Identifier
St000012:
Dyck paths
⟶ ℤ
(values match
St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2.)
Values
No modified entries
Description
The area of a Dyck path.
This is half the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly below the path do not contribute to this statistic.
1. Dyck paths are in bijection with area sequences $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$.
2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$
3. The area is equidistributed with St000005The bounce statistic of a Dyck path. and St000006The dinv of a Dyck path.. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.
This is half the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly below the path do not contribute to this statistic.
1. Dyck paths are in bijection with area sequences $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$.
2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$
3. The area is equidistributed with St000005The bounce statistic of a Dyck path. and St000006The dinv of a Dyck path.. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.
Diff Description
The area of a Dyck path.
This is half the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directlyabovebelow the axispath do not contribute to this statistic.
1. Dyck paths are in bijection with '''area sequences''' (a_1,\ldots,a_n) such that a_1 = 0, a_{k+1} \leq a_k + 1.
2. The generating function \mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)} satisfy the recurrence \mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).
3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of q,t-Catalan numbers.
This is half the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly
1. Dyck paths are in bijection with '''area sequences''' (a_1,\ldots,a_n) such that a_1 = 0, a_{k+1} \leq a_k + 1.
2. The generating function \mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)} satisfy the recurrence \mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).
3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of q,t-Catalan numbers.
References
[1] Haglund, J. The $q$,$t$-Catalan numbers and the space of diagonal harmonics MathSciNet:2371044
Code
def statistic(x):
return x.area()
Created
Sep 27, 2011 at 19:29 by Chris Berg
Updated
Nov 21, 2025 at 03:59 by Nadia Lafreniere
