Your data matches 361 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000920
(load all 58 compositions to match this statistic)
Mapping
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 1
[1,0,1,0]
 => 1
[1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => 1
[1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => 1
[1,1,1,0,0,0]
 => 2
[1,0,1,0,1,0,1,0]
 => 1
[1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => 2
[1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => 2
[1,1,1,0,0,1,0,0]
 => 2
[1,1,1,0,1,0,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => 2
[1,1,1,0,0,0,1,1,0,0]
 => 2
[1,1,1,0,0,1,0,0,1,0]
 => 2
[1,1,1,0,0,1,0,1,0,0]
 => 2
Description
The logarithmic height of a Dyck path. This is the floor of the binary logarithm of the usual height increased by one: $$ \lfloor\log_2(1+height(D))\rfloor $$
Matching statistic: St000201
(load all 30 compositions to match this statistic)
Mapping
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [.,.]
 => 1
[1,0,1,0]
 => [.,[.,.]]
 => 1
[1,1,0,0]
 => [[.,.],.]
 => 1
[1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => 1
[1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => 1
[1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => 1
[1,1,0,1,0,0]
 => [[[.,.],.],.]
 => 1
[1,1,1,0,0,0]
 => [[.,.],[.,.]]
 => 2
[1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => 1
[1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => 1
[1,0,1,1,0,0,1,0]
 => [.,[[.,[.,.]],.]]
 => 1
[1,0,1,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => 1
[1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => 2
[1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => 1
[1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => 1
[1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => 1
[1,1,0,1,0,1,0,0]
 => [[[[.,.],.],.],.]
 => 1
[1,1,0,1,1,0,0,0]
 => [[[.,.],[.,.]],.]
 => 2
[1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => 2
[1,1,1,0,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => 2
[1,1,1,0,1,0,0,0]
 => [[.,[.,.]],[.,.]]
 => 2
[1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,[.,.]],.]]]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [.,[[.,[.,[.,.]]],.]]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [.,[[[.,[.,.]],.],.]]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [.,[[.,.],[.,[.,.]]]]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [[.,[[.,[.,.]],.]],.]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [[.,[[[.,.],.],.]],.]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [[[.,[[.,.],.]],.],.]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [[[[[.,.],.],.],.],.]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [[[.,.],[[.,.],.]],.]
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [[.,.],[.,[.,[.,.]]]]
 => 2
[1,1,1,0,0,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => 2
[1,1,1,0,0,1,0,0,1,0]
 => [[.,.],[[.,[.,.]],.]]
 => 2
[1,1,1,0,0,1,0,1,0,0]
 => [[.,.],[[[.,.],.],.]]
 => 2
Description
The number of leaf nodes in a binary tree. Equivalently, the number of cherries [1] in the complete binary tree. The number of binary trees of size $n$, at least $1$, with exactly one leaf node for is $2^{n-1}$, see [2]. The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].
Matching statistic: St000298
(load all 14 compositions to match this statistic)
Mapping
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => ([],1)
 => 1
[1,0,1,0]
 => ([(0,1)],2)
 => 1
[1,1,0,0]
 => ([(0,1)],2)
 => 1
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 1
[1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2
[1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,1,0,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2
[1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[1,1,1,1,0,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[1,1,1,0,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[1,1,1,0,0,1,0,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
Description
The order dimension or Dushnik-Miller dimension of a poset. This is the minimal number of linear orderings whose intersection is the given poset.
Matching statistic: St000307
(load all 7 compositions to match this statistic)
Mapping
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => ([],1)
 => 1
[1,0,1,0]
 => ([(0,1)],2)
 => 1
[1,1,0,0]
 => ([(0,1)],2)
 => 1
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 1
[1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2
[1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,1,0,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2
[1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[1,1,1,1,0,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[1,1,1,0,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[1,1,1,0,0,1,0,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
Description
The number of rowmotion orbits of a poset. Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
Matching statistic: St000396
(load all 38 compositions to match this statistic)
Mapping
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [.,.]
