Your data matches 44 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001632
Mp00068: Permutations Simion-Schmidt mapPermutations
Mp00073: Permutations major-index to inversion-number bijectionPermutations
Mp00065: Permutations permutation posetPosets
St001632: Posets ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
=> 1
[2,1,3] => [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[3,1,2] => [3,1,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[2,3,1,4] => [2,4,1,3] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[2,4,1,3] => [2,4,1,3] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 0
[3,4,1,2] => [3,4,1,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[4,2,1,3] => [4,2,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 0
[4,3,1,2] => [4,3,1,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 0
[2,3,4,1,5] => [2,5,4,1,3] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 0
[2,3,5,1,4] => [2,5,4,1,3] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 0
[2,4,3,1,5] => [2,5,4,1,3] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 0
[2,4,5,1,3] => [2,5,4,1,3] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 0
[2,5,3,1,4] => [2,5,4,1,3] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 0
[2,5,4,1,3] => [2,5,4,1,3] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 0
[3,2,4,1,5] => [3,2,5,1,4] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 1
[3,2,5,1,4] => [3,2,5,1,4] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 1
[3,4,2,1,5] => [3,5,2,1,4] => [4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> 1
[3,4,5,1,2] => [3,5,4,1,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 0
[3,5,2,1,4] => [3,5,2,1,4] => [4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> 1
[3,5,4,1,2] => [3,5,4,1,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 0
[4,2,3,1,5] => [4,2,5,1,3] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[4,2,5,1,3] => [4,2,5,1,3] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[4,3,5,1,2] => [4,3,5,1,2] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 1
[4,5,1,2,3] => [4,5,1,3,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> 0
[4,5,1,3,2] => [4,5,1,3,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> 0
[4,5,2,1,3] => [4,5,2,1,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> 0
[4,5,3,1,2] => [4,5,3,1,2] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 0
[5,2,3,1,4] => [5,2,4,1,3] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> 1
[5,2,4,1,3] => [5,2,4,1,3] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> 1
[5,3,2,1,4] => [5,3,2,1,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> 0
[5,3,4,1,2] => [5,3,4,1,2] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> 1
[5,4,2,1,3] => [5,4,2,1,3] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> 0
[5,4,3,1,2] => [5,4,3,1,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> 0
[2,3,4,5,1,6] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,3,4,6,1,5] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,3,5,4,1,6] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,3,5,6,1,4] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,3,6,4,1,5] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,3,6,5,1,4] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,4,3,5,1,6] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,4,3,6,1,5] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,4,5,3,1,6] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,4,5,6,1,3] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,4,6,3,1,5] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,4,6,5,1,3] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,5,3,4,1,6] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,5,3,6,1,4] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,5,4,3,1,6] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Mp00069: Permutations complementPermutations
Mp00254: Permutations Inverse fireworks mapPermutations
Mp00160: Permutations graph of inversionsGraphs
St000264: Graphs ⟶ ℤResult quality: 33% values known / values provided: 36%distinct values known / distinct values provided: 33%
Values
[1,2] => [2,1] => [2,1] => ([(0,1)],2)
=> ? = 1 + 3
[2,1,3] => [2,3,1] => [1,3,2] => ([(1,2)],3)
=> ? = 1 + 3
[3,1,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? = 2 + 3
[2,3,1,4] => [3,2,4,1] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 2 + 3
[2,4,1,3] => [3,1,4,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 2 + 3
[3,2,1,4] => [2,3,4,1] => [1,2,4,3] => ([(2,3)],4)
=> ? = 0 + 3
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 2 + 3
[4,2,1,3] => [1,3,4,2] => [1,2,4,3] => ([(2,3)],4)
=> ? = 0 + 3
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? = 0 + 3
[2,3,4,1,5] => [4,3,2,5,1] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,3,5,1,4] => [4,3,1,5,2] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,4,3,1,5] => [4,2,3,5,1] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 0 + 3
[2,4,5,1,3] => [4,2,1,5,3] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,5,3,1,4] => [4,1,3,5,2] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 0 + 3
[2,5,4,1,3] => [4,1,2,5,3] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 0 + 3
[3,2,4,1,5] => [3,4,2,5,1] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 1 + 3
[3,2,5,1,4] => [3,4,1,5,2] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 1 + 3
[3,4,2,1,5] => [3,2,4,5,1] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? = 1 + 3
[3,4,5,1,2] => [3,2,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[3,5,2,1,4] => [3,1,4,5,2] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? = 1 + 3
[3,5,4,1,2] => [3,1,2,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 0 + 3
[4,2,3,1,5] => [2,4,3,5,1] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 2 + 3
