- FindStat

Your data matches 490 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000013
(load all 51 compositions to match this statistic)

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The height of a Dyck path. The height of a Dyck path

$D$ of semilength

$n$ is defined as the maximal height of a peak of

$D$ . The height of

$D$ at position

$i$ is the number of up-steps minus the number of down-steps before position

$i$ .

Matching statistic: St000676
(load all 20 compositions to match this statistic)

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength

$n$ with

$k$ up steps in odd positions and

$k$ returns to the main diagonal are counted by the binomial coefficient

$\binom{n-1}{k-1}$ [3,4].

Matching statistic: St001615
(load all 2 compositions to match this statistic)

Mapping

Mp00047: Ordered trees —to poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001615: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[]
 => ([],1)
 => ([],1)
 => 0
[[]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[],[]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[],[]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 1
[[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 1
[[],[],[],[],[[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[],[[]],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[],[[],[]]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[[],[]],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[[],[],[]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[],[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[],[],[]],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[],[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[[]],[],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[],[]],[],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[],[],[]],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[],[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[],[],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ([(0,1)],2)
 => 1

Description

The number of join prime elements of a lattice. An element

$x$ of a lattice

$L$ is join-prime (or coprime) if

$x \leq a \vee b$ implies

$x \leq a$ or

$x \leq b$ for every

$a, b \in L$ .

Matching statistic: St001617
(load all 2 compositions to match this statistic)

Mapping

Mp00047: Ordered trees —to poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001617: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[]
 => ([],1)
 => ([],1)
 => 0
[[]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[],[]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[],[]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 1
[[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 1
[[],[],[],[],[[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[],[[]],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[],[[],[]]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[[],[]],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[[],[],[]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[],[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[],[],[]],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[],[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[[]],[],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[],[]],[],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[],[],[]],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[],[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[],[],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ([(0,1)],2)
 => 1

Description

The dimension of the space of valuations of a lattice. A valuation, or modular function, on a lattice

$L$ is a function

$v:L\mapsto\mathbb R$ satisfying

$v(a\vee b) + v(a\wedge b) = v(a) + v(b).$ It was shown by Birkhoff [1, thm. X.2], that a lattice with a positive valuation must be modular. This was sharpened by Fleischer and Traynor [2, thm. 1], which states that the modular functions on an arbitrary lattice are in bijection with the modular functions on its modular quotient [[Mp00196]]. Moreover, Birkhoff [1, thm. X.2] showed that the dimension of the space of modular functions equals the number of subsets of projective prime intervals.

Matching statistic: St001622
(load all 2 compositions to match this statistic)

Mapping

Mp00047: Ordered trees —to poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001622: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[]
 => ([],1)
 => ([],1)
 => 0
[[]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[],[]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[],[]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 1
[[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 1
[[],[],[],[],[[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[],[[]],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[],[[],[]]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[[],[]],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[[],[],[]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[],[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[],[],[]],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[[],[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[[]],[],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[],[]],[],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[],[],[]],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[[],[],[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 2
[[],[],[],[],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ([(0,1)],2)
 => 1

Description

The number of join-irreducible elements of a lattice. An element

$j$ of a lattice

$L$ is '''join irreducible''' if it is not the least element and if

$j=x\vee y$ , then

$j\in\{x,y\}$ for all

$x,y\in L$ .

Matching statistic: St000550

Mapping

Mp00047: Ordered trees —to poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St000550: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[],[]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[],[]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[],[],[],[],[[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[],[[]],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[],[[],[]]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[[],[]],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[[],[],[]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[],[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[],[],[]],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[],[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[]],[],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[],[]],[],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[],[],[]],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[],[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[],[],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ([(0,1)],2)
 => 2 = 1 + 1

Description

The number of modular elements of a lattice. A pair

$(x, y)$ of elements of a lattice

$L$ is a modular pair if for every

$z\geq y$ we have that

$(y\vee x) \wedge z = y \vee (x \wedge z)$ . An element

$x$ is left-modular if

$(x, y)$ is a modular pair for every

$y\in L$ , and is modular if both

$(x, y)$ and

$(y, x)$ are modular pairs for every

$y\in L$ .