 => 1
[1,0,1,0]
 => [.,[.,.]]
 => 1
[1,1,0,0]
 => [[.,.],.]
 => 1
[1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => 1
[1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => 1
[1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => 1
[1,1,0,1,0,0]
 => [[[.,.],.],.]
 => 1
[1,1,1,0,0,0]
 => [[.,.],[.,.]]
 => 2
[1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => 1
[1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => 1
[1,0,1,1,0,0,1,0]
 => [.,[[.,[.,.]],.]]
 => 1
[1,0,1,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => 1
[1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => 2
[1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => 1
[1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => 1
[1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => 1
[1,1,0,1,0,1,0,0]
 => [[[[.,.],.],.],.]
 => 1
[1,1,0,1,1,0,0,0]
 => [[[.,.],[.,.]],.]
 => 2
[1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => 2
[1,1,1,0,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => 2
[1,1,1,0,1,0,0,0]
 => [[.,[.,.]],[.,.]]
 => 2
[1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,[.,.]],.]]]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [.,[[.,[.,[.,.]]],.]]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [.,[[[.,[.,.]],.],.]]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [.,[[.,.],[.,[.,.]]]]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [[.,[[.,[.,.]],.]],.]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [[.,[[[.,.],.],.]],.]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [[[.,[[.,.],.]],.],.]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [[[[[.,.],.],.],.],.]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [[[.,.],[[.,.],.]],.]
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [[.,.],[.,[.,[.,.]]]]
 => 2
[1,1,1,0,0,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => 2
[1,1,1,0,0,1,0,0,1,0]
 => [[.,.],[[.,[.,.]],.]]
 => 2
[1,1,1,0,0,1,0,1,0,0]
 => [[.,.],[[[.,.],.],.]]
 => 2
Description
The register function (or Horton-Strahler number) of a binary tree. This is different from the dimension of the associated poset for the tree $[[[.,.],[.,.]],[[.,.],[.,.]]]$: its register function is 3, whereas the dimension of the associated poset is 2.
Matching statistic: St000527
(load all 13 compositions to match this statistic)
Mapping
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000527: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => ([],1)
 => 1
[1,0,1,0]
 => ([(0,1)],2)
 => 1
[1,1,0,0]
 => ([(0,1)],2)
 => 1
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 1
[1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2
[1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,1,0,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2
[1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[1,1,1,1,0,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[1,1,1,0,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[1,1,1,0,0,1,0,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
Description
The width of the poset. This is the size of the poset's longest antichain, also called Dilworth number.
Matching statistic: St001597
(load all 8 compositions to match this statistic)
Mapping
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001597: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [[1],[]]
 => 1
[1,0,1,0]
 => [[1,1],[]]
 => 1
[1,1,0,0]
 => [[2],[]]
 => 1
[1,0,1,0,1,0]
 => [[1,1,1],[]]
 => 1
[1,0,1,1,0,0]
 => [[2,1],[]]
 => 1
[1,1,0,0,1,0]
 => [[2,2],[1]]
 => 1
[1,1,0,1,0,0]
 => [[3],[]]
 => 1
[1,1,1,0,0,0]
 => [[2,2],[]]
 => 2
[1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => 1
[1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => 1
[1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => 1
[1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => 1
[1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => 2
[1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => 1
[1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => 1
[1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => 1
[1,1,0,1,0,1,0,0]
 => [[4],[]]
 => 1
[1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => 2
[1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => 2
[1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => 2
[1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => 2
[1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1],[]]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [[2,1,1,1],[]]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1],[]]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1],[]]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1],[]]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [[4,4],[2]]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [[4,3],[1]]
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => 2
[1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => 2
[1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => 2
[1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => 2
Description
The Frobenius rank of a skew partition. This is the minimal number of border strips in a border strip decomposition of the skew partition.
Matching statistic: St000196
(load all 30 compositions to match this statistic)
Mapping
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [.,.]