[4,2,5,1,3] => [2,4,1,5,3] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 2 + 3
[4,3,2,1,5] => [2,3,4,5,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 0 + 3
[4,3,5,1,2] => [2,3,1,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 1 + 3
[4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[4,5,1,3,2] => [2,1,5,3,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 0 + 3
[4,5,2,1,3] => [2,1,4,5,3] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? = 0 + 3
[4,5,3,1,2] => [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? = 0 + 3
[5,2,3,1,4] => [1,4,3,5,2] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 1 + 3
[5,2,4,1,3] => [1,4,2,5,3] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 1 + 3
[5,3,2,1,4] => [1,3,4,5,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 0 + 3
[5,3,4,1,2] => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 1 + 3
[5,4,2,1,3] => [1,2,4,5,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 0 + 3
[5,4,3,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 0 + 3
[2,3,4,5,1,6] => [5,4,3,2,6,1] => [4,3,2,1,6,5] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,3,4,6,1,5] => [5,4,3,1,6,2] => [4,3,2,1,6,5] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,3,5,4,1,6] => [5,4,2,3,6,1] => [4,3,1,2,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,3,5,6,1,4] => [5,4,2,1,6,3] => [4,3,2,1,6,5] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,3,6,4,1,5] => [5,4,1,3,6,2] => [4,3,1,2,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,3,6,5,1,4] => [5,4,1,2,6,3] => [4,3,1,2,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,4,3,5,1,6] => [5,3,4,2,6,1] => [4,1,3,2,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,4,3,6,1,5] => [5,3,4,1,6,2] => [4,1,3,2,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,4,5,3,1,6] => [5,3,2,4,6,1] => [4,2,1,3,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,4,5,6,1,3] => [5,3,2,1,6,4] => [4,3,2,1,6,5] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,4,6,3,1,5] => [5,3,1,4,6,2] => [4,2,1,3,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,4,6,5,1,3] => [5,3,1,2,6,4] => [4,3,1,2,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,5,3,4,1,6] => [5,2,4,3,6,1] => [4,1,3,2,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,5,3,6,1,4] => [5,2,4,1,6,3] => [4,1,3,2,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,5,4,3,1,6] => [5,2,3,4,6,1] => [4,1,2,3,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 3
[2,5,4,6,1,3] => [5,2,3,1,6,4] => [4,1,3,2,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,5,6,3,1,4] => [5,2,1,4,6,3] => [4,2,1,3,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,5,6,4,1,3] => [5,2,1,3,6,4] => [4,2,1,3,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,6,3,4,1,5] => [5,1,4,3,6,2] => [4,1,3,2,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,6,3,5,1,4] => [5,1,4,2,6,3] => [4,1,3,2,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,6,4,3,1,5] => [5,1,3,4,6,2] => [4,1,2,3,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 3
[2,6,4,5,1,3] => [5,1,3,2,6,4] => [4,1,3,2,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,6,5,3,1,4] => [5,1,2,4,6,3] => [4,1,2,3,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 3
[2,6,5,4,1,3] => [5,1,2,3,6,4] => [4,1,2,3,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 3
[3,2,4,5,1,6] => [4,5,3,2,6,1] => [1,4,3,2,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[3,2,4,6,1,5] => [4,5,3,1,6,2] => [1,4,3,2,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[3,2,5,4,1,6] => [4,5,2,3,6,1] => [2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? = 0 + 3
[3,2,5,6,1,4] => [4,5,2,1,6,3] => [1,4,3,2,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[3,2,6,4,1,5] => [4,5,1,3,6,2] => [2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? = 0 + 3
[3,2,6,5,1,4] => [4,5,1,2,6,3] => [2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? = 0 + 3
[3,4,2,5,1,6] => [4,3,5,2,6,1] => [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
=> ? = 1 + 3
[3,4,2,6,1,5] => [4,3,5,1,6,2] => [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
=> ? = 1 + 3
[3,4,5,2,1,6] => [4,3,2,5,6,1] => [3,2,1,4,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[3,4,5,6,1,2] => [4,3,2,1,6,5] => [4,3,2,1,6,5] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[3,4,6,2,1,5] => [4,3,1,5,6,2] => [3,2,1,4,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[3,4,6,5,1,2] => [4,3,1,2,6,5] => [4,3,1,2,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[3,5,2,4,1,6] => [4,2,5,3,6,1] => [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
=> ? = 1 + 3
[3,5,2,6,1,4] => [4,2,5,1,6,3] => [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
=> ? = 1 + 3
[3,5,4,2,1,6] => [4,2,3,5,6,1] => [3,1,2,4,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ? = 0 + 3
[3,5,4,6,1,2] => [4,2,3,1,6,5] => [4,1,3,2,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[3,5,6,2,1,4] => [4,2,1,5,6,3] => [3,2,1,4,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[3,5,6,4,1,2] => [4,2,1,3,6,5] => [4,2,1,3,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[3,6,2,4,1,5] => [4,1,5,3,6,2] => [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
=> ? = 1 + 3
[3,6,2,5,1,4] => [4,1,5,2,6,3] => [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
=> ? = 1 + 3
[3,6,4,2,1,5] => [4,1,3,5,6,2] => [3,1,2,4,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ? = 0 + 3
[3,6,4,5,1,2] => [4,1,3,2,6,5] => [4,1,3,2,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[3,6,5,2,1,4] => [4,1,2,5,6,3] => [3,1,2,4,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ? = 0 + 3
[3,6,5,4,1,2] => [4,1,2,3,6,5] => [4,1,2,3,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 3
