Elementary proofs abound: simply choose x = y = 1 in the binomial expansion of (x + y). The reader is surely aware of many other proofs, including some combinatorial in nature. At the end of the previous century, the evaluation of these sums was trivialized by the work of H. Wilf and D. Zeilberger [7]. In the preface to the charming book [7], the authors begin with the phrase You’ve been up all...

We present new proofs and generalizations of unimodality of the q-binomial coefficients ( n k ) q as polynomials in q. We use an algebraic approach by interpreting the differences between numbers of certain partitions as Kronecker coefficients of representations of Sn. Other applications of this approach include strict unimodality of the diagonal q-binomial coefficients and unimodality of certa...

In this note we consider unimodality problems of sequences of multinomial coefficients and symmetric functions. The results presented here generalize our early results for binomial coefficients. We also give an answer to a question of Sagan about strong q-log-concavity of certain sequences of symmetric functions, which can unify many known results for q-binomial coefficients and q-Stirling numb...