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Your data matches 114 different statistics following compositions of up to 3 maps.
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Matching statistic: St001292
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001292: 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
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001292: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([],2)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
([(0,1)],2)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([],3)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
([(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
([(0,1),(0,2)],3)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
([(0,2),(2,1)],3)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(0,2),(1,2)],3)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
([(0,1),(0,2),(0,3)],4)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
([(0,2),(0,3),(3,1)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
([(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
([(0,3),(3,1),(3,2)],4)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
([(0,3),(1,3),(3,2)],4)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
([(0,3),(1,2)],4)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
([(0,3),(2,1),(3,2)],4)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 2
([(1,4),(3,2),(4,3)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 3
([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
Description
The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Here $A$ is the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]].
Matching statistic: St001526
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
St001526: Dyck paths ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 75%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
St001526: Dyck paths ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 75%
Values
([],1)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2 = 0 + 2
([],2)
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
([(0,1)],2)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2 = 0 + 2
([],3)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 1 + 2
([(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 1 + 2
([(0,1),(0,2)],3)
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
([(0,2),(2,1)],3)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2 = 0 + 2
([(0,2),(1,2)],3)
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
([(0,1),(0,2),(0,3)],4)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 1 + 2
([(0,2),(0,3),(3,1)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 1 + 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
([(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 2 + 2
([(0,3),(3,1),(3,2)],4)
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
([(0,3),(1,3),(3,2)],4)
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
([(0,3),(1,3),(2,3)],4)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 1 + 2
([(0,3),(1,2)],4)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 0 + 2
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 0 + 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
([(0,3),(2,1),(3,2)],4)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2 = 0 + 2
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 1 + 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 1 + 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 0 + 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 0 + 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 1 + 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 1 + 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 0 + 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 2 + 2
([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 1 + 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 1 + 2
([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 2
([(1,4),(3,2),(4,3)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 2
([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 1 + 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 1 + 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2 = 0 + 2
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 0 + 2
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 2 + 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 1 + 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 1 + 2
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 1 + 2
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 0 + 2
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 1 + 2
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 1 + 2
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0 + 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 2
([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 2
([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 2
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> [5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 1 + 2
([(0,4),(1,2),(1,3),(2,5),(3,4),(4,5)],6)
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 2
([(0,3),(0,4),(2,5),(3,5),(4,1),(4,2)],6)
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 2
([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
=> [5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 1 + 2
([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 2
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 2
([(0,5),(1,4),(4,2),(4,5),(5,3)],6)
=> [4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 1 + 2
([(0,3),(1,2),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 1 + 2
([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 2
([(0,5),(1,3),(1,5),(4,2),(5,4)],6)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 2
([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 2
([(0,5),(1,3),(3,4),(4,2),(4,5)],6)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 2
([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 2
([(0,3),(0,4),(3,5),(3,6),(4,5),(4,6),(5,2),(6,1)],7)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 1 + 2
([(0,1),(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0 + 2
([(0,6),(1,6),(4,2),(5,4),(6,3),(6,5)],7)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 2
([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 2