[4,2,3,5,1,6] => [3,5,4,2,6,1] => [1,4,3,2,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[4,2,3,6,1,5] => [3,5,4,1,6,2] => [1,4,3,2,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[4,2,5,3,1,6] => [3,5,2,4,6,1] => [2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? = 0 + 3
[4,2,5,6,1,3] => [3,5,2,1,6,4] => [1,4,3,2,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[4,2,6,3,1,5] => [3,5,1,4,6,2] => [2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? = 0 + 3
[4,2,6,5,1,3] => [3,5,1,2,6,4] => [2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? = 0 + 3
[4,3,5,6,1,2] => [3,4,2,1,6,5] => [1,4,3,2,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[4,5,6,2,1,3] => [3,2,1,5,6,4] => [3,2,1,4,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[4,5,6,3,1,2] => [3,2,1,4,6,5] => [3,2,1,4,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[5,2,3,1,4,6] => [2,5,4,6,3,1] => [1,3,2,6,5,4] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[5,2,3,1,6,4] => [2,5,4,6,1,3] => [1,4,3,6,2,5] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 3 = 0 + 3
[5,2,3,4,1,6] => [2,5,4,3,6,1] => [1,4,3,2,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[5,2,3,6,1,4] => [2,5,4,1,6,3] => [1,4,3,2,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[5,2,4,1,3,6] => [2,5,3,6,4,1] => [1,3,2,6,5,4] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[5,2,4,1,6,3] => [2,5,3,6,1,4] => [1,4,3,6,2,5] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 3 = 0 + 3
[5,2,4,6,1,3] => [2,5,3,1,6,4] => [1,4,3,2,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[5,2,6,1,3,4] => [2,5,1,6,4,3] => [1,3,2,6,5,4] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
Description
The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.
Mp00068: Permutations Simion-Schmidt mapPermutations
Mp00204: Permutations LLPSInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St001605: Integer partitions ⟶ ℤResult quality: 28% values known / values provided: 28%distinct values known / distinct values provided: 67%
Values
[1,2] => [1,2] => [1,1]
=> [1]
=> ? = 1 + 1
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? = 1 + 1
[3,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? = 2 + 1
[2,3,1,4] => [2,4,1,3] => [2,1,1]
=> [1,1]
=> ? = 2 + 1
[2,4,1,3] => [2,4,1,3] => [2,1,1]
=> [1,1]
=> ? = 2 + 1
[3,2,1,4] => [3,2,1,4] => [3,1]
=> [1]
=> ? = 0 + 1
[3,4,1,2] => [3,4,1,2] => [2,1,1]
=> [1,1]
=> ? = 2 + 1
[4,2,1,3] => [4,2,1,3] => [3,1]
=> [1]
=> ? = 0 + 1
[4,3,1,2] => [4,3,1,2] => [3,1]
=> [1]
=> ? = 0 + 1
[2,3,4,1,5] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> ? = 0 + 1
[2,3,5,1,4] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> ? = 0 + 1
[2,4,3,1,5] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> ? = 0 + 1
[2,4,5,1,3] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> ? = 0 + 1
[2,5,3,1,4] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> ? = 0 + 1
[2,5,4,1,3] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> ? = 0 + 1
[3,2,4,1,5] => [3,2,5,1,4] => [3,1,1]
=> [1,1]
=> ? = 1 + 1
[3,2,5,1,4] => [3,2,5,1,4] => [3,1,1]
=> [1,1]
=> ? = 1 + 1
[3,4,2,1,5] => [3,5,2,1,4] => [3,1,1]
=> [1,1]
=> ? = 1 + 1
[3,4,5,1,2] => [3,5,4,1,2] => [3,1,1]
=> [1,1]
=> ? = 0 + 1
[3,5,2,1,4] => [3,5,2,1,4] => [3,1,1]
=> [1,1]
=> ? = 1 + 1
[3,5,4,1,2] => [3,5,4,1,2] => [3,1,1]
=> [1,1]
=> ? = 0 + 1
[4,2,3,1,5] => [4,2,5,1,3] => [3,1,1]
=> [1,1]
=> ? = 2 + 1
[4,2,5,1,3] => [4,2,5,1,3] => [3,1,1]
=> [1,1]
=> ? = 2 + 1
[4,3,2,1,5] => [4,3,2,1,5] => [4,1]
=> [1]
=> ? = 0 + 1
[4,3,5,1,2] => [4,3,5,1,2] => [3,1,1]
=> [1,1]
=> ? = 1 + 1
[4,5,1,2,3] => [4,5,1,3,2] => [3,1,1]
=> [1,1]
=> ? = 0 + 1
[4,5,1,3,2] => [4,5,1,3,2] => [3,1,1]
=> [1,1]
=> ? = 0 + 1
[4,5,2,1,3] => [4,5,2,1,3] => [3,1,1]
=> [1,1]
=> ? = 0 + 1
[4,5,3,1,2] => [4,5,3,1,2] => [3,1,1]
=> [1,1]
=> ? = 0 + 1
[5,2,3,1,4] => [5,2,4,1,3] => [3,1,1]
=> [1,1]
=> ? = 1 + 1
[5,2,4,1,3] => [5,2,4,1,3] => [3,1,1]
=> [1,1]
=> ? = 1 + 1
[5,3,2,1,4] => [5,3,2,1,4] => [4,1]
=> [1]
=> ? = 0 + 1
[5,3,4,1,2] => [5,3,4,1,2] => [3,1,1]
=> [1,1]
=> ? = 1 + 1
[5,4,2,1,3] => [5,4,2,1,3] => [4,1]
=> [1]
=> ? = 0 + 1
[5,4,3,1,2] => [5,4,3,1,2] => [4,1]
=> [1]
=> ? = 0 + 1
[2,3,4,5,1,6] => [2,6,5,4,1,3] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[2,3,4,6,1,5] => [2,6,5,4,1,3] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[2,3,5,4,1,6] => [2,6,5,4,1,3] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[2,3,5,6,1,4] => [2,6,5,4,1,3] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[2,3,6,4,1,5] => [2,6,5,4,1,3] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[2,3,6,5,1,4] => [2,6,5,4,1,3] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[2,4,3,5,1,6] => [2,6,5,4,1,3] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[2,4,3,6,1,5] => [2,6,5,4,1,3] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[2,4,5,3,1,6] => [2,6,5,4,1,3] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[2,4,5,6,1,3] => [2,6,5,4,1,3] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[2,4,6,3,1,5] => [2,6,5,4,1,3] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[2,4,6,5,1,3] => [2,6,5,4,1,3] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[2,5,3,4,1,6] => [2,6,5,4,1,3] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[2,5,3,6,1,4] => [2,6,5,4,1,3] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[2,5,4,3,1,6] => [2,6,5,4,1,3] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[3,2,4,5,1,6] => [3,2,6,5,1,4] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[3,2,4,6,1,5] => [3,2,6,5,1,4] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[3,2,5,4,1,6] => [3,2,6,5,1,4] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[3,2,5,6,1,4] => [3,2,6,5,1,4] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[3,2,6,4,1,5] => [3,2,6,5,1,4] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[3,2,6,5,1,4] => [3,2,6,5,1,4] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[3,4,2,5,1,6] => [3,6,2,5,1,4] => [3,1,1,1]
