searching the database
Your data matches 265 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000632
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00047: Ordered trees —to poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000632: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> 0
[[]]
=> ([(0,1)],2)
=> 0
[[],[]]
=> ([(0,2),(1,2)],3)
=> 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> 0
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 2
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 2
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 3
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 3
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 3
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 1
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 3
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 1
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 1
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 1
[[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> 3
[[[],[],[[]]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> 2
[[[],[[]],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> 2
[[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 1
[[[[]],[],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> 2
[[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> 1
[[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 1
[[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> 2
[[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 1
[[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 1
[[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 1
[[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[[],[],[],[],[],[]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[[],[],[],[],[[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 4
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: St000068
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00047: Ordered trees —to poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> 1 = 0 + 1
[[]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2 = 1 + 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 3 = 2 + 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2 = 1 + 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 4 + 1
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2 = 1 + 1
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2 = 1 + 1
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2 = 1 + 1
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2 = 1 + 1
[[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> 4 = 3 + 1
[[[],[],[[]]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> 3 = 2 + 1
[[[],[[]],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> 3 = 2 + 1
[[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2 = 1 + 1
[[[[]],[],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> 3 = 2 + 1
[[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> 2 = 1 + 1
[[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2 = 1 + 1
[[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> 3 = 2 + 1
[[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2 = 1 + 1
[[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2 = 1 + 1
[[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2 = 1 + 1
[[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[[],[],[],[],[],[]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6 = 5 + 1
[[],[],[],[],[[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 5 = 4 + 1
Description
The number of minimal elements in a poset.
Matching statistic: St000071
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00047: Ordered trees —to poset⟶ Posets
St000071: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000071: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> 1 = 0 + 1
[[]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2 = 1 + 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 3 = 2 + 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2 = 1 + 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 4 + 1
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2 = 1 + 1
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2 = 1 + 1
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2 = 1 + 1
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2 = 1 + 1
[[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> 4 = 3 + 1
[[[],[],[[]]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> 3 = 2 + 1
[[[],[[]],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> 3 = 2 + 1
[[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2 = 1 + 1
[[[[]],[],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> 3 = 2 + 1
[[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> 2 = 1 + 1
[[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2 = 1 + 1
[[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> 3 = 2 + 1
[[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2 = 1 + 1
[[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2 = 1 + 1
[[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2 = 1 + 1
[[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[[],[],[],[],[],[]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6 = 5 + 1
[[],[],[],[],[[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 5 = 4 + 1
Description
The number of maximal chains in a poset.
Matching statistic: St000527
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00047: Ordered trees —to poset⟶ Posets
St000527: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000527: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> 1 = 0 + 1
[[]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2 = 1 + 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 3 = 2 + 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2 = 1 + 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 4 + 1
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2 = 1 + 1
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2 = 1 + 1
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 2 + 1
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2 = 1 + 1
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2 = 1 + 1
[[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> 4 = 3 + 1
[[[],[],[[]]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> 3 = 2 + 1
[[[],[[]],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> 3 = 2 + 1
[[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2 = 1 + 1
[[[[]],[],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> 3 = 2 + 1
[[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> 2 = 1 + 1
[[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2 = 1 + 1
[[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> 3 = 2 + 1
[[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2 = 1 + 1
[[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2 = 1 + 1
[[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2 = 1 + 1
[[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[[],[],[],[],[],[]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6 = 5 + 1
[[],[],[],[],[[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 5 = 4 + 1
Description
The width of the poset.
This is the size of the poset's longest antichain, also called Dilworth number.
