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arXiv · 1703.06382

Schur positivity and log-concavity related to longest increasing subsequences

Alice L. L. Gao·Matthew H. Y. Xie·Arthur L. B. Yang
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Abstract

Chen proposed a conjecture on the log-concavity of the generating function for the symmetric group with respect to the length of longest increasing subsequences of permutations. Motivated by Chen's log-concavity conjecture, B\'{o}na, Lackner and Sagan further studied similar problems by restricting the whole symmetric group to certain of its subsets. They obtained the log-concavity of the corresponding generating functions for these subsets by using the hook-length formula. In this paper, we generalize and prove their results by establishing the Schur positivity of certain symmetric functions. This also enables us to propose a new approach to Chen's original conjecture.

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Alice L. L. Gao, Matthew H. Y. Xie, Arthur L. B. Yang. 2017-03-19. Schur positivity and log-concavity related to longest increasing subsequences. https://arxiv.org/abs/1703.06382

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