- FindStat

Identifier

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001203: Dyck paths ⟶ ℤ (values match St000306The bounce count of a Dyck path.)

Values

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

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

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

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

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

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

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

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

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

[1,2,3,4] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 1

[1,2,4,3] => [1,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => 2

[1,3,2,4] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => 2

[1,3,4,2] => [1,0,1,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => 2

[1,4,2,3] => [1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => 3

[1,4,3,2] => [1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => 3

[2,1,3,4] => [1,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,0] => 2

[2,1,4,3] => [1,1,0,0,1,1,0,0] => [1,1,0,1,0,1,0,0] => 2

[2,3,1,4] => [1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 2

[2,3,4,1] => [1,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0] => 2

[2,4,1,3] => [1,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,0] => 3

[2,4,3,1] => [1,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,0] => 3

[3,1,2,4] => [1,1,1,0,0,0,1,0] => [1,0,1,1,0,1,0,0] => 3

[3,1,4,2] => [1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => 3

[3,2,1,4] => [1,1,1,0,0,0,1,0] => [1,0,1,1,0,1,0,0] => 3

[3,2,4,1] => [1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => 3

[3,4,1,2] => [1,1,1,0,1,0,0,0] => [1,1,0,0,1,0,1,0] => 3

[3,4,2,1] => [1,1,1,0,1,0,0,0] => [1,1,0,0,1,0,1,0] => 3

[4,1,2,3] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 4

[4,1,3,2] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 4

[4,2,1,3] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 4

[4,2,3,1] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 4

[4,3,1,2] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 4

[4,3,2,1] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 4

[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1

[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => 2

[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => 2

[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 3

[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 3

[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 2

[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 2

[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => 2

[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 2

[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,0,0] => 3

[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,0,0] => 3

[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => 3

[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => 3

[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => 3

[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => 3

[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0] => [1,1,0,0,1,0,1,1,0,0] => 3

[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0] => [1,1,0,0,1,0,1,1,0,0] => 3

[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 4

[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 4

[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 4

[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 4

[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 4

[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 4

[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,1,0,0,0,0] => 2

[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 2

[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => 2

[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,1,0,0] => 2

[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => 3

[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => 3

[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 2

[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,0,0] => 2

[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 2

[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 2

[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => 3

[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => 3

[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => 3

[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 3

[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => 3

[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 3

[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,1,0,0,1,0] => 3

[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,1,0,0,1,0] => 3

[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => 4

[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => 4

[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => 4

[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => 4

[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => 4

[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => 4

[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0] => [1,0,1,1,1,0,1,0,0,0] => 3

[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0] => 3

[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 3

[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => 3

[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0] => 3

[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0] => 3

[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => [1,0,1,1,1,0,1,0,0,0] => 3

[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0] => 3

[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 3

[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => 3

[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0] => 3

[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0] => 3

[3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => 3

[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => 3

[3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => 3

[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => 3

[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => 3

[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => 3

[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => 4

[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => 4

>>> Load all 5913 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

click to show known generating functions

Description

We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
In the list $L$ delete the first entry $c_0$ and substract from all other entries $n-1$ and then append the last element 1 (this was suggested by Christian Stump). The result is a Kupisch series of an LNakayama algebra.
Example:
[5,6,6,6,6] goes into [2,2,2,2,1].
Now associate to the CNakayama algebra with the above properties the Dyck path corresponding to the Kupisch series of the LNakayama algebra.
The statistic return the global dimension of the CNakayama algebra divided by 2.

Map

left-to-right-maxima to Dyck path

Description

The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.

Map

zeta map

Description

The zeta map on Dyck paths.
The zeta map $\zeta$ is a bijection on Dyck paths of semilength $n$.
It was defined in [1, Theorem 1], see also [2, Theorem 3.15] and sends the bistatistic (area, dinv) to the bistatistic (bounce, area). It is defined by sending a Dyck path $D$ with corresponding area sequence $a=(a_1,\ldots,a_n)$ to a Dyck path as follows:

First, build an intermediate Dyck path consisting of $d_1$ north steps, followed by $d_1$ east steps, followed by $d_2$ north steps and $d_2$ east steps, and so on, where $d_i$ is the number of $i-1$'s within the sequence $a$.
For example, given $a=(0,1,2,2,2,3,1,2)$, we build the path
$$NE\ NNEE\ NNNNEEEE\ NE.$$
Next, the rectangles between two consecutive peaks are filled. Observe that such the rectangle between the $k$th and the $(k+1)$st peak must be filled by $d_k$ east steps and $d_{k+1}$ north steps. In the above example, the rectangle between the second and the third peak must be filled by $2$ east and $4$ north steps, the $2$ being the number of $1$'s in $a$, and $4$ being the number of $2$'s. To fill such a rectangle, scan through the sequence a from left to right, and add east or north steps whenever you see a $k-1$ or $k$, respectively. So to fill the $2\times 4$ rectangle, we look for $1$'s and $2$'s in the sequence and see $122212$, so this rectangle gets filled with $ENNNEN$.
The complete path we obtain in thus
$$NENNENNNENEEENEE.$$