Human and constructive proof of combinatorial identities: an example from Romik

International audienceIt has become customary to prove binomial identities by means of the method for automated proofs as developed by Petkovšek, Wilf and Zeilberger. In this paper, we wish to emphasize the role of "human'' and constructive proofs in contrast with the somewhat lazy attitude of relaying on "automated'' proofs. As a meaningful example, we consider the four formulas by Romik, related to Motzkin and central trinomial numbers. We show that a proof of these identities can be obtained by using the method of coefficients, a human method only requiring hand computations

A strip-like tiling algorithm

AbstractWe extend our previous results on the connection between strip tiling problems and regular grammars by showing that an analogous algorithm is applicable to other tiling problems, not necessarily related to rectangular strips. We find generating functions for monomer and dimer tilings of T- and L-shaped figures, holed and slotted strips, diagonal strips and combinations of them, and show how analogous results can be obtained by using different pieces

Complementary Riordan arrays

Abstract 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

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We give an alternative proof of an identity that appeared recently in Integers. By using the concept of Riordan arrays we obtain a short, elementary proof