Abstract. 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.

A leaf of a plane tree is called an old leaf if it is the leftmost child of its parent, and it is called a young leaf otherwise. In this paper we enumerate plane trees with a given number of old leaves and young leaves. The formula is obtained combinatorially by presenting two bijections between plane trees and 2-Motzkin paths which map young leaves to red horizontal steps, and old leaves to up steps plus one. We derive some implications to the enumeration of restricted permutations with respect to certain statistics such as pairs of consecutive deficiencies, double descents, and ascending runs. Finally, our main bijection is applied to obtain refinements of two identities of Coker, involving refined Narayana numbers and the Catalan numbers.

We prove a conjecture of Liu and Wang on the q-log-convexity of the Narayana polynomials of type B. By using Pieri's rule and the Jacobi-Trudi identity for Schur functions, we obtain an expansion of a sum of products of elementary symmetric functions in terms of Schur functions with nonnegative coefficients. By the principal specialization this, leads to q-log-convexity. We also show that the linear transformation with respect to the triangular array of Narayana numbers of type B is log-convexity preserving.

We use both Abel's lemma on summation by parts and Zeilberger's algorithm to find recurrence relations for definite summations. The role of Abel's lemma can be extended to the case of linear difference operators with polynomial coefficients. This approach can be used to verify and discover identities involving harmonic numbers and derangement numbers. As examples, we use the Abel-Zeilberger algorithm to prove the Paule-Schneider identities, the Apéry-Schmidt-Strehl identity, Calkin's identity and some identities involving Fibonacci numbers.

A family of k-subsets A 1 , A 2 , . . . , A d on [n] = {1, 2, . . . , n} is called a (d, c)-cluster if the union A 1 ∪ A 2 ∪ · · · ∪ A d contains at most ck elements with c < d. Let F be a family of k-subsets of an n-element set. We show that for k ≥ 2 and n ≥ k + 2, if every (k, 2)-cluster of F is intersecting, then F contains no (k − 1)-dimensional simplices. This leads to an affirmative answer to Mubayi's conjecture for d = k based on Chvátal's simplex theorem. We also show that for any d satisfying 3 ≤ d ≤ k and n ≥ dk d−1 , if every (d, d+12 )-cluster is intersecting, then |F| ≤ n−1 k−1 with equality only when F is a complete star. This result is an extension of both Frankl's theorem and Mubayi's theorem.