Unimodality of generalized Gaussian coefficients

A combinatorial proof of the unimodality of the generalized q-Gaussian coefficients [ N λ ] q based on the explicit formula for Kostka-Foulkes polynomials is given. 1. Let us mention that the proof of the unimodality of the generalized Gaussian coefficients based on theoretic-representation considerations was given by E.B. Dynkin [1] (see also [2], [10], [11]). Recently K.O’Hara [6] gave a constructive proof of the unimodality of the Gaussian coefficient [ n+ k n ] q = s(k)(1, · · · , q ), and D. Zeilberger [12] derived some identity which may be consider as an “algebraization” of O’Hara’s construction. By induction this identity immediately implies the unimodality of [ n+ k n ] q . Using the observation (see Lemma 1) that the generalized Gaussian coefficient [ n λ′ ] q may be identified (up to degree q ) with the Kostka-Foulkes polynomial K λ̃,μ (q) (see Lemma 1), the proof of the unimodality of [ n λ′ ] q is a simple consequence of the exact formula for Kostka-Foulkes polynomials contained in [4]. Furthermore the expression for K λ̃,μ (q) in the case λ = (k) coincides with identity (KOH) from [8]. So we obtain a generalization and a combinatorial proof of (KOH) for arbitrary partition λ.