=> [1,1,1]
=> 2 = 1 + 1
[3,4,2,6,1,5] => [3,6,2,5,1,4] => [3,1,1,1]
=> [1,1,1]
=> 2 = 1 + 1
[3,5,2,4,1,6] => [3,6,2,5,1,4] => [3,1,1,1]
=> [1,1,1]
=> 2 = 1 + 1
[3,5,2,6,1,4] => [3,6,2,5,1,4] => [3,1,1,1]
=> [1,1,1]
=> 2 = 1 + 1
[3,6,2,4,1,5] => [3,6,2,5,1,4] => [3,1,1,1]
=> [1,1,1]
=> 2 = 1 + 1
[3,6,2,5,1,4] => [3,6,2,5,1,4] => [3,1,1,1]
=> [1,1,1]
=> 2 = 1 + 1
[4,2,3,5,1,6] => [4,2,6,5,1,3] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[4,2,3,6,1,5] => [4,2,6,5,1,3] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[4,2,5,3,1,6] => [4,2,6,5,1,3] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[4,2,5,6,1,3] => [4,2,6,5,1,3] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[4,2,6,3,1,5] => [4,2,6,5,1,3] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[4,2,6,5,1,3] => [4,2,6,5,1,3] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[4,3,5,6,1,2] => [4,3,6,5,1,2] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[4,3,6,5,1,2] => [4,3,6,5,1,2] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[4,5,2,3,1,6] => [4,6,2,5,1,3] => [3,1,1,1]
=> [1,1,1]
=> 2 = 1 + 1
[4,5,2,6,1,3] => [4,6,2,5,1,3] => [3,1,1,1]
=> [1,1,1]
=> 2 = 1 + 1
[4,5,3,6,1,2] => [4,6,3,5,1,2] => [3,1,1,1]
=> [1,1,1]
=> 2 = 1 + 1
[4,6,2,3,1,5] => [4,6,2,5,1,3] => [3,1,1,1]
=> [1,1,1]
=> 2 = 1 + 1
[4,6,2,5,1,3] => [4,6,2,5,1,3] => [3,1,1,1]
=> [1,1,1]
=> 2 = 1 + 1
[4,6,3,5,1,2] => [4,6,3,5,1,2] => [3,1,1,1]
=> [1,1,1]
=> 2 = 1 + 1
[5,2,3,1,4,6] => [5,2,6,1,4,3] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[5,2,3,1,6,4] => [5,2,6,1,4,3] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[5,2,3,4,1,6] => [5,2,6,4,1,3] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[5,2,3,6,1,4] => [5,2,6,4,1,3] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[5,2,4,1,3,6] => [5,2,6,1,4,3] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[5,2,4,1,6,3] => [5,2,6,1,4,3] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[5,2,4,3,1,6] => [5,2,6,4,1,3] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[5,2,4,6,1,3] => [5,2,6,4,1,3] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[5,2,6,1,3,4] => [5,2,6,1,4,3] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[5,2,6,1,4,3] => [5,2,6,1,4,3] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[5,2,6,3,1,4] => [5,2,6,4,1,3] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[5,2,6,4,1,3] => [5,2,6,4,1,3] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[5,3,4,1,2,6] => [5,3,6,1,4,2] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[5,3,4,1,6,2] => [5,3,6,1,4,2] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[5,3,4,6,1,2] => [5,3,6,4,1,2] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[5,3,6,1,2,4] => [5,3,6,1,4,2] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[5,3,6,1,4,2] => [5,3,6,1,4,2] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[5,3,6,4,1,2] => [5,3,6,4,1,2] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[5,6,2,1,3,4] => [5,6,2,1,4,3] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[5,6,2,1,4,3] => [5,6,2,1,4,3] => [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[5,6,2,3,1,4] => [5,6,2,4,1,3] => [3,1,1,1]
=> [1,1,1]
=> 2 = 1 + 1
[5,6,2,4,1,3] => [5,6,2,4,1,3] => [3,1,1,1]
=> [1,1,1]
=> 2 = 1 + 1
[5,6,3,4,1,2] => [5,6,3,4,1,2] => [3,1,1,1]
=> [1,1,1]
=> 2 = 1 + 1
Description
The number of colourings of a cycle such that the multiplicities of colours are given by a partition. Two colourings are considered equal, if they are obtained by an action of the cyclic group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Mp00088: Permutations Kreweras complementPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St001199: Dyck paths ⟶ ℤResult quality: 12% values known / values provided: 12%distinct values known / distinct values provided: 33%
Values
[1,2] => [2,1] => [1,1,0,0]
=> ? = 1 + 1
[2,1,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1 + 1
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> ? = 2 + 1
[2,3,1,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[3,2,1,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[3,4,1,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[4,2,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[4,3,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[2,3,4,1,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[2,3,5,1,4] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[2,4,3,1,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[2,4,5,1,3] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[2,5,3,1,4] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[2,5,4,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[3,2,4,1,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[3,2,5,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[3,4,2,1,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[3,4,5,1,2] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[3,5,2,1,4] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[3,5,4,1,2] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[4,2,3,1,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[4,2,5,1,3] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[4,3,2,1,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[4,3,5,1,2] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[4,5,1,2,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[4,5,1,3,2] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[4,5,2,1,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[4,5,3,1,2] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[5,2,3,1,4] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[5,2,4,1,3] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[5,3,2,1,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[5,3,4,1