# F50-07: Computer Algebra and Combinatorial Inequalities

The aim here is to develop new methods for proving combinatorial inequalities using a combination of symbolic algorithms and classical techniques from enumerative combinatorics. One particular aspect of this project part is to deal with open conjectures on $q$-inequalities, unimodality, and log-concavity. This class of problems has been subject to intensive investigations in different areas of mathematics and still there are many open conjectures left to be resolved. We expect that combining the efforts of two communities will further push the development of new results. From the symbolic computation point of view we plan to build on the existing approach developed by Gerhold and Kauers for proving inequalities on sequences involving a discrete parameter using Cylindrical Algebraic Decomposition. Furthermore we want to apply algorithms for symbolic summation as developed in the algorithmic combinatorics group at RISC. Combinatorially, we plan to build upon inductive and injective proofs and embed them systematically in symbolic methods. Asymptotic analysis as envisaged also within this SFB, may enter in combination with monotonicity statements obtained using symbolic computation. In general, the proposed project part will be carried out problem oriented in its starting phase, studying specific problems to derive methodological approaches. Besides theoretical results, a major goal is to implement the findings in a computer algebra system and make the programs publicly available.