Proof of a positivity conjecture on Schur functions

In the study of Zeilbergerʼs conjecture on an integer sequence related to the Catalan numbers, Lassalle proposed the following conjecture. Let ( t ) n denote the rising factorial, and let Λ R denote the algebra of symmetric functions with real coefficients. If φ is the homomorphism from Λ R to R defined by φ ( h n ) = 1 / ( ( t ) n n ! ) for some t > 0 , then for any Schur function s λ , the value φ ( s λ ) is positive. In this paper, we provide an affirmative answer to Lassalleʼs conjecture by using the Laguerre–Pólya–Schur theory of multiplier sequences.