- FindStat

Identifier

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001808: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 24

[1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 18

[1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 16

[1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 12

[1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 8

[1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 12

[1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => 9

[1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => 8

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 120

[1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => 96

[1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => 90

[1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => 72

[1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => 54

[1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => 80

[1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => 64

[1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => 60

[1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => 48

[1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => 36

[1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 40

[1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => 32

[1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => 24

[1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 16

[1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => 60

[1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => 48

[1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => 45

[1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => 36

[1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 27

[1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => 40

[1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => 32

[1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => 30

[1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => 24

[1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 18

[1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => 20

[1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => 16

[1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => 12

[1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 8

[1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => 20

[1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 16

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

[1,1,1,0,0,1,0,1,0,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => 12

[1,1,1,0,0,1,1,0,0,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => 9

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

[1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => 8

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 720

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,1,1,0,0,0] => 600

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,1,1,0,0,1,0,0] => 576

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,0,1,1,0,1,0,0,0] => 480

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,1,1,1,0,0,0,0] => 384

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,1,0,0] => 540

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,1,1,0,0,0] => 450

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,0,1,1,0,1,0,0,1,0,0] => 432

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,0,1,1,0,1,0,1,0,0,0] => 360

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,1,1,0,0,0,0] => 288

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,1,0,0,0,1,0,0] => 324

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,0,1,1,1,0,0,1,0,0,0] => 270

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,1,1,0,1,0,0,0,0] => 216

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0] => 162

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,1,0,0] => 480

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,1,0,1,1,0,0,0] => 400

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,1,0,0,1,0,0] => 384

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,1,1,0,1,0,0,0] => 320

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,0,1,1,0,0,1,1,1,0,0,0,0] => 256

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,0,1,1,0,1,0,0,1,0,1,0,0] => 360

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,1,1,0,0,0] => 300

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,1,0,0,1,0,0] => 288

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,1,0,1,0,1,0,1,0,0,0] => 240

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,1,1,0,0,0,0] => 192

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,1,1,0,0,0,1,0,0] => 216

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,1,1,0,0,1,0,0,0] => 180

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,0,1,1,0,1,1,0,1,0,0,0,0] => 144

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,1,0,1,1,1,0,0,0,0,0] => 108

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,1,0,0] => 240

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,1,1,0,0,0,1,1,0,0,0] => 200

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,1,1,0,0,1,0,0,1,0,0] => 192

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,1,0,0,0] => 160

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,1,1,0,0,1,1,0,0,0,0] => 128

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,1,1,0,1,0,0,0,1,0,0] => 144

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,1,0,1,0,0,1,0,0,0] => 120

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,1,1,0,1,0,1,0,0,0,0] => 96

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,1,0,1,1,0,0,0,0,0] => 72

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$F_{1} = q$

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

$F_{3} = q + q^{2} + q^{3} + q^{4} + q^{6}$

$F_{4} = q + q^{2} + q^{3} + 2\ q^{4} + q^{6} + 2\ q^{8} + q^{9} + 2\ q^{12} + q^{16} + q^{18} + q^{24}$

$F_{5} = q + q^{2} + q^{3} + 2\ q^{4} + q^{5} + q^{6} + 2\ q^{8} + q^{9} + q^{10} + 2\ q^{12} + q^{15} + 3\ q^{16} + q^{18} + 2\ q^{20} + 2\ q^{24} + q^{27} + q^{30} + 2\ q^{32} + 2\ q^{36} + 2\ q^{40} + q^{45} + 2\ q^{48} + q^{54} + 2\ q^{60} + q^{64} + q^{72} + q^{80} + q^{90} + q^{96} + q^{120}$

$F_{6} = q + q^{2} + q^{3} + 2\ q^{4} + q^{5} + 2\ q^{6} + 2\ q^{8} + q^{9} + q^{10} + 3\ q^{12} + q^{15} + 3\ q^{16} + 2\ q^{18} + 2\ q^{20} + 4\ q^{24} + q^{25} + q^{27} + 2\ q^{30} + 3\ q^{32} + 3\ q^{36} + 2\ q^{40} + q^{45} + 5\ q^{48} + q^{50} + 2\ q^{54} + 3\ q^{60} + 3\ q^{64} + 4\ q^{72} + q^{75} + 3\ q^{80} + q^{81} + 2\ q^{90} + 5\ q^{96} + 2\ q^{100} + 3\ q^{108} + 4\ q^{120} + 2\ q^{128} + q^{135} + 4\ q^{144} + q^{150} + 2\ q^{160} + 2\ q^{162} + 3\ q^{180} + 4\ q^{192} + 2\ q^{200} + 3\ q^{216} + q^{225} + 4\ q^{240} + q^{256} + 2\ q^{270} + 3\ q^{288} + 2\ q^{300} + q^{320} + q^{324} + 3\ q^{360} + 2\ q^{384} + q^{400} + q^{432} + q^{450} + 2\ q^{480} + q^{540} + q^{576} + q^{600} + q^{720}$

Description

The box weight or horizontal decoration of a Dyck path.
Let a Dyck path

$D = (d_1,d_2,\dots,d_n)$ with steps

$d_i \in \{N=(0,1),E=(1,0)\}$ be given.
For the

$i$ th step

$d_i \in D$ we define the weight

$\beta(d_i) = 1, \quad \text{ if } d_i=N,$
and

$\beta(d_i) = \sum_{k = 1}^{i} [\![ d_k = N]\!], \quad \text{ if } d_i=E,$
where we use the Iverson bracket

$[\![ A ]\!]$ that is equal to

$1$ if

$A$ is true, and

$0$ otherwise.
The box weight or horizontal deocration of

$D$ is defined as

$\prod_{i=1}^{n} \beta(d_i).$
The name describes the fact that between each

$E$ step and the line

$y=-1$ exactly one unit box is marked.

Map

prime Dyck path

Description

Return the Dyck path obtained by adding an initial up and a final down step.