Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St001813
(load all 4 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001813: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => ([],1)
 => 1
[1,0,1,0]
 => [1,2] => ([(0,1)],2)
 => 2
[1,1,0,0]
 => [2,1] => ([],2)
 => 1
[1,0,1,0,1,0]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 6
[1,0,1,1,0,0]
 => [1,3,2] => ([(0,1),(0,2)],3)
 => 3
[1,1,0,0,1,0]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => 4
[1,1,0,1,0,0]
 => [2,3,1] => ([(1,2)],3)
 => 2
[1,1,1,0,0,0]
 => [3,2,1] => ([],3)
 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 24
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => 12
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 16
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => 8
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 4
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 18
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 9
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 12
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 6
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => ([(1,2),(1,3)],4)
 => 3
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 8
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 4
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(2,3)],4)
 => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([],4)
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 120
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => 60
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 80
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 40
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => 20
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 90
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 45
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 60
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 30
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => 15
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 40
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 20
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => 10
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 5
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 96
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => 48
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => 64
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => 32
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 16
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 72
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => 36
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 48
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => 24
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
 => 12
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 32
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
 => 16
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
 => 8
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => 4
Description
The product of the sizes of the principal order filters in a poset.
Matching statistic: St001808
(load all 4 compositions to match this statistic)
Mapping
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001808: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => ? = 1
[1,0,1,0]
 => [1,0,1,0]
 => 2
[1,1,0,0]
 => [1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 6
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 3
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 4
[1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 24
[1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 12
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 16
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 8
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 4
[1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 18
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 9
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 12
[1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 6
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3
[1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 8
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 4
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 120
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 60
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 80
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 40
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 20
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 90
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 45
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 60
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 30
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 15
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 40
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 20
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 10
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 5
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 96
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 48
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 64
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 32
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 16
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 72
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 36
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 48
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 24
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 12
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 32
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 16
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 8
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 4
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 54
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.