,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[5,4,2,1,3] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[5,4,3,1,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[2,3,4,5,1,6] => [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[2,3,4,6,1,5] => [6,2,3,4,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[2,3,5,4,1,6] => [6,2,3,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[2,3,5,6,1,4] => [6,2,3,1,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[2,3,6,4,1,5] => [6,2,3,5,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[2,3,6,5,1,4] => [6,2,3,1,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[2,4,3,5,1,6] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[2,4,3,6,1,5] => [6,2,4,3,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[2,4,5,3,1,6] => [6,2,5,3,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[2,4,5,6,1,3] => [6,2,1,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[2,4,6,3,1,5] => [6,2,5,3,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[2,4,6,5,1,3] => [6,2,1,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[2,5,3,4,1,6] => [6,2,4,5,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[2,5,3,6,1,4] => [6,2,4,1,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[2,5,4,3,1,6] => [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[2,5,4,6,1,3] => [6,2,1,4,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[2,5,6,3,1,4] => [6,2,5,1,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[5,2,3,1,4,6] => [5,3,4,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 0 + 1
[5,2,3,1,6,4] => [5,3,4,1,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[5,2,4,1,3,6] => [5,3,6,4,2,1] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 0 + 1
[5,2,4,1,6,3] => [5,3,1,4,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[5,2,6,1,3,4] => [5,3,6,1,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 0 + 1
[5,2,6,1,4,3] => [5,3,1,6,2,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 0 + 1
[5,3,4,1,2,6] => [5,6,3,4,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[5,3,4,1,6,2] => [5,1,3,4,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[5,3,6,1,2,4] => [5,6,3,1,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[5,3,6,1,4,2] => [5,1,3,6,2,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 0 + 1
[5,4,6,1,2,3] => [5,6,1,3,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[5,4,6,1,3,2] => [5,1,6,3,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 0 + 1
[5,6,2,1,3,4] => [5,4,6,1,2,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 0 + 1
[5,6,2,1,4,3] => [5,4,1,6,2,3] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 0 + 1
[5,6,4,1,2,3] => [5,6,1,4,2,3] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[5,6,4,1,3,2] => [5,1,6,4,2,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 0 + 1
[6,4,5,1,2,3] => [5,6,1,3,4,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[6,4,5,1,3,2] => [5,1,6,3,4,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 0 + 1
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Mp00088: Permutations Kreweras complementPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St001198: Dyck paths ⟶ ℤResult quality: 12% values known / values provided: 12%distinct values known / distinct values provided: 33%
Values
[1,2] => [2,1] => [1,1,0,0]
=> ? = 1 + 2
[2,1,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1 + 2
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> ? = 2 + 2
[2,3,1,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 + 2
[2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ? = 2 + 2
[3,2,1,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[3,4,1,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? = 2 + 2
[4,2,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[4,3,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[2,3,4,1,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[2,3,5,1,4] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[2,4,3,1,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[2,4,5,1,3] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[2,5,3,1,4] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[2,5,4,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[3,2,4,1,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[3,2,5,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[3,4,2,1,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[3,4,5,1,2] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[3,5,2,1,4] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[3,5,4,1,2] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[4,2,3,1,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[4,2,5,1,3] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[4,3,2,1,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[4,3,5,1,2] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[4,5,1,2,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2 = 0 + 2
[4,5,1,3,2] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2 = 0 + 2