Matching statistic: St000551

Mapping

Mp00047: Ordered trees —to poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St000551: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[],[]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[],[]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[],[],[],[],[[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[],[[]],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[],[[],[]]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[[],[]],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[[],[],[]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[],[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[],[],[]],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[],[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[]],[],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[],[]],[],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[],[],[]],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[],[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[],[],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ([(0,1)],2)
 => 2 = 1 + 1

Description

The number of left modular elements of a lattice. A pair

$(x, y)$ of elements of a lattice

$L$ is a modular pair if for every

$z\geq y$ we have that

$(y\vee x) \wedge z = y \vee (x \wedge z)$ . An element

$x$ is left-modular if

$(x, y)$ is a modular pair for every

$y\in L$ .

Matching statistic: St001616

Mapping

Mp00047: Ordered trees —to poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[],[]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[],[]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[],[],[],[],[[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[],[[]],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[],[[],[]]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[[],[]],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[[],[],[]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[],[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[],[],[]],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[],[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[]],[],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[],[]],[],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[],[],[]],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[],[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[],[],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ([(0,1)],2)
 => 2 = 1 + 1

Description

The number of neutral elements in a lattice. An element

$e$ of the lattice

$L$ is neutral if the sublattice generated by

$e$ ,

$x$ and

$y$ is distributive for all

$x, y \in L$ .

Matching statistic: St001720
(load all 4 compositions to match this statistic)

Mapping

Mp00047: Ordered trees —to poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001720: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[]
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
[[]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[],[]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[],[]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[],[],[],[],[[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[],[[]],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[],[[],[]]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[[],[]],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[[],[],[]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[],[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[],[],[]],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[[],[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[]],[],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[],[]],[],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[],[],[]],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[[],[],[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[],[],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ([(0,1)],2)
 => 2 = 1 + 1

Description

The minimal length of a chain of small intervals in a lattice. An interval

$[a, b]$ is small if

$b$ is a join of elements covering

$a$ .

Matching statistic: St001619

Mapping

Mp00047: Ordered trees —to poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001619: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[]
 => ([],1)
 => ([],1)
 => 2 = 0 + 2
[[]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 3 = 1 + 2
[[],[]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 3 = 1 + 2
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 3 = 1 + 2
[[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[[],[]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 3 = 1 + 2
[[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[],[[],[]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 3 = 1 + 2
[[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[],[],[],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 3 = 1 + 2
[[],[],[],[],[[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[],[],[],[[]],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[],[],[],[[],[]]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[],[],[[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[],[],[[],[]],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[],[],[[],[],[]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[],[[]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[],[[],[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[],[[],[],[]],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[],[[],[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[[]],[],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[[],[]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[[],[],[]],[],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[[],[],[],[]],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[[],[],[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[[],[],[],[],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ([(0,1)],2)
 => 3 = 1 + 2

Description

The number of non-isomorphic sublattices of a lattice.

The following 480 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001666The number of non-isomorphic subposets of a lattice which are lattices. St000007The number of saliances of the permutation. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000025The number of initial rises of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000080The rank of the poset. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001613The binary logarithm of the size of the center of a lattice. St001621The number of atoms of a lattice. St001631The number of simple modules