 => 0 = 1 - 1
[1,0,1,0]
 => [.,[.,.]]
 => 0 = 1 - 1
[1,1,0,0]
 => [[.,.],.]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => [[[.,.],.],.]
 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [[.,.],[.,.]]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [.,[[.,[.,.]],.]]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [[[[.,.],.],.],.]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [[[.,.],[.,.]],.]
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [[.,[.,.]],[.,.]]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,[.,.]],.]]]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [.,[[.,[.,[.,.]]],.]]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [.,[[[.,[.,.]],.],.]]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [.,[[.,.],[.,[.,.]]]]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [[.,[[.,[.,.]],.]],.]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [[.,[[[.,.],.],.]],.]
 => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [[[.,[[.,.],.]],.],.]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [[[[[.,.],.],.],.],.]
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [[[.,.],[[.,.],.]],.]
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [[.,.],[.,[.,[.,.]]]]
 => 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [[.,.],[[.,[.,.]],.]]
 => 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [[.,.],[[[.,.],.],.]]
 => 1 = 2 - 1
Description
The number of occurrences of the contiguous pattern {{{[[.,.],[.,.]]}}} in a binary tree. Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by $2^{n-2}$.
Matching statistic: St000632
(load all 7 compositions to match this statistic)
Mapping
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0]
 => ([(0,1)],2)
 => 0 = 1 - 1
[1,1,0,0]
 => ([(0,1)],2)
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1 = 2 - 1
Description
The jump number of the poset. A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.
Matching statistic: St001022
(load all 21 compositions to match this statistic)
Mapping
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001022: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => 0 = 1 - 1
[1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0]
 => [1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
Description
Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path.
The following 351 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000010The length of the partition. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000159The number of distinct parts of the integer partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000346The number of coarsenings of a partition. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000482The (zero)-forcing number of a graph. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000544The cop number of a graph. St000679The pruning number of an ordered tree. St000758The length of the longest staircase fitting into an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000862The number of parts of the shifted shape of a permutation. St001029The size of the core of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001432The order dimension of the partition. St001484The number of singletons of an integer partition. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001732The number of peaks visible from the left. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. St001883The mutual visibility number of a graph. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000035The number of left outer peaks of a permutation. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000272The treewidth of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000386The number of factors DDU in a Dyck path. St000481The number of upper covers of a partition in dominance order. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000659The number of rises of length at least 2 of a Dyck path. St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001083The number of boxed occurrences of 132 in a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001331The size of the minimal feedback vertex set. St001333The cardinality of a minimal edge-isolating set of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001358The largest degree of a regular subgraph of a graph. St001393The induced matching number of a graph. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001469The holeyness of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001638The book thickness of a graph. St001644The dimension of a graph. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001743The discrepancy of a graph. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001792The arboricity of a graph. St001826The maximal number of leaves on a vertex of a graph. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001928The number of non-overlapping descents in a permutation. St001962The proper pathwidth of a graph. St000054The first entry of the permutation. St000058The order of a permutation. St000069The number of maximal elements of a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000172The Grundy number of a graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000378The diagonal inversion number of an integer partition. St000542The number of left-to-right-minima of a permutation. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000935The number of ordered refinements of an integer partition. St001057The Grundy value of the game of creating an independent set in a graph. St001116The game chromatic number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001389The number of partitions of the same length below the given integer partition. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001716The 1-improper chromatic number of a graph. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001963The tree-depth of a graph. St000028The number of stack-sorts needed to sort a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000142The number of even parts of a partition. St000157The number of descents of a standard tableau. St000185The weighted size of a partition. St000225Difference between largest and smallest parts in a partition. St000245The number of ascents of a permutation. St000306The bounce count of a Dyck path. St000336The leg major index of a standard tableau. St000352The Elizalde-Pak rank of a permutation. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000480The number of lower covers of a partition in dominance order. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000647The number of big descents of a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000672The number of minimal elements in Bruhat order not less than the permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000759The smallest missing part in an integer partition. St000834The number of right outer peaks of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000996The number of exclusive left-to-right maxima of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001092The number of distinct even parts of a partition. St001176The size of a partition minus its first part. St001214The aft of an integer partition. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001280The number of parts of an integer partition that are at least two. