We establish a congruence on sums of central $q$-binomial coefficients. From this $q$-congruence, we derive the divisibility of the $q$-trinomial coefficients introduced by Andrews and Baxter.
The alternating descent statistic on permutations was introduced by Chebikin as a variant of the descent statistic. We show that the alternating descent polynomials on permutations are unimodal via a five-term recurrence relation. We also found a qua
dratic recursion for the alternating major index $q$-analog of the alternating descent polynomials. As an interesting application of this quadratic recursion, we show that $(1+q)^{lfloor n/2rfloor}$ divides $sum_{piinmathfrak{S}_n}q^{rm{altmaj}(pi)}$, where $mathfrak{S}_n$ is the set of all permutations of ${1,2,ldots,n}$ and $rm{altmaj}(pi)$ is the alternating major index of $pi$. This leads us to discover a $q$-analog of $n!=2^{ell}m$, $m$ odd, using the statistic of alternating major index. Moreover, we study the $gamma$-vectors of the alternating descent polynomials by using these two recursions and the ${textbf{cd}}$-index. Further intriguing conjectures are formulated, which indicate that the alternating descent statistic deserves more work.
We show that the formalism of hybrid polynomials, interpolating between Hermite and Laguerre polynomials, is very useful in the study of Motzkin numbers and central trinomial coefficients. These sequences are identified as special values of hybrid po
lynomials, a fact which we use to derive their generalized forms and new identities satisfied by them.
We introduce a notion of $q$-deformed rational numbers and $q$-deformed continued fractions. A $q$-deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the $q$-deformed Pa
scal identitiy for the Gaussian binomial coefficients, but the Pascal triangle is replaced by the Farey graph. The coefficients of the polynomials defining the $q$-rational count quiver subrepresentations of the maximal indecomposable representation of the graph dual to the triangulation. Several other properties, such as total positivity properties, $q$-deformation of the Farey graph, matrix presentations and $q$-continuants are given, as well as a relation to the Jones polynomial of rational knots.
An $m$-general set in $AG(n,q)$ is a set of points such that any subset of size $m$ is in general position. A $3$-general set is often called a capset. In this paper, we study the maximum size of an $m$-general set in $AG(n,q)$, significantly improvi
ng previous results. When $m=4$ and $q=2$ we give a precise estimate, solving a problem raised by Bennett.
For an integer $qge2$, a $q$-recursive sequence is defined by recurrence relations on subsequences of indices modulo some powers of~$q$. In this article, $q$-recursive sequences are studied and the asymptotic behavior of their summatory functions is
analyzed. It is shown that every $q$-recursive sequence is $q$-regular in the sense of Allouche and Shallit and that a $q$-linear representation of the sequence can be computed easily by using the coefficients from the recurrence relations. Detailed asymptotic results for $q$-recursive sequences are then obtained based on a general result on the asymptotic analysis of $q$-regular sequences. Three particular sequences are studied in detail: We discuss the asymptotic behavior of the summatory functions of Sterns diatomic sequence, the number of non-zero elements in some generalized Pascals triangle and the number of unbordered factors in the Thue--Morse sequence. For the first two sequences, our analysis even leads to precise formulae{} without error terms.