This talk discusses several inequalities for partition statistics, including ranks and cranks for partitions, overpartitions, and k-colored partitions. After a brief review of Ramanujan’s congruences and Dyson’s rank, we summarize results on symmetry, unimodality, log-concavity, and log-subadditivity. We also outline some common methods, including combinatorial arguments, generating functions, modular forms, and the circle method. Finally, we mention recent work on asymptotic formulas, bounds, and Turán- and Laguerre-type inequalities.

张文静，湖南大学数学学院副教授，2019年博士毕业于天津大学应用数学中心，师从陈永川教授，主要研究组合数学、特殊函数与整数分拆，论文发表在Forum Math.，Canad. J. Math.，Proc. Amer. Math. Soc.，Adv. in Appl. Math.，J. Number Theory等国际期刊。主持国家青年基金和面上项目，湖南省青年基金和面上基金各一项，现任中国运筹学会图论与组合分会第六届理事会青年理事。