$S$ with

$dim Ext^1(S,A)=1$ in the incidence algebra

$A$ of the poset. St000189The number of elements in the poset. St000528The height of a poset. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001717The largest size of an interval in a poset. St000070The number of antichains in a poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St001664The number of non-isomorphic subposets of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St000521The number of distinct subtrees of an ordered tree. St000167The number of leaves of an ordered tree. St000397The Strahler number of a rooted tree. St000415The size of the automorphism group of the rooted tree underlying the ordered tree. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001058The breadth of the ordered tree. St000024The number of double up and double down steps of a Dyck path. St000298The order dimension or Dushnik-Miller dimension of a poset. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000381The largest part of an integer composition. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000527The width of the poset. St000542The number of left-to-right-minima of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000808The number of up steps of the associated bargraph. St000846The maximal number of elements covering an element of a poset. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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: St001471The magnitude of a Dyck path. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000035The number of left outer peaks of a permutation. St000053The number of valleys of the Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000141The maximum drop size of a permutation. St000211The rank of the set partition. St000245The number of ascents of a permutation. St000306The bounce count of a Dyck path. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000345The number of refinements of a partition. St000374The number of exclusive right-to-left minima of a permutation. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000834The number of right outer peaks of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000884The number of isolated descents of a permutation. St000935The number of ordered refinements of an integer partition. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001011Number of simple modules of projective dimension 2 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. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001389The number of partitions of the same length below the given integer partition. St001489The maximum of the number of descents and the number of inverse descents. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001651The Frankl number of a lattice. 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. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001928The number of non-overlapping descents in a permutation. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000038The product of the heights of the descending steps of a Dyck path. St000040The number of regions of the inversion arrangement of a permutation. St000058The order of a permutation. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000105The number of blocks in the set partition. St000110The number of permutations less than or equal to a permutation in left weak order. St000172The Grundy number of a graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000272The treewidth of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000327The number of cover relations in a poset. St000346The number of coarsenings of a partition. St000362The size of a minimal vertex cover of a graph. St000377The dinv defect of an integer partition. St000442The maximal area to the right of an up step of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000507The number of ascents of a standard tableau. St000536The pathwidth of a graph. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. 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. St000845The maximal number of elements covered by an element in a poset. St000883The number of longest increasing subsequences of a permutation. St000920The logarithmic height of a Dyck path. St001029The size of the core of a graph. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. St001176The size of a partition minus its first part. St001261The Castelnuovo-Mumford regularity of a graph. St001277The degeneracy 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. St001358The largest degree of a regular subgraph of a graph. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001670The connected partition number of a graph. St001792The arboricity of a graph. St001809The index of the step at the first peak of maximal height in a Dyck path. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001963The tree-depth of a graph. St000012The area of a Dyck path. St000016The number of attacking pairs of a standard tableau. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. 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. St000159The number of distinct parts of the integer partition. St000185The weighted size of a partition. St000209Maximum difference of elements in cycles. St000214The number of adjacencies of a permutation. St000237The number of small exceedances. St000291The number of descents of a binary word. St000336The leg major index of a standard tableau. St000340The number of non-final maximal constant sub-paths of length greater than one. St000352The Elizalde-Pak rank of a permutation. St000382The first part of an integer composition. St000387The matching number of a graph. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000439The position of the first down step of a Dyck path. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000535The rank-width of a graph. St000628The balance of a binary word. St000632The jump number of the poset. St000651The maximal size of a rise in a permutation. St000656The number of cuts of a poset. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000665The number of rafts of a permutation. St000670The reversal length of a permutation. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000734The last entry in the first row of a standard tableau. St000742The number of big ascents of a permutation after prepending zero. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000897The number of different multiplicities of parts of an integer partition. St000906The length of the shortest maximal chain in a poset. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000996The number of exclusive left-to-right maxima of a permutation. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001090The number of pop-stack-sorts needed to sort a permutation. St001092The number of distinct even parts of a partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. 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. St001333The cardinality of a minimal edge-isolating set of a graph. 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. St001393The induced matching number of a graph. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001462The number of factors of a standard tableaux under concatenation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001726The number of visible inversions of a permutation. St001743The discrepancy of a graph. St001801Half the number of preimage-image pairs of different parity in a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001961The sum of the greatest common divisors of all pairs of parts. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n-1}]$ by adding

$c_0$ to

$c_{n-1}$ . St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000166The depth minus 1 of an ordered tree. St000094The depth of an ordered tree. St000168The number of internal nodes of an ordered tree. St000328The maximum number of child nodes in a tree. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000919The number of maximal left branches of a binary tree. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000335The difference of lower and upper interactions. St000392The length of the longest run of ones in a binary word. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000982The length of the longest constant subword. St000991The number of right-to-left minima of a permutation. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001062The maximal size of a block of a set partition. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001372The length of a longest cyclic run of ones of a binary word. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001530The depth of a Dyck path. St000021The number of descents of a permutation. St000120The number of left tunnels of a Dyck path. St000154The sum of the descent bottoms of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000238The number of indices that are not small weak excedances. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000332The positive inversions of an alternating sign matrix. St000354The number of recoils 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. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000829The Ulam distance of a permutation to the identity permutation. St000933The number of multipartitions of sizes given by an integer partition. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . 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

