Reciprocity between Dyck paths and alternating sequences - heaps, orthogonal polynomials, …
Reciprocity is a much used (and misused) term in mathematics. The meaning of "reciprocity" in our context is the one introduced by Richard Stanley: if a sequence $(a_n)_{n\ge0}$ is extended to negative indices $n$, and if there is a combinatorial meaning for these "negative" terms of the sequence, then he speaks of a "combinatorial reciprocity law". The theme of this talk is a previously unobserved "reciprocity law" between Dyck paths and alternating sequences. As it turns out, this "reciprocity" extends to (certain) families of Dyck paths and alternating sequences, and it relates to several other (combinatorial and non-combinatorial) objects, including Viennot's theory of heaps, orthogonal polynomials, determinants, …