Identifier
Values
[1,0] => 10 => [1,2] => [1,0,1,1,0,0] => 1
[1,0,1,0] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 2
[1,1,0,0] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3
[1,0,1,0,1,0] => 101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 3
[1,0,1,1,0,0] => 101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => 4
[1,1,0,0,1,0] => 110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => 4
[1,1,0,1,0,0] => 110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => 4
[1,1,1,0,0,0] => 111000 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 6
[] => => [1] => [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
bounce path
Description
The bounce path determined by an integer composition.
Map
to binary word
Description
Return the Dyck word as binary word.
Map
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.