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001587Half of the largest even part of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001971The number of negative eigenvalues of the adjacency matrix of the graph. St000619The number of cyclic descents of a permutation. St000354The number of recoils of a permutation. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000624The normalized sum of the minimal distances to a greater element. St000829The Ulam distance of a permutation to the identity permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001592The maximal number of simple paths between any two different vertices of a graph. St000485The length of the longest cycle of a permutation. St000568The hook number of a binary tree. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001081The number of minimal length factorizations of a permutation into star transpositions. St001128The exponens consonantiae of a partition. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000288The number of ones in a binary word. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000353The number of inner valleys of a permutation. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000486The number of cycles of length at least 3 of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000628The balance of a binary word. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000730The maximal arc length of a set partition. St000753The Grundy value for the game of Kayles on a binary word. St000779The tier of a permutation. St000809The reduced reflection length of the permutation. St000919The number of maximal left branches of a binary tree. St000940The number of characters of the symmetric group whose value on the partition is zero. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St000454The largest eigenvalue of a graph if it is integral. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001323The independence gap of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000257The number of distinct parts of a partition that occur at least twice. St000183The side length of the Durfee square of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001330The hat guessing number of a graph. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001394The genus of a permutation. St001728The number of invisible descents of a permutation. St000358The number of occurrences of the pattern 31-2. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000264The girth of a graph, which is not a tree. St001498The normalised height of a Nakayama algebra with magnitude 1. St000640The rank of the largest boolean interval in a poset. St000360The number of occurrences of the pattern 32-1. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St000099The number of valleys of a permutation, including the boundary. St000325The width of the tree associated to a permutation. St000822The Hadwiger number of the graph. St001734The lettericity of a graph. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000397The Strahler number of a rooted tree. St000523The number of 2-protected nodes of a rooted tree. St000646The number of big ascents of a permutation. St001874Lusztig's a-function for the symmetric group. St000024The number of double up and double down steps of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000092The number of outer peaks of a permutation. St000308The height of the tree associated to a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000409The number of pitchforks in a binary tree. St000658The number of rises of length 2 of a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000889The number of alternating sign matrices with the same antidiagonal sums. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001203We 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: St001235The global dimension of the corresponding Comp-Nakayama algebra. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001471The magnitude of a Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001642The Prague dimension of a graph. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000154The sum of the descent bottoms of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000317The cycle descent number of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001427The number of descents of a signed permutation. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001812The biclique partition number of a graph. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000702The number of weak deficiencies of a permutation. St000216The absolute length of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000906The length of the shortest maximal chain in a poset. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001877Number of indecomposable injective modules with projective dimension 2. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St001513The number of nested exceedences of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000914The sum of the values of the Möbius function of a poset. St001964The interval resolution global dimension of a poset. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St000181The number of connected components of the Hasse diagram for the poset. St000314The number of left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000991The number of right-to-left minima of a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001530The depth of a Dyck path. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000252The number of nodes of degree 3 of a binary tree. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000357The number of occurrences of the pattern 12-3. St000443The number of long tunnels of a Dyck path. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001549The number of restricted non-inversions between exceedances. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001871The number of triconnected components of a graph. St001896The number of right descents of a signed permutations. St001960The number of descents of a permutation minus one if its first entry is not one. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001890The maximum magnitude of the Möbius function of a poset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001487The number of inner corners of a skew partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St000805The number of peaks of the associated bargraph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St000807The sum of the heights of the valleys of the associated bargraph. St001823The Stasinski-Voll length of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001060The distinguishing index of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001570The minimal number of edges to add to make a graph Hamiltonian. St000007The number of saliances of the permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000359The number of occurrences of the pattern 23-1. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000731The number of double exceedences of a permutation. St000635The number of strictly order preserving maps of a poset into itself. St001490The number of connected components of a skew partition. St000842The breadth of a permutation.