A note on two identities arising from enumeration of convex polyominoes

Motivated by some binomial coefficients identities encountered in our approach to the enumeration of convex polyominoes, we prove some more general identities of the same type, one of which turns out to be related to a strange evaluation of ${}_3F_2$ of Gessel and Stanton.Comment: 10 pages, to appear in J. Comput. Appl. Math; minor grammatical change

Factors of sums and alternating sums involving binomial coefficients and powers of integers

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers $n_1,..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer $r$, there holds {align*} \sum_{k=0}^{n_1}\epsilon^k (2k+1)^{2r+1}\prod_{i=1}^{m} {n_i+n_{i+1}+1\choose n_i-k} \equiv 0 \mod (n_1+n_m+1){n_1+n_m\choose n_1}, {align*} and conjecture that for any nonnegative integer $r$ and positive integer $s$ such that $r+s$ is odd, $$\sum_{k=0}^{n}\epsilon ^k (2k+1)^{r}({2n\choose n-k}-{2n\choose n-k-1})^{s} \equiv 0 \mod{{2n\choose n}},$$ where $\epsilon=\pm 1$.Comment: 14 pages, to appear in Int. J. Number Theor

Genus one partitions

We prove the conjecture by M. Yip stating that counting genus one partitions by the number of their elements and parts yields, up to a shift of indices, the same array of numbers as counting genus one rooted hypermonopoles. Our proof involves representing each genus one permutation by a four-colored noncrossing partition. This representation may be selected in a unique way for permutations containing no trivial cycles. The conclusion follows from a general generating function formula that holds for any class of permutations that is closed under the removal and reinsertion of trivial cycles. Our method also provides another way to count rooted hypermonopoles of genus one, and puts the spotlight on a class of genus one permutations that is invariant under an obvious extension of the Kreweras duality map to genus one permutations

Parameterized generic Galois groups for q-difference equations, followed by the appendix "The Galois D-groupoid of a q-difference system" by Anne Granier

We introduce the parameterized generic Galois group of a q-difference module, that is a differential group in the sense of Kolchin. It is associated to the smallest differential tannakian category generated by the q-difference module, equipped with the forgetful functor. Our previous results on the Grothendieck conjecture for q-difference equations lead to an adelic description of the parameterized generic Galois group, in the spirit of the Grothendieck-Katz's conjecture on p-curvatures. Using this description, we show that the Malgrange-Granier D-groupoid of a nonlinear q-difference system coincides, in the linear case, with the parameterized generic Galois group introduced here. The paper is followed by an appendix by A. Granier, that provides a quick introduction to the D-groupoid of a non-linear q-difference equation.Comment: The content of this paper was previously included in arXiv:1002.483

Multiple orthogonal polynomials, d-orthogonal polynomials, production matrices, and branched continued fractions

I analyze an unexpected connection between multiple orthogonal polynomials, d-orthogonal polynomials, production matrices and branched continued fractions. This work can be viewed as a partial extension of Viennot's combinatorial theory of orthogonal polynomials to the case where the production matrix is lower-Hessenberg but is not necessarily tridiagonal