Arthur L. B. Yang

Recurrence Relations for Strongly q-Log-Convex Polynomials

We consider a class of strongly q-log-convex polynomials based on a triangular recurrence relation with linear coefficients, and we show that the Bell polynomials, the Bessel polynomials, the Ramanujan polynomials and the Dowling polynomials are strongly q-log-convex. We also prove that the Bessel transformation preserves log-convexity. Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, P. R. China e-mail: chen@nankai.edu.cn wxw@cfc.nankai.edu.cn yang@nankai.edu.cn Received by the editors June 27, 2008. Published electronically February 10, 2011. This work was supported by the 973 Project, the PCSIRT Project of the Ministry of Education and the National Science Foundation of China. AMS subject classification: 05A20, 05E99.

Major index for 01-fillings of moon polyominoes

We propose a major index statistic on 01-fillings of moon polyominoes which, when specialized to certain shapes, reduces to the major index for permutations and set partitions. We consider the set F(M,s;A) of all 01-fillings of a moon polyomino M with given column sum s whose empty rows are A, and prove that this major index has the same distribution as the number of north-east chains, which are the natural extension of inversions (resp. crossings) for permutations (resp. set partitions). Hence our result generalizes the classical equidistribution results for the permutation statistics inv and maj. Two proofs are presented. The first is an algebraic one using generating functions, and the second is a bijection on 01-fillings of moon polyominoes in the spirit of Foatas second fundamental transformation on words and permutations.

The Flagged Cauchy Determinant

Abstract.We consider a flagged form of the Cauchy determinant, for which we provide a combinatorial interpretation in terms of nonintersecting lattice paths. In combination with the standard determinant for the enumeration of nonintersecting lattice paths, we are able to give a new proof of the Cauchy identity for Schur functions. Moreover, by choosing different starting and end points for the lattice paths, we are led to a lattice path proof of an identity of Gessel which expresses a Cauchy-like sum of Schur functions in terms of the complete symmetric functions.

The skew Schubert polynomials

We obtain a tableau definition of the skew Schubert polynomials named by Lascoux, which are defined as flagged double skew Schur functions. These polynomials are in fact Schubert polynomials in two sets of variables indexed by 321-avoiding permutations. From the divided difference definition of the skew Schubert polynomials, we construct a lattice path interpretation based on the Chen-Li-Louck pairing lemma. The lattice path explanation immediately leads to the determinantal definition and the tableau definition of the skew Schubert polynomials. For the case of a single variable set, the skew Schubert polynomials reduce to flagged skew Schur functions as studied by Wachs and by Billey, Jockusch, and Stanley. We also present a lattice path interpretation for the isobaric divided difference operators, and derive an expression of the flagged Schur function in terms of isobaric operators acting on a monomial. Moreover, we find lattice path interpretations for the Giambelli identity and the Lascoux-Pragacz identity for super-Schur functions. For the super-Lascoux-Pragacz identity, the lattice path construction is related to the code of the partition which determines the directions of the lines parallel to the y-axis in the lattice.

Melham's conjecture on odd power sums of fibonacci numbers

Abstract Ozeki and Prodinger showed that the odd power sum of the first several consecutive Fibonacci numbers of even order is equal to a polynomial evaluated at a certain Fibonacci number of odd order. We prove that this polynomial and its derivative both vanish at 1, and will be an integer polynomial after multiplying it by a product of the first consecutive Lucas numbers of odd order. This presents an affirmative answer to a conjecture of Melham.

The real-rootedness of generalized Narayana polynomials related to the Boros-Moll polynomials

In this paper, we prove the real-rootedness of a family of generalized Narayana polynomials, which arose in the study of the infinite log-concavity of the Boros-Moll polynomials. We establish certain recurrence relations for these Narayana polynomials, from which we derive the real-rootedness. To prove the real-rootedness, we use a sufficient condition, due to Liu and Wang, to determine whether two polynomials have interlaced zeros. The recurrence relations are verified with the help of the Mathematica package HolonomicFunctions. AMS Classification 2010: 05A15, 26C10

Total positivity of Narayana matrices

Abstract We prove the total positivity of the Narayana triangles of type A and type B , and thus affirmatively confirm a conjecture of Chen, Liang and Wang and a conjecture of Pan and Zeng. We also prove the strict total positivity of the Narayana squares of type A and type B .

Brenti's Open Problem on the Real-Rootedness of

We prove that, for any positive

q

q

-Eulerian Polynomials of Type

, the