[4,5,2,1,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[4,5,3,1,2] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[5,2,3,1,4] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[5,2,4,1,3] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[5,3,2,1,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[5,3,4,1,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[5,4,2,1,3] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[5,4,3,1,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[2,3,4,5,1,6] => [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,3,4,6,1,5] => [6,2,3,4,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,3,5,4,1,6] => [6,2,3,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,3,5,6,1,4] => [6,2,3,1,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,3,6,4,1,5] => [6,2,3,5,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,3,6,5,1,4] => [6,2,3,1,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,4,3,5,1,6] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,4,3,6,1,5] => [6,2,4,3,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,4,5,3,1,6] => [6,2,5,3,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,4,5,6,1,3] => [6,2,1,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,4,6,3,1,5] => [6,2,5,3,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,4,6,5,1,3] => [6,2,1,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,5,3,4,1,6] => [6,2,4,5,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,5,3,6,1,4] => [6,2,4,1,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,5,4,3,1,6] => [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,5,4,6,1,3] => [6,2,1,4,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,5,6,3,1,4] => [6,2,5,1,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[5,2,3,1,4,6] => [5,3,4,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2 = 0 + 2
[5,2,3,1,6,4] => [5,3,4,1,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[5,2,4,1,3,6] => [5,3,6,4,2,1] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2 = 0 + 2
[5,2,4,1,6,3] => [5,3,1,4,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[5,2,6,1,3,4] => [5,3,6,1,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2 = 0 + 2
[5,2,6,1,4,3] => [5,3,1,6,2,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2 = 0 + 2
[5,3,4,1,2,6] => [5,6,3,4,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 0 + 2
[5,3,4,1,6,2] => [5,1,3,4,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[5,3,6,1,2,4] => [5,6,3,1,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 0 + 2
[5,3,6,1,4,2] => [5,1,3,6,2,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2 = 0 + 2
[5,4,6,1,2,3] => [5,6,1,3,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 0 + 2
[5,4,6,1,3,2] => [5,1,6,3,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2 = 0 + 2
[5,6,2,1,3,4] => [5,4,6,1,2,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2 = 0 + 2
[5,6,2,1,4,3] => [5,4,1,6,2,3] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2 = 0 + 2
[5,6,4,1,2,3] => [5,6,1,4,2,3] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 0 + 2
[5,6,4,1,3,2] => [5,1,6,4,2,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2 = 0 + 2
[6,4,5,1,2,3] => [5,6,1,3,4,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 0 + 2
[6,4,5,1,3,2] => [5,1,6,3,4,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2 = 0 + 2
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Mp00088: Permutations Kreweras complementPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St001206: Dyck paths ⟶ ℤResult quality: 12% values known / values provided: 12%distinct values known / distinct values provided: 33%
Values
[1,2] => [2,1] => [1,1,0,0]
=> ? = 1 + 2
[2,1,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1 + 2
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> ? = 2 + 2
[2,3,1,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 + 2
[2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ? = 2 + 2
[3,2,1,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[3,4,1,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? = 2 + 2
[4,2,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[4,3,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[2,3,4,1,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[2,3,5,1,4] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[2,4,3,1,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[2,4,5,1,3] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[2,5,3,1,4] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[2,5,4,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[3,2,4,1,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[3,2,5,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[3,4,2,1,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[3,4,5,1,2] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[3,5,2,1,4] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[3,5,4,1,2] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[4,2,3,1,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[4,2,5,1,3] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[4,3,2,1,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[4,3,5,1,2] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[4,5,1,2,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2 = 0 + 2
[4,5,1,3,2] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2 = 0 + 2
[4,5,2,1,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[4,5,3,1,2] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[5,2,3,1,4] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[5,2,4,1,3] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[5,3,2,1,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[5,3,4,1,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[5,4,2,1,3] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[5,4,3,1,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[2,3,4,5,1,6] => [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,3,4,6,1,5] => [6,2,3,4,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,3,5,4,1,6] => [6,2,3,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,3,5,6,1,4] => [6,2,3,1,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,3,6,4,1,5] => [6,2,3,5,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,3,6,5,1,4] => [6,2,3,1,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,4,3,5,1,6] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,4,3,6,1,5] => [6,2,4,3,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,4,5,3,1,6] => [6,2,5,3,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,4,5,6,1,3] => [6,2,1,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,4,6,3,1,5] => [6,2,5,3,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,4,6,5,1,3] => [6,2,1,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,5,3,4,1,6] => [6,2,4,5,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,5,3,6,1,4] => [6,2,4,1,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,5,4,3,1,6] => [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,5,4,6,1,3] => [6,2,1,4,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,5,6,3,1,4] => [6,2,5,1,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[5,2,3,1,4,6] => [5,3,4,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2 = 0 + 2
