The “classical” parking functions, counted by the Cayley number (n+ 1) n− 1, carry a natural
permutation representation of the symmetric group S n in which the number of orbits is the …

We introduce a family of generalized Schr\" oder polynomials $ S_\tau (q, t, a) $, indexed by
triangular partitions $\tau $ and prove that $ S_\tau (q, t, a) $ agrees with the Poincar\'e …

We study the relationship between rational slope Dyck paths and invariant subsets in Z,
extending the work of the first two authors in the relatively prime case. We also find a …

Recent results have placed the classical shuffle conjecture of Haglund et al. in a broader
context of an infinite family of conjectures about parking functions in any rectangular lattice …

We give a new combinatorial proof of the well known result that the dinv of an $(m, n) $-Dyck
path is equal to the area of its sweep map image. The first proof of this remarkable identity …

Given a coprime pair $(m, n) $ of positive integers, rational Catalan numbers $\frac {1}{m+
n}\binom {m+ n}{m, n} $ counts two combinatorial objects: rational $(m, n) $-Dyck paths are …

We define generalized Schröder polynomials Sλ (q, t, a) for triangular partitions and prove
that these polynomials recover the triangular (q, t)-Catalan polynomials of [2] at a= 0 …

The well-known q, t-Catalan sequence has two combinatorial interpretations as weighted
sums of ordinary Dyck paths: one is Haglund's area-bounce formula, and the other is …

Given a coprime pair $(m, n) $ of positive integers, rational Catalan numbers $\dfrac {1}{m+
1}\begin {pmatrix} m+ n\\m, n\end {pmatrix} $ counts two combinatorial objects: rational $(m …