([(0,6),(1,6),(4,3),(5,2),(6,4),(6,5)],7)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 1 + 2
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(6,2),(6,3)],7)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0 + 2
([(0,5),(0,6),(1,5),(1,6),(2,4),(3,4),(5,3),(6,2)],7)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 1 + 2
([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0 + 2
([(0,5),(1,4),(1,5),(3,6),(4,3),(5,6),(6,2)],7)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 2
([(0,5),(0,6),(1,5),(1,6),(4,2),(4,3),(5,4),(6,4)],7)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0 + 2
([(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1)],7)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 1 + 2
([(0,5),(0,6),(1,5),(1,6),(2,3),(3,4),(5,2),(6,4)],7)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 2
([(0,3),(0,5),(3,6),(4,1),(4,6),(5,4),(6,2)],7)
=> [4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 1 + 2
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5),(4,6),(5,6)],7)
=> [5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 1 + 2
([(0,4),(1,3),(1,5),(3,6),(4,5),(5,6),(6,2)],7)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 2
([(0,2),(0,4),(1,5),(1,6),(2,5),(2,6),(3,1),(4,3)],7)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 2
([(0,2),(0,5),(2,6),(3,4),(4,1),(4,6),(5,3)],7)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 2
([(0,5),(1,3),(1,4),(3,6),(4,5),(5,6),(6,2)],7)
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 2
([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(6,4)],7)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 2
([(0,3),(0,5),(3,6),(4,2),(5,1),(5,6),(6,4)],7)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 2
([(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(6,1)],7)
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 2
([(0,5),(2,6),(3,1),(4,3),(4,6),(5,2),(5,4)],7)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 2
([(0,2),(1,5),(1,6),(2,3),(3,5),(3,6),(5,4),(6,4)],7)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 2
([(0,5),(1,4),(4,6),(5,6),(6,2),(6,3)],7)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 1 + 2
([(0,3),(1,2),(2,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 1 + 2
Description
The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001491
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 50%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 50%
Values
([],1)
=> [1]
=> [1]
=> 10 => 1 = 0 + 1
([],2)
=> [2]
=> [1,1]
=> 110 => 1 = 0 + 1
([(0,1)],2)
=> [1]
=> [1]
=> 10 => 1 = 0 + 1
([],3)
=> [3,3]
=> [2,2,2]
=> 11100 => ? = 1 + 1
([(1,2)],3)
=> [3]
=> [1,1,1]
=> 1110 => 2 = 1 + 1
([(0,1),(0,2)],3)
=> [2]
=> [1,1]
=> 110 => 1 = 0 + 1
([(0,2),(2,1)],3)
=> [1]
=> [1]
=> 10 => 1 = 0 + 1
([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> 110 => 1 = 0 + 1
([(0,1),(0,2),(0,3)],4)
=> [3,3]
=> [2,2,2]
=> 11100 => ? = 1 + 1
([(0,2),(0,3),(3,1)],4)
=> [3]
=> [1,1,1]
=> 1110 => 2 = 1 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 110 => 1 = 0 + 1
([(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 11110 => ? = 2 + 1
([(0,3),(3,1),(3,2)],4)
=> [2]
=> [1,1]
=> 110 => 1 = 0 + 1
([(0,3),(1,3),(3,2)],4)
=> [2]
=> [1,1]
=> 110 => 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> [3,3]
=> [2,2,2]
=> 11100 => ? = 1 + 1
([(0,3),(1,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 110110 => ? = 0 + 1
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> [2,2,1]
=> 11010 => ? = 0 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,3),(2,1),(3,2)],4)
=> [1]
=> [1]
=> 10 => 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 1110 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [2,2,2]
=> 11100 => ? = 1 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> [1,1]
=> 110 => 1 = 0 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,2]
=> [2,2,1,1]
=> 110110 => ? = 0 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 11010 => ? = 0 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2]
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,3]
=> [2,2,2]
=> 11100 => ? = 1 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [2,2]
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,3]
=> [2,2,2]
=> 11100 => ? = 1 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> [1,1]
=> 110 => 1 = 0 + 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1111010 => ? = 2 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 11010 => ? = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [2,2]
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> [1,1,1,1]
=> 11110 => ? = 2 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> [3,3,3,2]
=> 1110100 => ? = 1 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> [1,1,1]
=> 1110 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [5,3]
=> [2,2,2,1,1]
=> 1110110 => ? = 1 + 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1111010 => ? = 2 + 1
([(1,4),(3,2),(4,3)],5)
=> [5]
=> [1,1,1,1,1]
=> 111110 => ? = 3 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> [1,1]
=> 110 => 1 = 0 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 1110 => 2 = 1 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> [1,1,1]
=> 1110 => 2 = 1 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> [1]
=> 10 => 1 = 0 + 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> 110110 => ? = 0 + 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 11110 => ? = 2 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> [1,1]
=> 110 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [3,3]
=> [2,2,2]
=> 11100 => ? = 1 + 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [2,2]
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [2,2]
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [3,3]
=> [2,2,2]
=> 11100 => ? = 1 + 1
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [3,3]
=> [2,2,2]
=> 11100 => ? = 1 + 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [2,2]
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [2,2,1,1]
=> 110110 => ? = 0 + 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> [1,1]
=> 110 => 1 = 0 + 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [3,3]
=> [2,2,2]
=> 11100 => ? = 1 + 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [2,2]
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [3,2]
=> [2,2,1]
