searching the database
Your data matches 69 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: St000955
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000955: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000955: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
([(1,2)],3)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
([(0,2),(1,2)],3)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
([(2,3)],4)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
([(1,3),(2,3)],4)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
([(0,3),(1,2)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
([(3,4)],5)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
([(2,4),(3,4)],5)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
([(1,4),(2,3)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
([(4,5)],6)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
([(3,5),(4,5)],6)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
([(2,5),(3,5),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
([(2,5),(3,4)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
Description
Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra.
Matching statistic: St000264
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 + 1
([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 + 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(2,3)],4)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 + 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 + 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 1 + 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 1
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(4,5)],6)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 1
([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 + 1
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 1 + 1
([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 1 + 1
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 2 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 2 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ? = 1 + 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 2 + 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> 3 = 2 + 1
([(5,6)],7)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
([(4,6),(5,6)],7)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 1
([(3,6),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 1
([(3,6),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 1 + 1
([(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(2,3),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 1 + 1
([(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 2 + 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 2 + 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 2 + 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 2 + 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> ? = 1 + 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ? = 2 + 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 2 + 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 2 + 1
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001060
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Values
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
([(1,2)],3)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 + 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> ? = 1 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 2 + 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(3,4)],5)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(1,4),(2,3)],5)
=> ([],2)
=> ([],1)
=> ? = 1 + 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 2 + 1
([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 3 = 2 + 1
([(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 + 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(2,5),(3,4)],6)
=> ([],2)
=> ([],1)
=> ? = 1 + 1
([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 2 + 1
([(1,2),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 1
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ? = 1 + 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 3 = 2 + 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(5,6)],7)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
([(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 + 1
([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(3,6),(4,5)],7)
=> ([],2)
=> ([],1)
=> ? = 1 + 1
([(3,6),(4,5),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 2 + 1
([(2,3),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 1
([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(1,6),(2,5),(3,4)],7)
=> ([],3)
=> ([],1)
=> ? = 1 + 1
([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? = 2 + 1
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> ([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2 + 1
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2 + 1
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> ?
=> ? = 2 + 1
([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 2 + 1
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St001570
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Values
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ? = 1 - 2
([(1,2)],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 2
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 2
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> ? = 1 - 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 2 - 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(3,4)],5)
=> ([],1)
=> ([],1)
=> ? = 1 - 2
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(1,4),(2,3)],5)
=> ([],2)
=> ([],1)
=> ? = 1 - 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 2 - 2
([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 - 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 0 = 2 - 2
([(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 1 - 2
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
([(2,5),(3,4)],6)
=> ([],2)
=> ([],1)
=> ? = 1 - 2
([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 2 - 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 - 2
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ? = 1 - 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 0 = 2 - 2
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(5,6)],7)
=> ([],1)
=> ([],1)
=> ? = 1 - 2
([(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
([(3,6),(4,5)],7)
=> ([],2)
=> ([],1)
=> ? = 1 - 2
([(3,6),(4,5),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 2 - 2
([(2,3),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 - 2
([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
([(1,6),(2,5),(3,4)],7)
=> ([],3)
=> ([],1)
=> ? = 1 - 2
([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? = 2 - 2
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> ([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2 - 2
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2 - 2
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> ?