$eA$ . 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. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001874Lusztig's a-function for the symmetric group. St000109The number of elements less than or equal to the given element in Bruhat order. St000164The number of short pairs. St000201The number of leaf nodes in a binary tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000390The number of runs of ones in a binary word. St000418The number of Dyck paths that are weakly below a Dyck path. St000485The length of the longest cycle of a permutation. St000707The product of the factorials of the parts. St000744The length of the path to the largest entry in a standard Young tableau. St000822The Hadwiger number of the graph. St000844The size of the largest block in the direct sum decomposition of a permutation. St000877The depth of the binary word interpreted as a path. St000925The number of topologically connected components of a set partition. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001202Call 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 special CNakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001330The hat guessing number of a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001652The length of a longest interval of consecutive numbers. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001662The length of the longest factor of consecutive numbers in a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000055The inversion sum of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000161The sum of the sizes of the right subtrees of a binary tree. St000224The sorting index of a permutation. St000288The number of ones in a binary word. St000292The number of ascents of a binary word. St000297The number of leading ones in a binary word. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000386The number of factors DDU in a Dyck path. St000389The number of runs of ones of odd length in a binary word. St000391The sum of the positions of the ones in a binary word. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000503The maximal difference between two elements in a common block. St000539The number of odd inversions of a permutation. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000640The rank of the largest boolean interval in a poset. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000730The maximal arc length of a set partition. St000753The Grundy value for the game of Kayles on a binary word. St000792The Grundy value for the game of ruler on a binary word. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000984The number of boxes below precisely one peak. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). 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. St001139The number of occurrences of hills of size 2 in a Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001427The number of descents of a signed permutation. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001592The maximal number of simple paths between any two different vertices of a graph. St001812The biclique partition number of a graph. 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). St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000702The number of weak deficiencies of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000061The number of nodes on the left branch of a binary tree. St000082The number of elements smaller than a binary tree in Tamari order. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001346The number of parking functions that give the same permutation. St001531Number of partial orders contained in the poset determined by the Dyck path. St001959The product of the heights of the peaks of a Dyck path. St000216The absolute length of a permutation. St000472The sum of the ascent bottoms of a permutation. St001480The number of simple summands of the module J^2/J^3. 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

$eA$ . St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000264The girth of a graph, which is not a tree. St000667The greatest common divisor of the parts of the partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St001175The size of a partition minus the hook length of the base cell. St001586The number of odd parts smaller than the largest even part in an integer partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000754The Grundy value for the game of removing nestings in a perfect matching. St000260The radius of a connected graph. St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001060The distinguishing index of a graph. St000699The toughness times the least common multiple of 1,. St001570The minimal number of edges to add to make a graph Hamiltonian. St001118The acyclic chromatic index of a graph. St001545The second Elser number of a connected graph. St000456The monochromatic index of a connected graph. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. 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. St000455The second largest eigenvalue of a graph if it is integral. St000478Another weight of a partition according to Alladi. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000464The Schultz index of a connected graph. St000862The number of parts of the shifted shape of a permutation. St001394The genus of a permutation. St000331The number of upper interactions of a Dyck path. St000619The number of cyclic descents of a permutation. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001674The number of vertices of the largest induced star graph in the graph. St001589The nesting number of a perfect matching. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000746The number of pairs with odd minimum in a perfect matching. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001323The independence gap of a graph. St001590The crossing number of a perfect matching. St000317The cycle descent number of a permutation. St000353The number of inner valleys of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St001152The number of pairs with even minimum in a perfect matching. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001684The reduced word complexity of a permutation. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001569The maximal modular displacement of a permutation. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St000630The length of the shortest palindromic decomposition of a binary word. St001896The number of right descents of a signed permutations. 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

$eA$ . St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000299The number of nonisomorphic vertex-induced subtrees. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000758The length of the longest staircase fitting into an integer composition. St000942The number of critical left to right maxima of the parking functions. St000983The length of the longest alternating subword. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001171The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001712The number of natural descents of a standard Young tableau. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000396The register function (or Horton-Strahler number) of a binary tree. St000366The number of double descents of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St001820The size of the image of the pop stack sorting operator. St000932The number of occurrences of the pattern UDU in a Dyck path. St001864The number of excedances of a signed permutation. St001626The number of maximal proper sublattices of a lattice. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000174The flush statistic of a semistandard tableau. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000486The number of cycles of length at least 3 of a permutation. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001728The number of invisible descents of a permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion.