- FindStat

Identifier

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
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

[1,0] => [1] => [1] => [1,0] => 0

[1,0,1,0] => [1,1] => [2] => [1,1,0,0] => 1

[1,1,0,0] => [2] => [1] => [1,0] => 0

[1,0,1,0,1,0] => [1,1,1] => [3] => [1,1,1,0,0,0] => 3

[1,0,1,1,0,0] => [1,2] => [1,1] => [1,0,1,0] => 0

[1,1,0,0,1,0] => [2,1] => [1,1] => [1,0,1,0] => 0

[1,1,0,1,0,0] => [3] => [1] => [1,0] => 0

[1,1,1,0,0,0] => [3] => [1] => [1,0] => 0

[1,0,1,0,1,0,1,0] => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0] => 6

[1,0,1,0,1,1,0,0] => [1,1,2] => [2,1] => [1,1,0,0,1,0] => 1

[1,0,1,1,0,0,1,0] => [1,2,1] => [1,1,1] => [1,0,1,0,1,0] => 0

[1,0,1,1,0,1,0,0] => [1,3] => [1,1] => [1,0,1,0] => 0

[1,0,1,1,1,0,0,0] => [1,3] => [1,1] => [1,0,1,0] => 0

[1,1,0,0,1,0,1,0] => [2,1,1] => [1,2] => [1,0,1,1,0,0] => 1

[1,1,0,0,1,1,0,0] => [2,2] => [2] => [1,1,0,0] => 1

[1,1,0,1,0,0,1,0] => [3,1] => [1,1] => [1,0,1,0] => 0

[1,1,0,1,0,1,0,0] => [4] => [1] => [1,0] => 0

[1,1,0,1,1,0,0,0] => [4] => [1] => [1,0] => 0

[1,1,1,0,0,0,1,0] => [3,1] => [1,1] => [1,0,1,0] => 0

[1,1,1,0,0,1,0,0] => [4] => [1] => [1,0] => 0

[1,1,1,0,1,0,0,0] => [4] => [1] => [1,0] => 0

[1,1,1,1,0,0,0,0] => [4] => [1] => [1,0] => 0

[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0] => 10

[1,0,1,0,1,0,1,1,0,0] => [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0] => 3

[1,0,1,0,1,1,0,0,1,0] => [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0] => 1

[1,0,1,0,1,1,0,1,0,0] => [1,1,3] => [2,1] => [1,1,0,0,1,0] => 1

[1,0,1,0,1,1,1,0,0,0] => [1,1,3] => [2,1] => [1,1,0,0,1,0] => 1

[1,0,1,1,0,0,1,0,1,0] => [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0] => 1

[1,0,1,1,0,0,1,1,0,0] => [1,2,2] => [1,2] => [1,0,1,1,0,0] => 1

[1,0,1,1,0,1,0,0,1,0] => [1,3,1] => [1,1,1] => [1,0,1,0,1,0] => 0

[1,0,1,1,0,1,0,1,0,0] => [1,4] => [1,1] => [1,0,1,0] => 0

[1,0,1,1,0,1,1,0,0,0] => [1,4] => [1,1] => [1,0,1,0] => 0

[1,0,1,1,1,0,0,0,1,0] => [1,3,1] => [1,1,1] => [1,0,1,0,1,0] => 0

[1,0,1,1,1,0,0,1,0,0] => [1,4] => [1,1] => [1,0,1,0] => 0

[1,0,1,1,1,0,1,0,0,0] => [1,4] => [1,1] => [1,0,1,0] => 0

[1,0,1,1,1,1,0,0,0,0] => [1,4] => [1,1] => [1,0,1,0] => 0

[1,1,0,0,1,0,1,0,1,0] => [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0] => 3