=> ? = 2 - 2
([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 2 - 2
Description
The minimal number of edges to add to make a graph Hamiltonian.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Matching statistic: St001704
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([],1)
=> ([],1)
=> ? = 1
([(1,2)],3)
=> ([(1,2)],3)
=> ([],2)
=> ([],1)
=> ? = 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ? = 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(2,3)],4)
=> ([(2,3)],4)
=> ([],3)
=> ([],1)
=> ? = 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> ? = 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 2
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> ? = 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 2
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(3,4)],5)
=> ([(3,4)],5)
=> ([],4)
=> ([],1)
=> ? = 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ? = 2
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ? = 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? = 2
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> ? = 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ([],1)
=> ? = 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ? = 1
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([(4,5)],6)
=> ([(4,5)],6)
=> ([],5)
=> ([],1)
=> ? = 1
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> ? = 2
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ? = 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ? = 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 2
([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> ? = 1
([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([],3)
=> ([],1)
=> ? = 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ? = 1
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ? = 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([],2)
=> ([],1)
=> ? = 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ? = 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ? = 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(5,6)],7)
=> ([(5,6)],7)
=> ([],6)
=> ([],1)
=> ? = 1
([(4,6),(5,6)],7)
=> ([(4,6),(5,6)],7)
=> ([],5)
=> ([],1)
=> ? = 2
([(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ? = 2
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 2
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([],2)
=> ([],1)
=> ? = 2
([(3,6),(4,5)],7)
=> ([(3,6),(4,5)],7)
=> ([],5)
=> ([],1)
=> ? = 1
([(3,6),(4,5),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([],4)
=> ([],1)
=> ? = 2
([(2,3),(4,6),(5,6)],7)
=> ([(2,3),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ? = 1
([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 2
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([],2)
=> ([],1)
=> ? = 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,6),(2,6),(3,5),(4,5)],7)
=> ([],3)
=> ([],1)
=> ? = 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([],2)
=> ([],1)
=> ? = 2
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> ([],4)
=> ([],1)
=> ? = 1
([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 2
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 2
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 2
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ?
=> ? = 2
Description
The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph.
The deck of a graph is the multiset of induced subgraphs obtained by deleting a single vertex.
The graph reconstruction conjecture states that the deck of a graph with at least three vertices determines the graph.
This statistic is only defined for graphs with at least two vertices, because there is only a single graph of the given size otherwise.
Matching statistic: St000455
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([],1)
=> ([],1)
=> ? = 1 - 3
([(1,2)],3)
=> ([(1,2)],3)
=> ([],2)
=> ([],1)
=> ? = 1 - 3
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ? = 2 - 3
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(2,3)],4)
=> ([(2,3)],4)
=> ([],3)
=> ([],1)
=> ? = 1 - 3
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> ? = 2 - 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 2 - 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> ? = 1 - 3
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 2 - 3
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> -1 = 2 - 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(3,4)],5)
=> ([(3,4)],5)
=> ([],4)
=> ([],1)
=> ? = 1 - 3
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ? = 2 - 3
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ? = 2 - 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? = 2 - 3
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> ? = 1 - 3
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ([],1)
=> ? = 2 - 3
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ? = 1 - 3
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> -1 = 2 - 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> -1 = 2 - 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ? = 2 - 3
([(4,5)],6)
=> ([(4,5)],6)
=> ([],5)
=> ([],1)
=> ? = 1 - 3
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> ? = 2 - 3
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ? = 2 - 3
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ? = 2 - 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 2 - 3
([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> ? = 1 - 3
([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([],3)
=> ([],1)
=> ? = 2 - 3
([(1,2),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ? = 1 - 3
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ? = 2 - 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 2 - 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> -1 = 2 - 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([],2)
=> ([],1)
=> ? = 2 - 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 2 - 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 2 - 3
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ? = 1 - 3
