2017/03/18 by Alice L. L. Gao, Gao, Alice L. L., Matthew H. Y. Xie +3
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.1703.06382
Chen proposed a conjecture on the log-concavity of the generating function\nfor the symmetric group with respect to the length of longest increasing\nsubsequences of permutations. Motivated by Chen's log-concavity conjecture,\nB 'ona, Lackner and Sagan further studied similar problems by restricting the\nwhole symmetric group to certain of its subsets. They obtained the\nlog-concavity of the corresponding generating functions for these subsets by\nusing the hook-length formula. In this paper, we generalize and prove their\nresults by establishing the Schur positivity of certain symmetric functions.\nThis also enables us to propose a new approach to Chen's original conjecture.\n