- FindStat

Identifier

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001126: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

[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] => 1

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

[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] => 0

[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] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 5

[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] => 3

[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] => 3

[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] => 3

[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] => 2

[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] => 2

[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] => 3

[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] => 1

[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] => 3

[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] => 3

[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] => 1

[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] => 1

[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] => 2

[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] => 2

[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] => 2

[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] => 2

[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] => 2

[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] => 2

[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] => 1

[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] => 1

[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] => 1

[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] => 1

[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] => 1

[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] => 1

[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] => 3

[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] => 1

[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] => 1

[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] => 1

[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] => 0

[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] => 0

[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] => 3

[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] => 1

[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] => 3

[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] => 3

[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] => 1

[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] => 1

[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] => 1

[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] => 1

[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] => 1

[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] => 1

[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] => 1

[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] => 1

[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] => 0

[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] => 0

[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] => 0

[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] => 0

[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] => 0

[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] => 0

[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] => 2

[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] => 0

[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] => 2

[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] => 2

[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] => 0

[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] => 0

[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] => 2

[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] => 0

[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] => 2

[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] => 2

[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] => 0

[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] => 0

[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] => 2

[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] => 2

[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] => 2

[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] => 2

[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] => 2

[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] => 2

[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] => 0

[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] => 0

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$F_{1} = q$

$F_{2} = 1 + q^{2}$

$F_{3} = 2 + 3\ q + q^{3}$

$F_{4} = 9 + 8\ q + 6\ q^{2} + q^{4}$

$F_{5} = 44 + 45\ q + 20\ q^{2} + 10\ q^{3} + q^{5}$

$F_{6} = 265 + 264\ q + 135\ q^{2} + 40\ q^{3} + 15\ q^{4} + q^{6}$

$F_{7} = 1854 + 1855\ q + 924\ q^{2} + 315\ q^{3} + 70\ q^{4} + 21\ q^{5} + q^{7}$

Description

Number of simple module that are 1-regular in the corresponding Nakayama algebra.

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
$NENNENNNENEEENEE.$

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.