Matching statistic: St000024
(load all 77 compositions to match this statistic)
(load all 77 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> []
=> []
=> 0
[[]]
=> [1,0]
=> [1,0]
=> 0
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
[[[],[],[[]]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[[[],[[]],[]]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[[[],[[[]]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[[[[]],[],[]]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[[[[]],[[]]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[[[[[]]],[]]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[[[[],[],[]]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[[[[],[[]]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[[[[[]],[]]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[[[[[],[]]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[[[[[[]]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 4
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000052
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> []
=> [1,0]
=> 0
[[]]
=> [1,0]
=> [1,1,0,0]
=> 0
[[],[]]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[[[]]]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 4
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 3
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 3
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 3
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 3
[[[],[],[[]]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 2
[[[],[[]],[]]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 2
[[[],[[[]]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 1
[[[[]],[],[]]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 2
[[[[]],[[]]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1
[[[[[]]],[]]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1
[[[[],[],[]]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 2
[[[[],[[]]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 1
[[[[[]],[]]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1
[[[[[],[]]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[[[[[[]]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 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]
=> 5
[[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> 4
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000053
(load all 79 compositions to match this statistic)
(load all 79 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> []
=> [1,0]
=> 0
[[]]
=> [1,0]
=> [1,1,0,0]
=> 0
[[],[]]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[[[]]]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 4
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 3
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 3
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 3
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 3
[[[],[],[[]]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 2
[[[],[[]],[]]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 2
[[[],[[[]]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 1
[[[[]],[],[]]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 2
[[[[]],[[]]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1
[[[[[]]],[]]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1
[[[[],[],[]]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 2
[[[[],[[]]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 1
[[[[[]],[]]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1
[[[[[],[]]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[[[[[[]]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 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]
=> 5
[[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> 4
Description
The number of valleys of the Dyck path.
Matching statistic: St000245
(load all 43 compositions to match this statistic)
(load all 43 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> []
=> [] => 0
[[]]
=> [1,0]
=> [1] => 0
[[],[]]
=> [1,0,1,0]
=> [1,2] => 1
[[[]]]
=> [1,1,0,0]
=> [2,1] => 0
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,2,3] => 2
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 3
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 4
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 3
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 3
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 3
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 3
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 3
[[[],[],[[]]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 2
[[[],[[]],[]]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 2
[[[],[[[]]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 1
[[[[]],[],[]]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 2
[[[[]],[[]]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 1
[[[[[]]],[]]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 1
[[[[],[],[]]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => 2
[[[[],[[]]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => 1
[[[[[]],[]]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => 1
[[[[[],[]]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => 1
[[[[[[]]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 0
[[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 5
[[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 4
Description
The number of ascents of a permutation.
Matching statistic: St000672
(load all 35 compositions to match this statistic)
(load all 35 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> []
=> [] => 0
[[]]
=> [1,0]
=> [1] => 0
[[],[]]
=> [1,0,1,0]
=> [1,2] => 1
[[[]]]
=> [1,1,0,0]
=> [2,1] => 0
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,2,3] => 2
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 3
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 4
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 3
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 3
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 3
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 3
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 3
[[[],[],[[]]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 2
[[[],[[]],[]]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 2
[[[],[[[]]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 1
[[[[]],[],[]]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 2
[[[[]],[[]]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 1
[[[[[]]],[]]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 1
[[[[],[],[]]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => 2
[[[[],[[]]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => 1
[[[[[]],[]]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => 1
[[[[[],[]]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => 1
[[[[[[]]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 0
[[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 5
[[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 4
Description
The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Matching statistic: St000996
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> []
=> [] => 0
[[]]
=> [1,0]
=> [1] => 0
[[],[]]
=> [1,0,1,0]
=> [2,1] => 1
[[[]]]
=> [1,1,0,0]
=> [1,2] => 0
[[],[],[]]
=> [1,0,1,0,1,0]
=> [2,3,1] => 2
[[],[[]]]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,3,2] => 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [3,1,2] => 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 3
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 4
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 3
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 3
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 2
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 3
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 3
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 3
[[[],[],[[]]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 2
[[[],[[]],[]]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 2
[[[],[[[]]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 1
[[[[]],[],[]]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 2
[[[[]],[[]]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => 1
[[[[[]]],[]]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 1
[[[[],[],[]]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 2
[[[[],[[]]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => 1
[[[[[]],[]]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => 1
[[[[[],[]]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 1
[[[[[[]]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => 5
[[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => 4
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
