On doubly symmetric Dyck words

In this paper we consider doubly symmetric Dyck words, i.e. Dyck words which are fixed by two symmetry operations α and β introduced in [1]. We study combinatorial properties of doubly symmetric Dyck words, leading to the definition of two recursive algorithms to build these words. As a consequence we have a representation of doubly symmetric Dyck words as vectors of integers, called track vectors. Finally, we show some bijections between a subfamily of doubly symmetric Dyck words and a subfamily of integer partitions. The computation of the sequence fn of doubly symmetric Dyck words of semi-length n shows surprising properties giving rise to some conjectures.