), no. 6, 1353–1361. MR4529928. doi: 10.5802/alco.253.
  5. G. Beck and S. Chern, Reciprocity between partitions and compositions, Ramanujan J. 59 (2022), no. 3, 803–812. MR4496532. doi: 10.1007/s11139-021-00541-5.
  6. S. Chern, On a congruence involving harmonic series and Bernoulli numbers, Int. J. Number Theory 18 (2022), no. 8, 1691–1712. MR4439569. doi: 10.1142/S1793042122500865.
  7. S. Chern and A. J. Yee, Diagonal hooks and a Schmidt-type partition identity, Electron. J. Combin. 29 (2022), no. 2, Paper No. 2.10, 14 pp. MR4410927. doi: 10.37236/10803.
  8. S. Chern, Parity considerations for drops in cycles on $\{1,2,\ldots,n\}$, Discrete Math. 345 (2022), no. 8, Paper No. 112934, 9 pp. MR4409977. doi: 10.1016/j.disc.2022.112934.
  9. S. Chern and D. Tang, Vanishing coefficients and identities concerning Ramanujan's parameters, Ramanujan J. 57 (2022), no. 4, 1367–1385. MR4394011. doi: 10.1007/s11139-021-00385-z.
  10. N. Chen, S. Chern, Y. Fan, and E. X. W. Xia, Some generating functions and inequalities for the Andrews–Stanley partition functions, Proc. Edinb. Math. Soc. (2) 65 (2022), no. 1, 120–135. MR4393364. doi: 10.1017/S0013091521000833.
  11. G. E. Andrews, S. Chern, and Z. Li, Linked partition ideals and the Alladi–Schur theorem, J. Combin. Theory Ser. A 189 (2022), Paper No. 105614, 19 pp. MR4389005. doi: 10.1016/j.jcta.2022.105614. (Mathematica codes: link)
  12. S. Chern, Further results on biases in integer partitions, Bull. Korean Math. Soc. 59 (2022), no. 1, 111–117. MR4372179. doi: 10.4134/BKMS.b210126.
  13. S. Chern, Weighted partition rank and crank moments. II. Odd-order moments, Ramanujan J. 57 (2022), no. 2, 471–485. MR4372228. doi: 10.1007/s11139-020-00365-9.
  14. S. Chern, Weighted partition rank and crank moments. III. A list of Andrews–Beck type congruences modulo 5, 7, 11 and 13, Int. J. Number Theory 18 (2022), no. 1, 141–163. MR4369797. doi: 10.1142/S1793042122500117.
  15. S. Chern and D. Tang, 5-Dissections and sign patterns of Ramanujan's parameter and its companion, Czechoslovak Math. J. 71(146) (2021), no. 4, 1115–1128. MR4339114. doi: 10.21136/CMJ.2021.0218-20.
  16. S. Chern, Note on partitions with even parts below odd parts, Math. Notes 110 (2021), no. 3, 454–457. MR4335386. doi: 10.1134/S0001434621090145.
  17. S. Chern, Asymptotics for the Taylor coefficients of certain infinite products, Ramanujan J. 55 (2021), no. 3, 987–1014. MR4286771. doi: 10.1007/s11139-020-00273-y.
  18. S. Chern, Partitions and the maximal excludant, Electron. J. Combin. 28 (2021), no. 3, Paper No. 3.13, 9 pp. MR4281690. MR4281690. doi: 10.37236/8736.
  19. S. Chern, 1-Shell totally symmetric plane partitions (TSPPs) modulo powers of 5, Ramanujan J. 55 (2021), no. 2, 713–731. MR4260169. doi: 10.1007/s11139-020-00306-6.
  20. S. Chern and D. Tang, The Rogers–Ramanujan continued fraction and related eta-quotient representations, Bull. Aust. Math. Soc. 103 (2021), no. 2, 248–259. MR4229493. doi: 10.1017/S0004972720000933.
  21. S. Chern, Note on the square-root partition into distinct parts, Ramanujan J. 54 (2021), no. 2, 449–461. MR4204766. doi: 10.1007/s11139-019-00191-8.
  22. S. Chern and M. D. Hirschhorn, Some results concerning partitions with designated summands, Ramanujan J. 54 (2021), no. 2, 385–395. MR4204762. doi: 10.1007/s11139-019-00198-1.
  23. G. E. Andrews and S. Chern, A proof of Lin's conjecture on inversion sequences avoiding patterns of relation triples, J. Combin. Theory Ser. A 179 (2021), Paper No. 105388, 20 pp. MR4190575. doi: 10.1016/j.jcta.2020.105388.
  24. S. Chern and D. Tang, Vanishing coefficients in quotients of theta functions of modulus five, Bull. Aust. Math. Soc. 102 (2020), no. 3, 387–398. MR4176682. doi: 10.1017/S0004972720000271.
  25. C. Wang and S. Chern, Some basic hypergeometric transformations and Rogers–Ramanujan type identities, Integral Transforms Spec. Funct. 31 (2020), no. 11, 873–890. MR4161758. doi: 10.1080/10652469.2020.1760267.