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ? = 2 - 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> -1 = 2 - 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ? = 2 - 3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> -1 = 2 - 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(5,6)],7)
=> ([(5,6)],7)
=> ([],6)
=> ([],1)
=> ? = 1 - 3
([(4,6),(5,6)],7)
=> ([(4,6),(5,6)],7)
=> ([],5)
=> ([],1)
=> ? = 2 - 3
([(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ? = 2 - 3
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 2 - 3
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([],2)
=> ([],1)
=> ? = 2 - 3
([(3,6),(4,5)],7)
=> ([(3,6),(4,5)],7)
=> ([],5)
=> ([],1)
=> ? = 1 - 3
([(3,6),(4,5),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([],4)
=> ([],1)
=> ? = 2 - 3
([(2,3),(4,6),(5,6)],7)
=> ([(2,3),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ? = 1 - 3
([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2 - 3
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 2 - 3
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 2 - 3
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([],2)
=> ([],1)
=> ? = 2 - 3
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2 - 3
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,6),(2,6),(3,5),(4,5)],7)
=> ([],3)
=> ([],1)
=> ? = 2 - 3
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2 - 3
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([],2)
=> ([],1)
=> ? = 2 - 3
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2 - 3
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2 - 3
([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> ([],4)
=> ([],1)
=> ? = 1 - 3
([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2 - 3
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 2 - 3
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 2 - 3
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> -1 = 2 - 3
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> -1 = 2 - 3
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000781
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 50%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 50%
Values
([(0,1)],2)
=> [1]
=> [1]
=> []
=> ? = 1 - 1
([(1,2)],3)
=> [1]
=> [1]
=> []
=> ? = 1 - 1
([(0,2),(1,2)],3)
=> [1,1]
=> [2]
=> []
=> ? = 2 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
([(2,3)],4)
=> [1]
=> [1]
=> []
=> ? = 1 - 1
([(1,3),(2,3)],4)
=> [1,1]
=> [2]
=> []
=> ? = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 1
([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> []
=> ? = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(3,4)],5)
=> [1]
=> [1]
=> []
=> ? = 1 - 1
([(2,4),(3,4)],5)
=> [1,1]
=> [2]
=> []
=> ? = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> []
=> ? = 1 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [3]
=> []
=> ? = 1 - 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(4,5)],6)
=> [1]
=> [1]
=> []
=> ? = 1 - 1
([(3,5),(4,5)],6)
=> [1,1]
=> [2]
=> []
=> ? = 2 - 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(2,5),(3,4)],6)
=> [1,1]
=> [2]
=> []
=> ? = 1 - 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [3]
=> []
=> ? = 1 - 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [3]
=> []
=> ? = 1 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [2,2,2]
=> [2,2]
=> 1 = 2 - 1
([(5,6)],7)
=> [1]
=> [1]
=> []
=> ? = 1 - 1
([(4,6),(5,6)],7)
=> [1,1]
=> [2]
=> []
=> ? = 2 - 1
([(3,6),(4,6),(5,6)],7)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(3,6),(4,5)],7)
=> [1,1]
=> [2]
=> []
=> ? = 1 - 1
([(3,6),(4,5),(5,6)],7)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 1
([(2,3),(4,6),(5,6)],7)
=> [1,1,1]
=> [3]
=> []
=> ? = 1 - 1
([(4,5),(4,6),(5,6)],7)
=> [3]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 1 = 2 - 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 1 = 2 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
Description
The number of proper colouring schemes of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
Matching statistic: St001901
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 50%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 50%
Values
([(0,1)],2)
=> [1]
=> [1]
=> []
=> ? = 1 - 1
([(1,2)],3)
=> [1]
=> [1]
=> []
=> ? = 1 - 1
([(0,2),(1,2)],3)
=> [1,1]
=> [2]
=> []
=> ? = 2 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
([(2,3)],4)
=> [1]
=> [1]
=> []
=> ? = 1 - 1
([(1,3),(2,3)],4)
=> [1,1]
=> [2]
=> []
=> ? = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 1
([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> []
=> ? = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(3,4)],5)
=> [1]
=> [1]
=> []
=> ? = 1 - 1
([(2,4),(3,4)],5)
=> [1,1]
=> [2]
=> []
=> ? = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> []
=> ? = 1 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [3]
=> []
=> ? = 1 - 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(4,5)],6)
=> [1]
=> [1]
=> []
=> ? = 1 - 1
([(3,5),(4,5)],6)
=> [1,1]
=> [2]
=> []
=> ? = 2 - 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(2,5),(3,4)],6)
=> [1,1]
=> [2]
=> []
=> ? = 1 - 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [3]
=> []
=> ? = 1 - 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [3]
=> []
=> ? = 1 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [2,2,2]
=> [2,2]
=> 1 = 2 - 1
([(5,6)],7)
=> [1]
=> [1]
=> []
=> ? = 1 - 1
([(4,6),(5,6)],7)
=> [1,1]
=> [2]
=> []
=> ? = 2 - 1
([(3,6),(4,6),(5,6)],7)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(3,6),(4,5)],7)
=> [1,1]
=> [2]
=> []
=> ? = 1 - 1
([(3,6),(4,5),(5,6)],7)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 1
([(2,3),(4,6),(5,6)],7)
=> [1,1,1]
=> [3]
=> []
=> ? = 1 - 1
([(4,5),(4,6),(5,6)],7)
=> [3]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 1 = 2 - 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 1 = 2 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
Description
The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition.
