Identifier
-
Mp00231:
Integer compositions
—bounce path⟶
Dyck paths
St000012: Dyck paths ⟶ ℤ (values match St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2.)
Values
[1] => [1,0] => 0
[1,1] => [1,0,1,0] => 0
[2] => [1,1,0,0] => 1
[1,1,1] => [1,0,1,0,1,0] => 0
[1,2] => [1,0,1,1,0,0] => 1
[2,1] => [1,1,0,0,1,0] => 1
[3] => [1,1,1,0,0,0] => 3
[1,1,1,1] => [1,0,1,0,1,0,1,0] => 0
[1,1,2] => [1,0,1,0,1,1,0,0] => 1
[1,2,1] => [1,0,1,1,0,0,1,0] => 1
[1,3] => [1,0,1,1,1,0,0,0] => 3
[2,1,1] => [1,1,0,0,1,0,1,0] => 1
[2,2] => [1,1,0,0,1,1,0,0] => 2
[3,1] => [1,1,1,0,0,0,1,0] => 3
[4] => [1,1,1,1,0,0,0,0] => 6
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 3
[1,4] => [1,0,1,1,1,1,0,0,0,0] => 6
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0] => 4
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 3
[3,2] => [1,1,1,0,0,0,1,1,0,0] => 4
[4,1] => [1,1,1,1,0,0,0,0,1,0] => 6
[5] => [1,1,1,1,1,0,0,0,0,0] => 10
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 3
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 6
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 4
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 3
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 4
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 6
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 10
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 2
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 4
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 2
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 3
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 4
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 7
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 3
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 4
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 4
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 6
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 6
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 7
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 10
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 15
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 0
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 1
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => 1
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 3
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => 1
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 2
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => 3
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 6
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => 2
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => 2
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => 4
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => 3
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => 4
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => 6
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 10
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => 2
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => 2
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => 4
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => 2
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 3
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => 4
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 7
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => 3
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => 4
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => 4
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 6
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => 6
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 7
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 10
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 15
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => 1
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => 2
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => 2
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => 4
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => 2
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => 3
>>> Load all 223 entries. <<<
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The area of a Dyck path.
This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic.
1. Dyck paths are 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 the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic.
1. Dyck paths are 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.
Map
bounce path
Description
The bounce path determined by an integer composition.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!