[1,1,0,0,1,0,1,1,0,0] => [2,1,2] => [1,1,1] => [1,0,1,0,1,0] => 0

[1,1,0,0,1,1,0,0,1,0] => [2,2,1] => [2,1] => [1,1,0,0,1,0] => 1

[1,1,0,0,1,1,0,1,0,0] => [2,3] => [1,1] => [1,0,1,0] => 0

[1,1,0,0,1,1,1,0,0,0] => [2,3] => [1,1] => [1,0,1,0] => 0

[1,1,0,1,0,0,1,0,1,0] => [3,1,1] => [1,2] => [1,0,1,1,0,0] => 1

[1,1,0,1,0,0,1,1,0,0] => [3,2] => [1,1] => [1,0,1,0] => 0

[1,1,0,1,0,1,0,0,1,0] => [4,1] => [1,1] => [1,0,1,0] => 0

[1,1,0,1,0,1,0,1,0,0] => [5] => [1] => [1,0] => 0

[1,1,0,1,0,1,1,0,0,0] => [5] => [1] => [1,0] => 0

[1,1,0,1,1,0,0,0,1,0] => [4,1] => [1,1] => [1,0,1,0] => 0

[1,1,0,1,1,0,0,1,0,0] => [5] => [1] => [1,0] => 0

[1,1,0,1,1,0,1,0,0,0] => [5] => [1] => [1,0] => 0

[1,1,0,1,1,1,0,0,0,0] => [5] => [1] => [1,0] => 0

[1,1,1,0,0,0,1,0,1,0] => [3,1,1] => [1,2] => [1,0,1,1,0,0] => 1

[1,1,1,0,0,0,1,1,0,0] => [3,2] => [1,1] => [1,0,1,0] => 0

[1,1,1,0,0,1,0,0,1,0] => [4,1] => [1,1] => [1,0,1,0] => 0

[1,1,1,0,0,1,0,1,0,0] => [5] => [1] => [1,0] => 0

[1,1,1,0,0,1,1,0,0,0] => [5] => [1] => [1,0] => 0

[1,1,1,0,1,0,0,0,1,0] => [4,1] => [1,1] => [1,0,1,0] => 0

[1,1,1,0,1,0,0,1,0,0] => [5] => [1] => [1,0] => 0

[1,1,1,0,1,0,1,0,0,0] => [5] => [1] => [1,0] => 0

[1,1,1,0,1,1,0,0,0,0] => [5] => [1] => [1,0] => 0

[1,1,1,1,0,0,0,0,1,0] => [4,1] => [1,1] => [1,0,1,0] => 0

[1,1,1,1,0,0,0,1,0,0] => [5] => [1] => [1,0] => 0

[1,1,1,1,0,0,1,0,0,0] => [5] => [1] => [1,0] => 0

[1,1,1,1,0,1,0,0,0,0] => [5] => [1] => [1,0] => 0

[1,1,1,1,1,0,0,0,0,0] => [5] => [1] => [1,0] => 0

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 15

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 6

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 3

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0] => 3

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0] => 3

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 2

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0] => 2

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0] => 1

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,4] => [2,1] => [1,1,0,0,1,0] => 1

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,4] => [2,1] => [1,1,0,0,1,0] => 1

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0] => 1

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,4] => [2,1] => [1,1,0,0,1,0] => 1

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,4] => [2,1] => [1,1,0,0,1,0] => 1

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,4] => [2,1] => [1,1,0,0,1,0] => 1

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 0

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0] => 1

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,2,3] => [1,1,1] => [1,0,1,0,1,0] => 0

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,2,3] => [1,1,1] => [1,0,1,0,1,0] => 0

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0] => 1

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,2] => [1,1,1] => [1,0,1,0,1,0] => 0

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,4,1] => [1,1,1] => [1,0,1,0,1,0] => 0

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,5] => [1,1] => [1,0,1,0] => 0

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,5] => [1,1] => [1,0,1,0] => 0

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,4,1] => [1,1,1] => [1,0,1,0,1,0] => 0

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,5] => [1,1] => [1,0,1,0] => 0

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,5] => [1,1] => [1,0,1,0] => 0

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,5] => [1,1] => [1,0,1,0] => 0

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0] => 1

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,3,2] => [1,1,1] => [1,0,1,0,1,0] => 0

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,1] => [1,1,1] => [1,0,1,0,1,0] => 0

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,5] => [1,1] => [1,0,1,0] => 0

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,5] => [1,1] => [1,0,1,0] => 0

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,1] => [1,1,1] => [1,0,1,0,1,0] => 0

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,5] => [1,1] => [1,0,1,0] => 0

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,5] => [1,1] => [1,0,1,0] => 0

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,5] => [1,1] => [1,0,1,0] => 0

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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.

Map

delta morphism

Description

Apply the delta morphism to an integer composition.
The delta morphism of a composition $C$ is the compositions composed of the lengths of consecutive runs of the same integer in $C$.

Map

touch composition

Description

Sends a Dyck path to its touch composition given by the composition of lengths of its touch points.

Map

bounce path

Description

The bounce path determined by an integer composition.