Matching statistic: St001934
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 50%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 50%
Values
([(0,1)],2)
=> [1]
=> [1]
=> []
=> ? = 1 - 1
([(1,2)],3)
=> [1]
=> [1]
=> []
=> ? = 1 - 1
([(0,2),(1,2)],3)
=> [1,1]
=> [2]
=> []
=> ? = 2 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
([(2,3)],4)
=> [1]
=> [1]
=> []
=> ? = 1 - 1
([(1,3),(2,3)],4)
=> [1,1]
=> [2]
=> []
=> ? = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 1
([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> []
=> ? = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(3,4)],5)
=> [1]
=> [1]
=> []
=> ? = 1 - 1
([(2,4),(3,4)],5)
=> [1,1]
=> [2]
=> []
=> ? = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> []
=> ? = 1 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [3]
=> []
=> ? = 1 - 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(4,5)],6)
=> [1]
=> [1]
=> []
=> ? = 1 - 1
([(3,5),(4,5)],6)
=> [1,1]
=> [2]
=> []
=> ? = 2 - 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(2,5),(3,4)],6)
=> [1,1]
=> [2]
=> []
=> ? = 1 - 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [3]
=> []
=> ? = 1 - 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [3]
=> []
=> ? = 1 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [2,2,2]
=> [2,2]
=> 1 = 2 - 1
([(5,6)],7)
=> [1]
=> [1]
=> []
=> ? = 1 - 1
([(4,6),(5,6)],7)
=> [1,1]
=> [2]
=> []
=> ? = 2 - 1
([(3,6),(4,6),(5,6)],7)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(3,6),(4,5)],7)
=> [1,1]
=> [2]
=> []
=> ? = 1 - 1
([(3,6),(4,5),(5,6)],7)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 1
([(2,3),(4,6),(5,6)],7)
=> [1,1,1]
=> [3]
=> []
=> ? = 1 - 1
([(4,5),(4,6),(5,6)],7)
=> [3]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 1 = 2 - 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 1 = 2 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 1 = 2 - 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
Description
The number of monotone factorisations of genus zero of a permutation of given cycle type.
A monotone factorisation of genus zero of a permutation $\pi\in\mathfrak S_n$ with $\ell$ cycles, including fixed points, is a tuple of $r = n - \ell$ transpositions
$$
(a_1, b_1),\dots,(a_r, b_r)
$$
with $b_1 \leq \dots \leq b_r$ and $a_i < b_i$ for all $i$, whose product, in this order, is $\pi$.
For example, the cycle $(2,3,1)$ has the two factorizations $(2,3)(1,3)$ and $(1,2)(2,3)$.
