Ever since Kaplansky and Riordan [9-l I] developed their elegant theory of permutations with restricted position, a great variety of combinatorial problems have been explicitly solved, for example, see Dillon and, Roselle [l], Goldman, Joichi, and White [4-S], and Foata, Schutzenberger [2; 31. Spurred by the oft-noted analogy between the case of finite boolean algebras and that of lattices of v...

By using the method of Riordan arrays we nd the asymptotic value of the expected storage utilization in a model of uniform B-tree-like structures proposed by Gupta and Srinivasan, thus solving the occupancy problem for that model.

In the case of two combinatorial polynomials, we show that they can exhibited as moments of paramaterized families of orthogonal polynomials, and hence derive their Hankel transforms. Exponential Riordan arrays are the main vehicles used for this.

We study families of generalized Catalan numbers, defined by convolution recurrence equations. We explore their relations to series reversion, Riordan array transforms, and in a special case, to Somos-4 sequences via the mechanism of the Hankel transform.

Burr and Erdős conjectured that for each k, ` ∈ Z+ such that kZ + ` contains even integers, there exists ck(`) such that any graph of average degree at least ck(`) contains a cycle of length ` mod k. This conjecture was proved by Bollobás, and many successive improvements of upper bounds on ck(`) appear in the literature. In this short note, for 1 6 ` 6 k, we show that ck(`) is proportional to ...

In 1987, Kolaitis, Prömel and Rothschild proved that, for every fixed r ∈ N, almost every n-vertex Kr+1-free graph is r-partite. In this paper we extend this result to all functions r = r(n) with r 6 (log n). The proof combines a new (close to sharp) supersaturation version of the Erdős–Simonovits stability theorem, the hypergraph container method, and a counting technique developed by Balogh, ...