[5,2,3,1,6,4] => [5,3,4,1,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[5,2,4,1,3,6] => [5,3,6,4,2,1] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2 = 0 + 2
[5,2,4,1,6,3] => [5,3,1,4,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[5,2,6,1,3,4] => [5,3,6,1,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2 = 0 + 2
[5,2,6,1,4,3] => [5,3,1,6,2,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2 = 0 + 2
[5,3,4,1,2,6] => [5,6,3,4,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 0 + 2
[5,3,4,1,6,2] => [5,1,3,4,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[5,3,6,1,2,4] => [5,6,3,1,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 0 + 2
[5,3,6,1,4,2] => [5,1,3,6,2,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2 = 0 + 2
[5,4,6,1,2,3] => [5,6,1,3,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 0 + 2
[5,4,6,1,3,2] => [5,1,6,3,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2 = 0 + 2
[5,6,2,1,3,4] => [5,4,6,1,2,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2 = 0 + 2
[5,6,2,1,4,3] => [5,4,1,6,2,3] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2 = 0 + 2
[5,6,4,1,2,3] => [5,6,1,4,2,3] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 0 + 2
[5,6,4,1,3,2] => [5,1,6,4,2,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2 = 0 + 2
[6,4,5,1,2,3] => [5,6,1,3,4,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 0 + 2
[6,4,5,1,3,2] => [5,1,6,3,4,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2 = 0 + 2
Description
The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
Matching statistic: St000175
Mp00088: Permutations Kreweras complementPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00027: Dyck paths to partitionInteger partitions
St000175: Integer partitions ⟶ ℤResult quality: 12% values known / values provided: 12%distinct values known / distinct values provided: 33%
Values
[1,2] => [2,1] => [1,1,0,0]
=> []
=> ? = 1
[2,1,3] => [3,2,1] => [1,1,1,0,0,0]
=> []
=> ? = 1
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> []
=> ? = 2
[2,3,1,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 2
[2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 2
[3,2,1,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 0
[3,4,1,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 2
[4,2,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 0
[4,3,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 0
[2,3,4,1,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[2,3,5,1,4] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[2,4,3,1,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[2,4,5,1,3] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[2,5,3,1,4] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[2,5,4,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[3,2,4,1,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[3,2,5,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[3,4,2,1,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[3,4,5,1,2] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[3,5,2,1,4] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[3,5,4,1,2] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[4,2,3,1,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 2
[4,2,5,1,3] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 2
[4,3,2,1,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[4,3,5,1,2] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[4,5,1,2,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 0
[4,5,1,3,2] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 0
[4,5,2,1,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[4,5,3,1,2] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[5,2,3,1,4] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[5,2,4,1,3] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[5,3,2,1,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[5,3,4,1,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[5,4,2,1,3] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[5,4,3,1,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[2,3,4,5,1,6] => [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,3,4,6,1,5] => [6,2,3,4,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,3,5,4,1,6] => [6,2,3,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,3,5,6,1,4] => [6,2,3,1,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,3,6,4,1,5] => [6,2,3,5,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,3,6,5,1,4] => [6,2,3,1,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,4,3,5,1,6] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,4,3,6,1,5] => [6,2,4,3,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,4,5,3,1,6] => [6,2,5,3,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,4,5,6,1,3] => [6,2,1,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,4,6,3,1,5] => [6,2,5,3,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,4,6,5,1,3] => [6,2,1,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,5,3,4,1,6] => [6,2,4,5,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,5,3,6,1,4] => [6,2,4,1,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,5,4,3,1,6] => [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,5,4,6,1,3] => [6,2,1,4,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,5,6,3,1,4] => [6,2,5,1,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[5,2,3,1,4,6] => [5,3,4,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 0
