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Log concave sequences of symmetric functions and analogs of the Jacobi-Trudi determinants. (English) Zbl 0769.05097
A sequence \(a_ 0, a_ 1, a_ 2, \dots\) is called log concave if it satisfies \(a_{k-1}a_{k+1}\leq a_ k^ 2\) for all \(k>0\). This notion can be generalized to polynomials in several variables. It is proved that elementary symmetric functions and complete homogeneous symmetric functions produce log concave sequences, when a specialization which is also log concave is taken. This implies log concavity properties of Gaussian coefficients and Stirling numbers of the first and second kinds, recently obtained by L. Butler, C. Krattenthaler, P. Leroux, and the author. The proofs are based on the Gessel-Viennot interpretation of determinants. Some analogues concerning the PF (Pólya frequency) property are also given. As pointed out by the author (private communication), recently F.Brenti [Séries formelles et combinatoire algebrique, Montréal 87-94 (1992)] gave a combinatorial interpretation for some determinants involving elementary and homogeneous symmetric functions, thus giving an answer to Question 7 of the present paper.

MSC:
05E05 Symmetric functions and generalizations
05A15 Exact enumeration problems, generating functions
05A10 Factorials, binomial coefficients, combinatorial functions
11B65 Binomial coefficients; factorials; \(q\)-identities
11B73 Bell and Stirling numbers
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