q have been investigated in relation to questions of representation theory concerning the growth of Kronecker coefficients. Further, one of the results of this note, Theorem 2.2, has also been motivated by, and finds a first useful application in the study of the unimodality of partitions with distinct parts that are contained inside certain Ferrers diagrams (see our own paper [10]). For m = ⌊a...

We give a combinatorial proof that k l-k-1 l + t q q q q a polynomial in q with nonnegative coefficients for nonnegative integers a, b, k, lwith a>~b and l~>k. In particular, for a=b=n and l=k, this implies the q-log-concavity of the Gaussian binomial coefficients k , which was conjectured q by BUTLER (Proc.

We prove a q-analog of a classical binomial congruence due to Ljunggren which states that ap bp ! ≡ a b ! modulo p for primes p > 5. This congruence subsumes and builds on earlier congruences by Babbage, Wolstenholme and Glaisher for which we recall existing q-analogs. Our congruence generalizes an earlier result of Clark. Résumé. Nous démontrons un q-analogue d’une congruence binomiale classiq...

Abstract. Several new transformations for q-binomial coefficients are found, which have the special feature that the kernel is a polynomial with nonnegative coefficients. By studying the group-like properties of these positivity preserving transformations, as well as their connection with the Bailey lemma, many new summation and transformation formulas for basic hypergeometric series are found....

We present a lower bound on the Kronecker coefficients of the symmetric group via the characters of Sn, which we apply to obtain various explicit estimates. Notably, we extend Sylvester’s unimodality of q-binomial coefficients ( n k ) q as polynomials in q to derive sharp bounds on the differences of their consecutive coefficients.

OF THE DISSERTATION A treatise on the binomial theorem by PATRICK DEVLIN Dissertation Director: Jeff Kahn This dissertation discusses four problems taken from various areas of combinatorics— stability results, extremal set systems, information theory, and hypergraph matchings. Though diverse in content, the unifying theme throughout is that each proof relies on the machinery of probabilistic co...

Let us note that every non empty loop structure which is add-left-cancelable and add-rightcancelable is also add-cancelable and every non empty loop structure which is add-cancelable is also add-left-cancelable and add-right-cancelable. Let us note that every non empty loop structure which is Abelian and add-right-cancelable is also add-left-cancelable and every non empty loop structure which i...

In [9], the author introduced a digital version of this theorem where the exponents appearing in (1) are viewed as sums of digits. To illustrate this, consider the binomial theorem for N = 2: (x + y) = xy + xy + xy + xy. (2) It is easy to verify that (2) is equivalent to (x+ y) = xy + xy + xy + xy, (3) where s(k) denotes the sum of digits of k expressed in binary. For example, s(3) = s(1 · 2 + ...