Matching statistic: St001593
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001593: Integer partitions ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 50%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001593: Integer partitions ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 50%
Values
([(0,1)],2)
=> [1]
=> [1]
=> []
=> ? = 1 - 2
([(1,2)],3)
=> [1]
=> [1]
=> []
=> ? = 1 - 2
([(0,2),(1,2)],3)
=> [1,1]
=> [2]
=> []
=> ? = 2 - 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> [1,1]
=> 0 = 2 - 2
([(2,3)],4)
=> [1]
=> [1]
=> []
=> ? = 1 - 2
([(1,3),(2,3)],4)
=> [1,1]
=> [2]
=> []
=> ? = 2 - 2
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 2
([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> []
=> ? = 1 - 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 2
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1]
=> 0 = 2 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 2 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0 = 2 - 2
([(3,4)],5)
=> [1]
=> [1]
=> []
=> ? = 1 - 2
([(2,4),(3,4)],5)
=> [1,1]
=> [2]
=> []
=> ? = 2 - 2
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 2
([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> []
=> ? = 1 - 2
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [3]
=> []
=> ? = 1 - 2
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> [1,1]
=> 0 = 2 - 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 2 - 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 2 - 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0 = 2 - 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 2 - 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 2 - 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 2 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0 = 2 - 2
([(4,5)],6)
=> [1]
=> [1]
=> []
=> ? = 1 - 2
([(3,5),(4,5)],6)
=> [1,1]
=> [2]
=> []
=> ? = 2 - 2
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 2
([(2,5),(3,4)],6)
=> [1,1]
=> [2]
=> []
=> ? = 1 - 2
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 2
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [3]
=> []
=> ? = 1 - 2
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1]
=> [1,1]
=> 0 = 2 - 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 2 - 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 2 - 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0 = 2 - 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 2 - 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 2
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [3]
=> []
=> ? = 1 - 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 2 - 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 2 - 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 2 - 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0 = 2 - 2
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 2 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0 = 2 - 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [2,2,2]
=> [2,2]
=> 0 = 2 - 2
([(5,6)],7)
=> [1]
=> [1]
=> []
=> ? = 1 - 2
([(4,6),(5,6)],7)
=> [1,1]
=> [2]
=> []
=> ? = 2 - 2
([(3,6),(4,6),(5,6)],7)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 2
([(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 2
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 2
([(3,6),(4,5)],7)
=> [1,1]
=> [2]
=> []
=> ? = 1 - 2
([(3,6),(4,5),(5,6)],7)
=> [1,1,1]
=> [3]
=> []
=> ? = 2 - 2
([(2,3),(4,6),(5,6)],7)
=> [1,1,1]
=> [3]
=> []
=> ? = 1 - 2
([(4,5),(4,6),(5,6)],7)
=> [3]
=> [1,1,1]
=> [1,1]
=> 0 = 2 - 2
([(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 2
([(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 2 - 2
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 2 - 2
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> [1,1,1,1]
=> [4]
=> []
=> ? = 2 - 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 2 - 2
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0 = 2 - 2
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 2 - 2
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 2 - 2
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 2 - 2
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 2 - 2
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 0 = 2 - 2
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0 = 2 - 2
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 2 - 2
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 0 = 2 - 2
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0 = 2 - 2
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 0 = 2 - 2
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 0 = 2 - 2
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
Description
This is the number of standard Young tableaux of the given shifted shape.
For an integer partition $\lambda = (\lambda_1,\dots,\lambda_k)$, the shifted diagram is obtained by moving the $i$-th row in the diagram $i-1$ boxes to the right, i.e.,
$$\lambda^∗ = \{(i, j) | 1 \leq i \leq k, i \leq j \leq \lambda_i + i − 1 \}.$$
In particular, this statistic is zero if and only if $\lambda_{i+1} = \lambda_i$ for some $i$.
The following 59 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001118The acyclic chromatic index of a graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000454The largest eigenvalue of a graph if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the 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. St000567The sum of the products of all pairs of parts. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001651The Frankl number of a lattice. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000379The number of Hamiltonian cycles in a graph. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000997The even-odd crank of an integer partition. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000934The 2-degree of an integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000741The Colin de Verdière graph invariant. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000456The monochromatic index of a connected graph. St001545The second Elser number of a connected graph.
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!