Emeric Deutsch

Dyck path enumeration

Abstract An elementary technique is used for the enumeration of Dyck paths according to various parameters. For several of the considered parameters the generating function is expressed in terms of the Narayana function. Many enumeration results are obtained, some of which involve the Fine numbers.

A survey of the Fine numbers

Abstract The Fine numbers and the Catalan numbers are intimately related. Two manifestations are the identity C n =2F n +F n−1 , n⩾1 , and the generating function identities F=C/(1+zC), C=F/(1−zF) . In this paper we collect and organize the previous literature, present many new settings, and develop the theory and generating functions as well as asymptotics. Among the topics developed are the hill-killer involution, the set of all path pairs, and some new results about noncrossing partitions.

Derivatives of the perron root at an essentially nonnegative matrix and the group inverse of an M-matrix

Abstract The results of the work in this paper can be divided into two major parts. In the first part we develop formulae, using group inverses of irreducible M-matrices, for the second-order partial derivatives with respect to the entries of the Perron root r at an essentially nonnegative and irreducible matrix A. We use these formulae to obtain explicit expressions for the matrices ( ∂ 2 r(A) ∂ ij 2 ) and ( ∂ 2 r(A) ∂ ii ∂ jj ) and examine these expressions at some special cases when, for example, in addition to A being essentially nonnegative and irreducible, A is row stochastic or A is of rank 1 and so on. In the second major part of this paper we investigate, for a singular and irreducible M-matrix Q, a certain matrix G which is a positive diagonal scaling of the group inverse Q#, of Q. We show, for instance, that zτ Gz ⩾ 0 for all real vectors z and use this result and some of the formulae obtained in the first part of the paper to provide a matrix theoretic proof of a theorem due to J. E. Cohen (Math. Proc, Cambridge Philos. Soc., 86 (1979), 343–350), which states that for an essentially nonnegative and irreducible matrix A, the Perron root is a convex function of the main diagonal of A.

A bijection between ordered trees and 2-Motzkin paths and its many consequences

A new bijection between ordered trees and 2-Motzkin paths is presented, together with its numerous consequences regarding ordered trees as well as other combinatorial structures such as Dyck paths, bushes, {0, 1, 2}-trees, Schroder paths, RNA secondary structures, noncrossing partitions, Fine paths, and Davenport-Schinzel sequences.

A bijection on Dyck paths and its consequences

Abstract A bijection is introduced in the set of all Dyck paths of semilength n from which it follows that (i) the parameters ‘height of the first peak’ and ‘number of returns’ have the same distribution and (ii) the parameter ‘number of high peaks’ has the Narayana distribution.

An involution on Dyck paths and its consequences

Abstract An involution is introduced in the set of all Dyck paths of semilength n from which one re-obtains easily the equidistribution of the parameters ‘number of valleys’ and ‘number of doublerises’ and also the equidistribution of the parameters ‘height of the first peak’ and ‘number of returns’.

On the first and second order derivatives of the Perron vector

Abstract In a previous work [5] the authors developed formulas for the second order partial derivatives of the Perron root as a function of the matrix entries at an essentially nonnegative and irreducible matrix. These formulas, which involve the group generalized inverse of an associated M -matrix, were used to investigate the concavity and convexity of the Perron root as a function of the entries. The authors now combine the above results together with an approach taken in an earlier joint paper [6] of the second author with L. Elsner and C. Johnson, and they develop formulas for the second order derivatives of an appropriately normalized Perron vector with respect to the matrix entries, which again are given in terms the group generalized inverse of an associated M -matrix. Convexity properties of the Perron vector as a function of the entries of the matrix are then examined. In addition, formulas for the first derivative of the Perron vector resulting from different normalizations of this eigenvector are also given. A by-product of one of these formulas yields that the group generalized inverse of a singular and irreducible M -matrix can be diagonally scaled to a matrix which is entrywise column diagonally dominant.

Statistics on non-crossing trees

A non-crossing tree is a tree drawn on the plane having as vertices a set of points on the boundary of a circle, and whose edges are straight line segments and do not cross. Continuing previous research on non-crossing trees, we study several new statistics: number of end-points and boundary edges, number of vertices of a given degree, maximum degree, height, and path-length. In some cases, we obtain closed formulae while in others we deduce asymptotic estimates. Our approach is based on generating functions and on several bijections between NC-trees and various other combinatorial objects.

Diagonally convex directed polyominoes and even trees: a bijection and related issues

We present a simple bijection between diagonally convex directed (DCD) polyominoes with n diagonals and plane trees with 2n edges in which every vertex has even degree (even trees), which specializes to a bijection between parallelogram polyominoes and full binary trees. Next we consider a natural definition of symmetry for DCD-polyominoes, even trees, ternary trees, and non-crossing trees, and show that the number of symmetric objects of a given size is the same in all four cases.

Bounded groups and norm-Hermitian matrices

Abstract An elementary proof is given that a bounded multiplicative group of complex (real) n × n nonsingular matrices is similar to a unitary (orthogonal) group. Given a norm on a complex n -space, it follows that there exists a nonsingular n × n matrix L (the Loewner-John matrix for the norm) such that LHL −1 is Hermitian for every norm-Hermitian H . Numerous applications of this result are given.