=> 11010 => ? = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
=> [4,4,2,2]
=> [4,4,2,2]
=> 11001100 => ? = 1 + 1
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> [2,2,2,2]
=> [4,4]
=> 110000 => ? = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [4,4]
=> [2,2,2,2]
=> 111100 => ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> [2,2]
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> [5,4]
=> [2,2,2,2,1]
=> 1111010 => ? = 2 + 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [3,2]
=> [2,2,1]
=> 11010 => ? = 0 + 1
([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
=> [5,4]
=> [2,2,2,2,1]
=> 1111010 => ? = 2 + 1
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> [5,3]
=> [2,2,2,1,1]
=> 1110110 => ? = 1 + 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [3]
=> [1,1,1]
=> 1110 => 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,5),(3,4),(4,5)],6)
=> [4,4,3]
=> [3,3,3,2]
=> 1110100 => ? = 1 + 1
([(0,3),(0,4),(2,5),(3,5),(4,1),(4,2)],6)
=> [4,4,3]
=> [3,3,3,2]
=> 1110100 => ? = 1 + 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> [4]
=> [1,1,1,1]
=> 11110 => ? = 2 + 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [2]
=> [1,1]
=> 110 => 1 = 0 + 1
([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
=> [5,3]
=> [2,2,2,1,1]
=> 1110110 => ? = 1 + 1
([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> [3,3,3]
=> [3,3,3]
=> 111000 => ? = 1 + 1
([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
=> [5,4]
=> [2,2,2,2,1]
=> 1111010 => ? = 2 + 1
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5]
=> [1,1,1,1,1]
=> 111110 => ? = 3 + 1
([(0,5),(1,4),(4,2),(4,5),(5,3)],6)
=> [4,3,3]
=> [3,3,3,1]
=> 1110010 => ? = 1 + 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> [2]
=> [1,1]
=> 110 => 1 = 0 + 1
([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> [2,2,1,1]
=> 110110 => ? = 0 + 1
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> 11110 => ? = 2 + 1
([(0,3),(1,2),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,4,2,2]
=> [4,4,2,2]
=> 11001100 => ? = 1 + 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> [1,1]
=> 110 => 1 = 0 + 1
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> 11110 => ? = 2 + 1
([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [5,4]
=> [2,2,2,2,1]
=> 1111010 => ? = 2 + 1
([(0,5),(1,3),(1,5),(4,2),(5,4)],6)
=> [5,4]
=> [2,2,2,2,1]
=> 1111010 => ? = 2 + 1
([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
=> [4,4]
=> [2,2,2,2]
=> 111100 => ? = 2 + 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> [2,2]
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> [3,2]
=> [2,2,1]
=> 11010 => ? = 0 + 1
([(0,5),(3,2),(4,1),(5,3),(5,4)],6)
=> [4,2]
=> [2,2,1,1]
=> 110110 => ? = 0 + 1
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [3]
=> [1,1,1]
=> 1110 => 2 = 1 + 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> [1]
=> 10 => 1 = 0 + 1
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> [1,1,1]
=> 1110 => 2 = 1 + 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [2]
=> [1,1]
=> 110 => 1 = 0 + 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [3]
=> [1,1,1]
=> 1110 => 2 = 1 + 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
=> [2,2]
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,6),(1,6),(2,5),(3,5),(4,2),(4,3),(6,4)],7)
=> [2,2]
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> [2,2]
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,6),(1,6),(4,5),(5,2),(5,3),(6,4)],7)
=> [2,2]
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> [2]
=> [1,1]
=> 110 => 1 = 0 + 1
([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> [2]
=> [1,1]
=> 110 => 1 = 0 + 1
([(0,6),(1,6),(2,5),(3,5),(5,4),(6,2),(6,3)],7)
=> [2,2]
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,2),(5,4),(6,4)],7)
=> [2,2]
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> [2]
=> [1,1]
=> 110 => 1 = 0 + 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St001532
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
([],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,2)],3)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2 = 1 + 1
([(0,1),(0,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
([(0,1),(0,2),(0,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2 = 1 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 3 = 2 + 1
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 1 = 0 + 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 1 = 0 + 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 1 = 0 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> 2 = 1 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1 = 0 + 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> ? = 2 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 1 = 0 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 3 = 2 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=> ([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=> ? = 1 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
=> ? = 2 + 1
([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ? = 3 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1 = 0 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 2 = 1 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 2 = 1 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 1 = 0 + 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 3 = 2 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> 2 = 1 + 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 1 = 0 + 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 1 = 0 + 1
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> 2 = 1 + 1
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ? = 1 + 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ? = 0 + 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 1 = 0 + 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 0 + 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> 2 = 1 + 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 1 = 0 + 1
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
=> ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,2),(6,1),(8,5),(8,6)],9)
=> ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,2),(6,1),(8,5),(8,6)],9)
=> ? = 1 + 1
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> ([(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(8,1),(8,2),(9,3),(9,4)],10)
=> ?
=> ? = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,2),(8,1),(8,5)],9)
=> ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,2),(8,1),(8,5)],9)
=> ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ? = 0 + 1
([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> ? = 2 + 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
=> ? = 2 + 1
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=> ([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=> ? = 1 + 1
([(0,3),(0,4),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=> ([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=> ? = 1 + 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 3 = 2 + 1
([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> ([(0,4),(0,6),(1,8),(2,7),(3,7),(4,8),(5,3),(6,1),(8,2),(8,5)],9)
=> ([(0,4),(0,6),(1,8),(2,7),(3,7),(4,8),(5,3),(6,1),(8,2),(8,5)],9)
=> ? = 1 + 1
([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
=> ? = 2 + 1
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ? = 3 + 1
([(0,5),(1,4),(4,2),(4,5),(5,3)],6)
=> ([(0,3),(0,5),(1,6),(2,6),(3,7),(4,2),(4,7),(5,4),(7,1)],8)
=> ([(0,3),(0,5),(1,6),(2,6),(3,7),(4,2),(4,7),(5,4),(7,1)],8)
=> ? = 1 + 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 0 + 1
([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> ? = 0 + 1
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ? = 2 + 1
([(0,3),(1,2),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,2),(6,1),(8,3),(8,4)],9)
=> ([(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,2),(6,1),(8,3),(8,4)],9)
=> ? = 1 + 1
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> ? = 2 + 1
([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> ? = 2 + 1
([(0,5),(1,3),(1,5),(4,2),(5,4)],6)
=> ([(0,3),(0,5),(1,6),(2,6),(3,7),(4,2),(5,1),(5,7),(7,4)],8)
=> ([(0,3),(0,5),(1,6),(2,6),(3,7),(4,2),(5,1),(5,7),(7,4)],8)
=> ? = 2 + 1
([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
=> ([(0,4),(0,6),(1,8),(2,7),(3,7),(4,8),(5,1),(6,5),(8,2),(8,3)],9)
=> ([(0,4),(0,6),(1,8),(2,7),(3,7),(4,8),(5,1),(6,5),(8,2),(8,3)],9)
=> ? = 2 + 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 0 + 1
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0 + 1
([(0,5),(3,2),(4,1),(5,3),(5,4)],6)
=> ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ? = 0 + 1
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ? = 1 + 1
([(0,5),(1,3),(3,4),(4,2),(4,5)],6)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(4,1),(4,6),(5,3),(6,7)],8)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(4,1),(4,6),(5,3),(6,7)],8)
=> ? = 2 + 1
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> ? = 1 + 1
([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ? = 3 + 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ? = 0 + 1
([(0,3),(0,4),(3,5),(3,6),(4,5),(4,6),(5,2),(6,1)],7)
=> ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,2),(6,1),(8,5),(8,6)],9)
=> ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,2),(6,1),(8,5),(8,6)],9)
=> ? = 1 + 1
([(0,1),(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> ([(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(8,1),(8,2),(9,3),(9,4)],10)
=> ?
=> ? = 0 + 1
([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> ([(0,2),(0,3),(0,4),(2,7),(3,7),(4,7),(5,1),(6,5),(7,6)],8)
=> ([(0,2),(0,3),(0,4),(2,7),(3,7),(4,7),(5,1),(6,5),(7,6)],8)
=> ? = 1 + 1
([(0,6),(1,6),(2,5),(3,5),(4,2),(4,3),(6,4)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ? = 0 + 1
([(0,6),(1,6),(4,2),(5,4),(6,3),(6,5)],7)
=> ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,2),(8,1),(8,5)],9)
=> ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,2),(8,1),(8,5)],9)
=> ? = 2 + 1
([(0,6),(1,6),(4,5),(5,2),(5,3),(6,4)],7)
=> ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
=> ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
=> ? = 0 + 1
([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,3),(0,6),(1,7),(3,7),(4,5),(5,1),(6,4),(7,2)],8)
=> ([(0,3),(0,6),(1,7),(3,7),(4,5),(5,1),(6,4),(7,2)],8)
=> ? = 3 + 1
([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 0 + 1
([(0,6),(1,6),(2,5),(3,5),(5,4),(6,2),(6,3)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ? = 0 + 1
([(0,6),(1,6),(4,3),(5,2),(6,4),(6,5)],7)
=> ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,2),(6,1),(8,5),(8,6)],9)
=> ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,2),(6,1),(8,5),(8,6)],9)
=> ? = 1 + 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(6,2),(6,3)],7)
=> ([(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(8,1),(8,2),(9,3),(9,4)],10)
=> ?