The following 255 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000010The length of the partition. St000069The number of maximal elements of a poset. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001581The achromatic number of a graph. St000021The number of descents of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000157The number of descents of a standard tableau. St000171The degree of the graph. St000225Difference between largest and smallest parts in a partition. St000272The treewidth of a graph. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000362The size of a minimal vertex cover of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000536The pathwidth of a graph. St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St000778The metric dimension of a graph. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001120The length of a longest path in a graph. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001644The dimension of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001777The number of weak descents in an integer composition. St001949The rigidity index of a graph. St000013The height of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000147The largest part of an integer partition. St000167The number of leaves of an ordered tree. St000288The number of ones in a binary word. St000363The number of minimal vertex covers of a graph. St000378The diagonal inversion number of an integer partition. St000451The length of the longest pattern of the form k 1 2. St000469The distinguishing number of a graph. St000482The (zero)-forcing number of a graph. St000676The number of odd rises of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000820The number of compositions obtained by rotating the composition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001058The breadth of the ordered tree. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001330The hat guessing number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001883The mutual visibility number of a graph. St000328The maximum number of child nodes in a tree. St000168The number of internal nodes of an ordered tree. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000166The depth minus 1 of an ordered tree. St000094The depth of an ordered tree. St000521The number of distinct subtrees of an ordered tree. St000141The maximum drop size of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000292The number of ascents of a binary word. 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. St000864The number of circled entries of the shifted recording tableau of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000062The length of the longest increasing subsequence of the permutation. St000164The number of short pairs. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000291The number of descents of a binary word. 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. St000390The number of runs of ones in a binary word. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000912The number of maximal antichains in a poset. 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. St001304The number of maximally independent sets of vertices of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St000080The rank of the poset. St000120The number of left tunnels of a Dyck path. St000211The rank of the set partition. St000234The number of global ascents of a permutation. St000238The number of indices that are not small weak excedances. St000306The bounce count of a Dyck path. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000332The positive inversions of an alternating sign matrix. St000386The number of factors DDU in a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001176The size of a partition minus its first part. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001358The largest degree of a regular subgraph of a graph. 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. St001742The difference of the maximal and the minimal degree in a graph. St001812The biclique partition number of a graph. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001962The proper pathwidth of a graph. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000035The number of left outer peaks of a permutation. St000056The decomposition (or block) number of a permutation. St000058The order of a permutation. St000153The number of adjacent cycles of a permutation. St000201The number of leaf nodes in a binary tree. St000237The number of small exceedances. St000335The difference of lower and upper interactions. St000381The largest part of an integer composition. St000389The number of runs of ones of odd length in a binary word. St000392The length of the longest run of ones in a binary word. St000444The length of the maximal rise of a Dyck path. St000507The number of ascents of a standard tableau. St000528The height of a poset. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000742The number of big ascents of a permutation after prepending zero. St000808The number of up steps of the associated bargraph. St000822The Hadwiger number of the graph. St000925The number of topologically connected components of a set partition. 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. 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. 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001372The length of a longest cyclic run of ones of a binary word. St001389The number of partitions of the same length below the given integer partition. St001461The number of topologically connected components of the chord diagram of a permutation. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001642The Prague dimension of a graph. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001717The largest size of an interval in a poset. St001725The harmonious chromatic number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001963The tree-depth of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000083The number of left oriented leafs of a binary tree except the first one. 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. St000829The Ulam distance of a permutation to the identity permutation. St000216The absolute length of a permutation. St000730The maximal arc length of a set partition. St000809The reduced reflection length of the permutation. St001419The length of the longest palindromic factor beginning with a one of a binary word. St000061The number of nodes on the left branch of a binary tree. St000485The length of the longest cycle of a permutation. St000668The least common multiple of the parts of the partition. St000702The number of weak deficiencies of a permutation. St000708The product of the parts of an integer partition. St001062The maximal size of a block of a set partition. St000159The number of distinct parts of the integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001323The independence gap of a graph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000619The number of cyclic descents of a permutation. St000741The Colin de Verdière graph invariant. St000711The number of big exceedences of a permutation. St000746The number of pairs with odd minimum in a perfect matching. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St001119The length of a shortest maximal path in a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St000087The number of induced subgraphs. St000286The number of connected components of the complement of a graph. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000926The clique-coclique number of a graph. St001316The domatic number of a graph. St001342The number of vertices in the center of a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001746The coalition number of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St001152The number of pairs with even minimum in a perfect matching. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001589The nesting number of a perfect matching. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001668The number of points of the poset minus the width of the poset. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001863The number of weak excedances of a signed permutation. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001712The number of natural descents of a standard Young tableau. 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. St000942The number of critical left to right maxima of the parking functions. St001773The number of minimal elements in Bruhat order not less than the signed permutation. 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 $eA$. 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 $eA$. St000932The number of occurrences of the pattern UDU in a Dyck path. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!