In this paper, we define a class of semiorders (or unit interval orders) that arose in the context of polyhedral combinatorics. In the first section of the paper, we will present a pure counting argument equating the number of these interesting (connected and irredundant) semiorders on n + 1 elements with the nth Riordan number. In the second section, we will make explicit the relationship betw...

Recently, the concept of the complementary array of a Riordan array (or recursive matrix) has been introduced. Here we generalize the concept and distinguish between dual and complementary arrays. We show a number of properties of these arrays, how they are computed and their relation with inversion. Finally, we use them to find explicit formulas for the elements of many recursive matrices.

Abstract. We give a combinatorial proof of a Touchard-Riordan-like formula discovered by the first author. As a consequence we find a connection between his formula and Jacobi’s triple product identity. We then give a combinatorial analog of Jacobi’s triple product identity by showing that a finite sum can be interpreted as a generating function of weighted Schröder paths, so that the triple pr...

In this commentary I examine conceptual and methodological issues in Dr. Riordan’s (2012) presentation of three cases treated using Mitchell’s integrative model that combines aspects of Kernberg and Kohut’s technical stances in the treatment of narcissism. Riordan proposes that Mitchell’s integrative model provides a useful blend of Kernberg’s focus on defense and Kohut’s focus on empathy. He t...

We consider some Riordan arrays related to binary words avoiding a pattern which can be easily studied by means of an A-matrix rather than their A-sequence. Both concepts allow us to define every element as a linear combination of other elements in the array; the A-sequence is unique while the A-matrix is not. However, for the problems considered in the present paper, we show that the A-matrix ...

We study a number of characteristics of the inverses of the elements of a family of Pascal-like matrices that are defined by Riordan arrays. We give several forms of the bivariate generating function of these inverses, along with four different closedform expressions for the general element of the inverse. We study the row sums and the diagonal sums of the inverses, and the first column sequenc...