=> ? = 0 + 1
Description
The leading coefficient of the Poincare polynomial of the poset cone.
For a poset $P$ on $\{1,\dots,n\}$, let $\mathcal K_P = \{\vec x\in\mathbb R^n| x_i < x_j \text{ for } i < _P j\}$. Furthermore let $\mathcal L(\mathcal A)$ be the intersection lattice of the braid arrangement $A_{n-1}$ and let $\mathcal L^{int} = \{ X \in \mathcal L(\mathcal A) | X \cap \mathcal K_P \neq \emptyset \}$.
Then the Poincare polynomial of the poset cone is $Poin(t) = \sum_{X\in\mathcal L^{int}} |\mu(0, X)| t^{codim X}$.
This statistic records its leading coefficient.
Matching statistic: St000264
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? = 0 + 2
([],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0 + 2
([(0,1)],2)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? = 0 + 2
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0 + 2
([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 0 + 2
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0 + 2
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0 + 2
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 + 2
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0 + 2
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0 + 2
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0 + 2
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 2
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 0 + 2
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0 + 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0 + 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0 + 2
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 2
([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 2
([(1,4),(3,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 3 + 2
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0 + 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> ([],1)
=> ? = 0 + 2
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0 + 2
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0 + 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 2
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 2
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> ([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 2
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0 + 2
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0 + 2
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 2
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
=> ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 2
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ? = 0 + 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 2 + 2
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 2
([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 2 + 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(0,4),(1,2),(1,3),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(0,3),(0,4),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(0,5),(1,3),(1,4),(3,6),(4,5),(5,6),(6,2)],7)
=> ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
([(0,5),(2,6),(3,6),(4,1),(4,3),(5,2),(5,4)],7)
=> ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(0,4),(1,5),(2,5),(2,6),(3,1),(3,6),(4,2),(4,3)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(0,3),(0,6),(1,5),(1,6),(3,5),(4,2),(5,4),(6,4)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St000046
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000046: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000046: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> []
=> ? = 0
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? = 0
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? = 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> []
=> ? = 0
([(0,2),(2,1)],3)
=> [3]
=> []
=> ?
=> ? = 0
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? = 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> []
=> ? = 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? = 0
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? = 2
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? = 0
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? = 0
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? = 0
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? = 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? = 0
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ?
=> ? = 0
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? = 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> []
=> ? = 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? = 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> []
=> ? = 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? = 2
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> []
=> ? = 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 2
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? = 3
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? = 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? = 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? = 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? = 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> []
=> ? = 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [1]
=> []
=> ? = 0
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
=> [3,3]
=> [3]
=> []
=> ? = 1
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> [3,3]
=> [3]
=> []
=> ? = 0
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? = 2
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 2
([(0,4),(1,2),(1,3),(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,3),(0,4),(2,5),(3,5),(4,1),(4,2)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(1,3),(1,4),(3,6),(4,5),(5,6),(6,2)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(2,6),(3,6),(4,1),(4,3),(5,2),(5,4)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
Description
The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition.
Matching statistic: St000137
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000137: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000137: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> []
=> ? = 0
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? = 0
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? = 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> []
=> ? = 0
([(0,2),(2,1)],3)
=> [3]
=> []
=> ?
=> ? = 0
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? = 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> []
=> ? = 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? = 0
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? = 2
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? = 0
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? = 0
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? = 0
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? = 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? = 0
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ?
=> ? = 0
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? = 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> []
=> ? = 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? = 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> []
=> ? = 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? = 2
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> []
=> ? = 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 2
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? = 3
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? = 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? = 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? = 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? = 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> []
=> ? = 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [1]
=> []
=> ? = 0
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
=> [3,3]
=> [3]
=> []
=> ? = 1
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> [3,3]
=> [3]
=> []
=> ? = 0
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? = 2
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 2
([(0,4),(1,2),(1,3),(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,3),(0,4),(2,5),(3,5),(4,1),(4,2)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(1,3),(1,4),(3,6),(4,5),(5,6),(6,2)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(2,6),(3,6),(4,1),(4,3),(5,2),(5,4)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
Description
The Grundy value of an integer partition.