  26. S. Chern, Linked partition ideals, directed graphs and $q$-multi-summations, Electron. J. Combin. 27 (2020), no. 3, Paper No. 3.33, 29 pp. MR4245146. doi: 10.37236/9446.
  27. M. Bian, S. Chern, D. D. M. Sang, and E. X. W. Xia, Ramanujan's theta function identities and the relations between sums of squares and sums of triangular numbers, Int. J. Number Theory 16 (2020), no. 6, 1275–1294. MR4120476. doi: 10.1142/S1793042120500669.
  28. S. Chern, D. Tang, and E. X. W. Xia, Arithmetic properties of 7-regular partition triples, Indian J. Pure Appl. Math. 51 (2020), no. 2, 717–733. MR4115620. doi: 10.1007/s13226-020-0426-4.
  29. S. Chern and Z. Li, Linked partition ideals and Kanade–Russell conjectures, Discrete Math. 343 (2020), no. 7, Paper No. 111876, 24 pp. MR4072958. doi: 10.1016/j.disc.2020.111876.
  30. S. Chern, S. Fu, and D. Tang, Multi-dimensional $q$-summations and multi-colored partitions, Ramanujan J. 51 (2020), no. 2, 297–306. MR4056853. doi: 10.1007/s11139-018-0079-7.
  31. S. Chern, Unlimited parity alternating partitions, Quaest. Math. 42 (2019), no. 9, 1345–1352. MR4040863. doi: 10.2989/16073606.2018.1520320.
  32. S. Chern and M. D. Hirschhorn, Partitions into distinct parts modulo powers of 5, Ann. Comb. 23 (2019), no. 3-4, 659–682. MR4039555. doi: 10.1007/s00026-019-00439-6. Also in: George E. Andrews—80 Years of Combinatory Analysis, 305–328, Birkhäuser/Springer, Cham, 2021. MR4221109. doi: 10.1007/978-3-030-57050-7_17.
  33. S. Chern, An extension of a formula of Jovovic, Integers 19 (2019), Paper No. A47, 7 pp. MR4017188.
  34. S. Chern, Note on sums involving the Euler function, Bull. Aust. Math. Soc. 100 (2019), no. 2, 194–200. MR4001535. doi: 10.1017/S0004972719000066; Correction, 103 (2021), no. 1, 174–175. MR4205772. doi: 10.1017/S0004972720000349.
  35. S. Chern, On a problem of George Andrews concerning partitions with even parts below odd parts, Afr. Mat. 30 (2019), no. 5-6, 691–695. MR3993626. doi: 10.1007/s13370-019-00676-1.
  36. S. Chern, D. Tang, and L. Wang, Some inequalities for Garvan's bicrank function of 2-colored partitions, Acta Arith. 190 (2019), no. 2, 171–191. MR3984264. doi: 10.4064/aa180507-7-11.
  37. T. Cai, H. Zhong, and S. Chern, A congruence involving the quotients of Euler and its applications. III (in Chinese), Acta Math. Sin. (Chin. Ser.) 62 (2019), no. 4, 529–540. MR3970566. (English translation)
  38. C. Wang and S. Chern, Some $q$-transformation formulas and Hecke type identities, Int. J. Number Theory 15 (2019), no. 7, 1349–1367. MR3982813. doi: 10.1142/S1793042119500751.
  39. S. Chern, Combinatorial proof of an identity of Andrews and Yee, Ramanujan J. 49 (2019), no. 3, 505–513. MR3979687. doi: 10.1007/s11139-018-0012-0.
  40. S. Chern, A further look at the truncated pentagonal number theorem, Acta Arith. 189 (2019), no. 4, 397–403. MR3962837. doi: 10.4064/aa180718-6-11.
  41. S. Chern and S. Qiu, Partitions, geometric progressions and a Putnam problem, Math. Gaz. 103 (2019), no. 557, 337–343. MR3961649. doi: 10.1017/mag.2019.71.
  42. S. Chern, On a conjecture of George Beck. II, Math. Student 88 (2019), no. 1-2, 159–164. MR3930713.
  43. S. Chern and L.-J. Hao, Congruences for two restricted overpartitions, Proc. Indian Acad. Sci. Math. Sci. 129 (2019), no. 3, Paper No. 31, 16 pp. MR3938513. doi: 10.1007/s12044-019-0474-z.
  44. W. Lin, S. Li, S. Chern, and J. E. Zhang, Pricing VIX derivatives with free stochastic volatility model, Rev. Deriv. Res. 22 (2019), no. 1, 41–75. doi: 10.1007/s11147-018-9145-y.
  45. S. Chern, Asymptotics for the Fourier coefficients of eta-quotients, J. Number Theory 199 (2019), 168–191. MR3926193. doi: 10.1016/j.jnt.2018.11.008.
  46. S. Chern and L.-J. Hao, Congruences for partition functions related to mock theta functions, Ramanujan J. 48 (2019), no. 2, 369–384. MR3911794. doi: 10.1007/s11139-017-9979-1.
