The $q,t$-Catalan polynomial $C_n(q,t)$ [2] is the Dyck-path generating function in (1), using the area-sequence convention in (5).
Parameters
$n$
— semilength (a nonnegative integer)
Formulas
(1)
$C_n(q,t)=\sum_{D\in\operatorname{Dyck}_n} q^{\operatorname{dinv}(D)}t^{\operatorname{area}(D)}$, summed over Dyck paths $D$ of semilength $n$ [1].
(2)
$C_n(1,1)=\frac{1}{n+1}\binom{2n}{n}$.
(3)
$C_n(q,t)=C_n(t,q)$.
(4)
$C_0(q,t)=C_1(q,t)=1$ and $C_2(q,t)=q+t$.
Comments
(5)
For $n=0$ the area sequence is empty. For $n>0$, an area sequence is $(a_1,\ldots,a_n)$ with $a_1=0$ and $0\leq a_{i+1}\leq a_i+1$ for $1\leq i<n$. Its area is $\sum_i a_i$, and its dinv is the number of pairs $i<j$ with $a_i-a_j\in\{0,1\}$.
Programs
(P1)
Sage
from sage.combinat.q_analogues import qt_catalan_number
qt_catalan_number(7)
References
[1]
J. Haglund, The q,t-Catalan Numbers and the Space of Diagonal Harmonics, University Lecture Series 41, American Mathematical Society, Providence, RI, 2008.
[2]
A. M. Garsia and M. Haiman, A remarkable q,t-Catalan sequence and q-Lagrange inversion, J. Algebraic Combin. 5 (1996), 191-244.
[3]
L. Carlitz and J. Riordan, Two element lattice permutation numbers and their q-generalization, Duke Math. J. 31 (1964), 371-388.
[4]
P. A. MacMahon, Combinatory Analysis, Vol. II, Cambridge University Press, Cambridge, 1916.
Carlitz-Riordan $q$-Catalan numbers — stores the specialisation $C_n(q,1)$, equivalently the area generating function of Dyck paths. A reader holding that one-variable polynomial is likely to want the Carlitz-Riordan table; a reader holding the two-variable dinv-area polynomial wants this table.
MacMahon $q$-Catalan numbers — stores $\widetilde C_n(q)=q^{\binom n2}C_n(q,q^{-1})$, equivalently the Gaussian binomial quotient. A reader holding that one-variable quotient is likely to want the MacMahon table; a reader holding the full two-variable polynomial wants this table.
Every entry is an exact polynomial with integer coefficients. The generator computes $C_n(q,t)$ with Sage's qt_catalan_number[5].
more
Before the values were written, the rows were checked against an independent enumeration of Dyck-path area sequences using the dinv and area statistics. The checks also verified (3), (2), and the small cases in (4) over the stored range.