Consider the two-player game on an integer partition.
In each move, a player removes either a box, or a 2x2-configuration of boxes such that the resulting diagram is still a partition.
The first player that cannot move lose. This happens exactly when the empty partition is reached.
The grundy value of an integer partition is defined as the grundy value of this two-player game as defined in [1].
This game was described to me during Norcom 2013, by Urban Larsson, and it seems to be quite difficult to give a good description of the partitions with Grundy value 0.
Matching statistic: St000207
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> []
=> ? = 0
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? = 0
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? = 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> []
=> ? = 0
([(0,2),(2,1)],3)
=> [3]
=> []
=> ?
=> ? = 0
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? = 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> []
=> ? = 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? = 0
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? = 2
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? = 0
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? = 0
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? = 0
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? = 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? = 0
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ?
=> ? = 0
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? = 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> []
=> ? = 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? = 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> []
=> ? = 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? = 2
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> []
=> ? = 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 2
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? = 3
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? = 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? = 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? = 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? = 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> []
=> ? = 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [1]
=> []
=> ? = 0
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
=> [3,3]
=> [3]
=> []
=> ? = 1
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> [3,3]
=> [3]
=> []
=> ? = 0
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? = 2
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 2
([(0,4),(1,2),(1,3),(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,3),(0,4),(2,5),(3,5),(4,1),(4,2)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(1,3),(1,4),(3,6),(4,5),(5,6),(6,2)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(2,6),(3,6),(4,1),(4,3),(5,2),(5,4)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has all vertices in integer lattice points.
Matching statistic: St000208
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> []
=> ? = 0
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? = 0
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? = 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> []
=> ? = 0
([(0,2),(2,1)],3)
=> [3]
=> []
=> ?
=> ? = 0
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? = 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> []
=> ? = 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? = 0
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? = 2
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? = 0
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? = 0
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? = 0
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? = 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? = 0
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ?
=> ? = 0
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? = 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> []
=> ? = 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? = 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> []
=> ? = 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? = 2
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> []
=> ? = 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 2
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? = 3
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? = 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? = 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? = 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? = 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> []
=> ? = 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [1]
=> []
=> ? = 0
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
=> [3,3]
=> [3]
=> []
=> ? = 1
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> [3,3]
=> [3]
=> []
=> ? = 0
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? = 2
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 2
([(0,4),(1,2),(1,3),(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,3),(0,4),(2,5),(3,5),(4,1),(4,2)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(1,3),(1,4),(3,6),(4,5),(5,6),(6,2)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(2,6),(3,6),(4,1),(4,3),(5,2),(5,4)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has only integer lattice points as vertices.
See also [[St000205]], [[St000206]] and [[St000207]].
Matching statistic: St000460
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> []
=> ? = 0
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? = 0
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? = 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> []
=> ? = 0
([(0,2),(2,1)],3)
=> [3]
=> []
=> ?
=> ? = 0
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? = 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> []
=> ? = 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? = 0
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? = 2
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? = 0
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? = 0
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? = 0
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? = 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? = 0
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ?
=> ? = 0
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? = 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> []
=> ? = 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? = 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> []
=> ? = 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? = 2
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> []
=> ? = 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 2
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? = 3
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? = 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? = 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? = 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? = 0
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? = 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> []
=> ? = 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [1]
=> []
=> ? = 0
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
=> [3,3]
=> [3]
=> []
=> ? = 1
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> [3,3]
=> [3]
=> []
=> ? = 0
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? = 2
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> [4,2]
=> [2]
=> []
=> ? = 0
([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? = 2
([(0,4),(1,2),(1,3),(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,3),(0,4),(2,5),(3,5),(4,1),(4,2)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(1,3),(1,4),(3,6),(4,5),(5,6),(6,2)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(2,6),(3,6),(4,1),(4,3),(5,2),(5,4)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
Description
The hook length of the last cell along the main diagonal of an integer partition.
The following 104 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St000145The Dyson rank of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000474Dyson's crank of a partition. St001306The number of induced paths on four vertices in a graph. St001350Half of the Albertson index of a graph. St001353The number of prime nodes in the modular decomposition of a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St000535The rank-width of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001694The number of maximal dissociation sets in a graph. St001716The 1-improper chromatic number of a graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph.
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