  47. S. Chern and D. Tang, On certain weighted 7-colored partitions, Ramanujan J. 48 (2019), no. 2, 305–322. MR3911790. doi: 10.1007/s11139-017-9978-2.
  48. S. Chern, On the power mean of a sum analogous to the Kloosterman sum, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 62(110) (2019), no. 1, 77–92. MR3930926.
  49. S. Chern, Note on the truncated generalizations of Gauss's square exponent theorem, Rocky Mountain J. Math. 48 (2018), no. 7, 2211–2222. MR3892131. doi: 10.1216/RMJ-2018-48-7-2211.
  50. S. Chern and M. G. Dastidar, Some congruences modulo 5 and 25 for overpartitions, Ramanujan J. 47 (2018), no. 2, 435–445. MR3863649. doi: 10.1007/s11139-017-9960-z. (Appendix: An auxiliary result on the double coset $\Gamma_0(N)\backslash\Gamma/\Gamma_\infty$, download)
  51. P. Adansie, S. Chern, and E. X. W. Xia, New infinite families of congruences for the number of tagged parts over partitions with designated summands, Int. J. Number Theory 14 (2018), no. 7, 1935–1942. MR3831401. doi: 10.1142/S1793042118501154.
  52. S. Chern, S. Fu, and D. Tang, Some inequalities for $k$-colored partition functions, Ramanujan J. 46 (2018), no. 3, 713–725. MR3826751. doi: 10.1007/s11139-017-9989-z.
  53. S. Chern, On a conjecture of George Beck, Int. J. Number Theory 14 (2018), no. 3, 647–651. MR3786639. doi: 10.1142/S1793042118500392.
  54. S. Chern, T. Cai, and H. Zhong, On the cardinality and sum of reciprocals of primitive sequences, Adv. Math. (China) 47 (2018), no. 1, 150–154. MR3816359. doi: 10.11845/sxjz.2016046b. (Personal copy)
  55. S. Chern and A. J. Yee, Overpartitions with bounded part differences, European J. Combin. 70 (2018), 317–324. MR3779621. doi: 10.1016/j.ejc.2018.01.003.
  56. S. Chern and M. G. Dastidar, Congruences and recursions for the cubic partition, Ramanujan J. 44 (2017), no. 3, 559–566. MR3723441. doi: 10.1007/s11139-016-9852-7.
  57. S. Chern, Arithmetic properties for cubic partition pairs modulo powers of 3, Acta Math. Sin. (Engl. Ser.) 33 (2017), no.11, 1504–1512. MR3712396. doi: 10.1007/s10114-017-7052-z.
  58. S. Chern, Remarks on the distribution of the primitive roots of a prime, Funct. Approx. Comment. Math. 57 (2017), no. 1, 39–46. MR3704224. doi: 10.7169/facm/1612.
  59. S. Chern, An overpartition analogue of partitions with bounded differences between largest and smallest parts, Discrete Math. 340 (2017), no. 12, 2834–2839. MR3698071. doi: 10.1016/j.disc.2017.08.003.
  60. S. Chern, A curious identity and its applications to partitions with bounded part differences, New Zealand J. Math. 47 (2017), 23–26. MR3691619.
  61. S. Chern, Distribution of reducible polynomials with a given coefficient set, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 60(108) (2017), no. 2, 141–146. MR3676583.
  62. S. Chern, New congruences for $\ell$-regular overpartitions, Integers 17 (2017), Paper No. A22, 8 pp. MR3657408.
  63. S. Chern, Congruences for 1-shell totally symmetric plane partitions, Integers 17 (2017), Paper No. A21, 7 pp. MR3657407.
  64. W. Lin, S. Li, X. Luo, and S. Chern, Consistent pricing of VIX and equity derivatives with the 4/2 stochastic volatility plus jumps model, J. Math. Anal. Appl. 447 (2017), no. 2, 778–797. MR3573114. doi: 10.1016/j.jmaa.2016.10.039.
  65. S. Chern, Integral right triangle and rhombus pairs with a common area and a common perimeter, Forum Geom. 16 (2016), 25–27. MR3474530.
  66. S. Chern, New congruences for 2-color partitions, J. Number Theory 163 (2016), 474–481. MR3459582. doi: 10.1016/j.jnt.2015.12.020.
  67. S. Chern, Formulas for partition $k$-tuples with $t$-cores, J. Math. Anal. Appl. 437 (2016), no. 2, 841–852. MR3456201. doi: 10.1016/j.jmaa.2016.01.040.
  68. S. Chern, A note on balancing binomial coefficients, Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), no. 8, 110–111. MR3403941. doi: 10.3792/pjaa.91.110. (Appendix: Programs for "A note on balancing binomial coefficients", download)
  69. S. Chern and A. Cui, Fibonacci numbers close to a power of 2, Fibonacci Quart. 52 (2014), no. 4, 344–348. MR3276060.
  70. S. Chern, Fermat numbers in multinomial coefficients, J. Integer Seq. 17 (2014), no. 3, Paper No. 14.3.2, 5 pp. MR3168684.