searching the database
Your data matches 67 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: St001034
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 2
[1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0]
=> 3
[1,1,0,1,0,0]
=> 3
[1,1,1,0,0,0]
=> 4
[1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,0]
=> 4
[1,0,1,1,1,0,0,0]
=> 5
[1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,1,0,0,0]
=> 5
[1,1,1,0,0,0,1,0]
=> 5
[1,1,1,0,0,1,0,0]
=> 5
[1,1,1,0,1,0,0,0]
=> 6
[1,1,1,1,0,0,0,0]
=> 6
[1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> 6
[1,0,1,1,1,0,0,0,1,0]
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> 6
[1,1,0,1,1,0,0,0,1,0]
=> 6
[1,1,0,1,1,0,0,1,0,0]
=> 6
[1,1,1,0,0,0,1,0,1,0]
=> 6
[1,1,1,0,0,0,1,1,0,0]
=> 6
[1,1,1,0,0,1,0,0,1,0]
=> 6
[1,1,1,0,0,1,0,1,0,0]
=> 6
Description
The area of the parallelogram polyomino associated with the Dyck path.
The (bivariate) generating function is given in [1].
Matching statistic: St000395
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 3
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 4
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 5
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 5
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 5
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 5
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 6
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 6
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 6
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 6
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 6
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 6
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 6
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 6
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 6
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 6
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St001717
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001717: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001717: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> ([],1)
=> 1
[1,0,1,0]
=> ([(0,1)],2)
=> 2
[1,1,0,0]
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
[1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
[1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
[1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
[1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
Description
The largest size of an interval in a poset.
Matching statistic: St001300
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001300: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001300: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> ([],1)
=> 0 = 1 - 1
[1,0,1,0]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4 = 5 - 1
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4 = 5 - 1
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4 = 5 - 1
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5 = 6 - 1
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5 = 6 - 1
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 5 = 6 - 1
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 5 = 6 - 1
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 5 = 6 - 1
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 5 = 6 - 1
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 5 = 6 - 1
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 5 = 6 - 1
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 5 = 6 - 1
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 5 = 6 - 1
[1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5 = 6 - 1
[1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5 = 6 - 1
[1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5 = 6 - 1
[1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5 = 6 - 1
Description
The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset.
Matching statistic: St000228
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> ([],1)
=> [1]
=> 1
[1,0,1,0]
=> ([(0,1)],2)
=> [2]
=> 2
[1,1,0,0]
=> ([(0,1)],2)
=> [2]
=> 2
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 4
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 5
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 5
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 5
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 5
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 6
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 6
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 6
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 6
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 6
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 6
[1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [5,1]
=> 6
[1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [5,1]
=> 6
[1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [5,1]
=> 6
[1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [5,1]
=> 6
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000229
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000229: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00151: Permutations —to cycle type⟶ Set partitions
St000229: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => {{1}}
=> 1
[1,0,1,0]
=> [1,2] => {{1},{2}}
=> 2
[1,1,0,0]
=> [2,1] => {{1,2}}
=> 2
[1,0,1,0,1,0]
=> [1,2,3] => {{1},{2},{3}}
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> 3
[1,1,0,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> 3
[1,1,0,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 3
[1,1,1,0,0,0]
=> [3,2,1] => {{1,3},{2}}
=> 4
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 4
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 4
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => {{1},{2,4},{3}}
=> 5
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 4
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 4
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 4
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => {{1,2,4},{3}}
=> 5
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => {{1,3},{2},{4}}
=> 5
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => {{1,3,4},{2}}
=> 5
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => {{1,4},{2},{3}}
=> 6
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => {{1,4},{2,3}}
=> 6
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 6
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => {{1,2,3,5},{4}}
=> 6
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => {{1,2,4},{3},{5}}
=> 6
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => {{1,2,4,5},{3}}
=> 6
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 6
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 6
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => {{1,3,4},{2},{5}}
=> 6
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => {{1,3,4,5},{2}}
=> 6
Description
Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition.
This is, for a set partition P={B1,…,Bk} of {1,…,n}, the statistic is
d(P)=∑i(max(Bi)−min(Bi)+1).
This statistic is called ''dimension index'' in [2]
Matching statistic: St000394
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000394: 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
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,1,0,0]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 4
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 5
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 5
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 5
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 6
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 6
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 6
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 6
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 6
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 6
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 6
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 6
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 6
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 6
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000479
Values
[1,0]
=> ([],1)
=> ([],1)
=> 1
[1,0,1,0]
=> ([(0,1)],2)
=> ([],2)
=> 2
[1,1,0,0]
=> ([(0,1)],2)
=> ([],2)
=> 2
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 4
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 5
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 5
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 5
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 5
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 6
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 6
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 6
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 6
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 6
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 6
[1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 6
[1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 6
[1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 6
[1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 6
Description
The Ramsey number of a graph.
This is the smallest integer n such that every two-colouring of the edges of the complete graph Kn contains a (not necessarily induced) monochromatic copy of the given graph. [1]
Thus, the Ramsey number of the complete graph Kn is the ordinary Ramsey number R(n,n). Very few of these numbers are known, in particular, it is only known that 43≤R(5,5)≤48. [2,3,4,5]
Matching statistic: St001318
Values
[1,0]
=> ([],1)
=> ([],1)
=> 1
[1,0,1,0]
=> ([(0,1)],2)
=> ([],2)
=> 2
[1,1,0,0]
=> ([(0,1)],2)
=> ([],2)
=> 2
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 4
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 5
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 5
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 5
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 5
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 6
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 6
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 6
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 6
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 6
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 6
[1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 6
[1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 6
[1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 6
[1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 6
Description
The number of vertices of the largest induced subforest with the same number of connected components of a graph.
Matching statistic: St001321
Values
[1,0]
=> ([],1)
=> ([],1)
=> 1
[1,0,1,0]
=> ([(0,1)],2)
=> ([],2)
=> 2
[1,1,0,0]
=> ([(0,1)],2)
=> ([],2)
=> 2
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 4
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 5
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 5
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 5
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 5
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 6
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 6
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 6
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 6
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 6
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 6
[1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 6
[1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 6
[1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 6
[1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 6
Description
The number of vertices of the largest induced subforest of a graph.
The following 57 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001342The number of vertices in the center of a graph. St001622The number of join-irreducible elements of a lattice. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000548The number of different non-empty partial sums of an integer partition. St000728The dimension of a set partition. St000874The position of the last double rise in a Dyck path. St001348The bounce of the parallelogram polyomino associated with the Dyck path. 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. St000171The degree of the graph. St000438The position of the last up step in a Dyck path. St001120The length of a longest path in a graph. St000656The number of cuts of a poset. St001645The pebbling number of a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St000189The number of elements in the poset. St001725The harmonious chromatic number of a graph. St001875The number of simple modules with projective dimension at most 1. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. 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. St000018The number of inversions of a permutation. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St000029The depth of a permutation. St000197The number of entries equal to positive one in the alternating sign matrix. St000224The sorting index of a permutation. St001278The number of indecomposable modules that are fixed by τΩ1 composed with its inverse in the corresponding Nakayama algebra. St001726The number of visible inversions of a permutation. St000030The sum of the descent differences of a permutations. St000809The reduced reflection length of the permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001869The maximum cut size of a graph. St000912The number of maximal antichains in a poset. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001894The depth of a signed permutation. St000680The Grundy value for Hackendot on posets. St000223The number of nestings in the permutation. St001727The number of invisible inversions of a permutation. St000245The number of ascents of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001554The number of distinct nonempty subtrees of a binary tree. St000670The reversal length of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000703The number of deficiencies of a permutation.
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!