[5,2,3,1,6,4] => [5,3,4,1,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 0
[5,2,4,1,3,6] => [5,3,6,4,2,1] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 0
[5,2,4,1,6,3] => [5,3,1,4,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 0
[5,2,6,1,3,4] => [5,3,6,1,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 0
[5,2,6,1,4,3] => [5,3,1,6,2,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 0
[5,3,4,1,2,6] => [5,6,3,4,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 0
[5,3,4,1,6,2] => [5,1,3,4,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 0
[5,3,6,1,2,4] => [5,6,3,1,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 0
[5,3,6,1,4,2] => [5,1,3,6,2,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 0
[5,4,6,1,2,3] => [5,6,1,3,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 0
[5,4,6,1,3,2] => [5,1,6,3,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 0
[5,6,2,1,3,4] => [5,4,6,1,2,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 0
[5,6,2,1,4,3] => [5,4,1,6,2,3] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 0
[5,6,4,1,2,3] => [5,6,1,4,2,3] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 0
[5,6,4,1,3,2] => [5,1,6,4,2,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 0
[6,4,5,1,2,3] => [5,6,1,3,4,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 0
[6,4,5,1,3,2] => [5,1,6,3,4,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 0
Description
Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. Given a partition $\lambda$ with $r$ parts, the number of semi-standard Young-tableaux of shape $k\lambda$ and boxes with values in $[r]$ grows as a polynomial in $k$. This follows by setting $q=1$ in (7.105) on page 375 of [1], which yields the polynomial $$p(k) = \prod_{i < j}\frac{k(\lambda_j-\lambda_i)+j-i}{j-i}.$$ The statistic of the degree of this polynomial. For example, the partition $(3, 2, 1, 1, 1)$ gives $$p(k) = \frac{-1}{36} (k - 3) (2k - 3) (k - 2)^2 (k - 1)^3$$ which has degree 7 in $k$. Thus, $[3, 2, 1, 1, 1] \mapsto 7$. This is the same as the number of unordered pairs of different parts, which follows from: $$\deg p(k)=\sum_{i < j}\begin{cases}1& \lambda_j \neq \lambda_i\\0&\lambda_i=\lambda_j\end{cases}=\sum_{\stackrel{i < j}{\lambda_j \neq \lambda_i}} 1$$
Matching statistic: St000205
Mp00088: Permutations Kreweras complementPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00027: Dyck paths to partitionInteger partitions
St000205: Integer partitions ⟶ ℤResult quality: 12% values known / values provided: 12%distinct values known / distinct values provided: 33%
Values
[1,2] => [2,1] => [1,1,0,0]
=> []
=> ? = 1
[2,1,3] => [3,2,1] => [1,1,1,0,0,0]
=> []
=> ? = 1
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> []
=> ? = 2
[2,3,1,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 2
[2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 2
[3,2,1,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 0
[3,4,1,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 2
[4,2,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 0
[4,3,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 0
[2,3,4,1,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[2,3,5,1,4] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[2,4,3,1,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[2,4,5,1,3] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[2,5,3,1,4] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[2,5,4,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[3,2,4,1,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[3,2,5,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[3,4,2,1,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[3,4,5,1,2] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[3,5,2,1,4] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[3,5,4,1,2] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[4,2,3,1,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 2
[4,2,5,1,3] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 2
[4,3,2,1,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[4,3,5,1,2] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[4,5,1,2,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 0
[4,5,1,3,2] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 0
[4,5,2,1,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[4,5,3,1,2] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[5,2,3,1,4] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[5,2,4,1,3] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[5,3,2,1,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[5,3,4,1,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[5,4,2,1,3] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[5,4,3,1,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[2,3,4,5,1,6] => [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,3,4,6,1,5] => [6,2,3,4,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,3,5,4,1,6] => [6,2,3,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,3,5,6,1,4] => [6,2,3,1,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,3,6,4,1,5] => [6,2,3,5,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,3,6,5,1,4] => [6,2,3,1,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,4,3,5,1,6] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,4,3,6,1,5] => [6,2,4,3,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,4,5,3,1,6] => [6,2,5,3,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,4,5,6,1,3] => [6,2,1,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,4,6,3,1,5] => [6,2,5,3,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,4,6,5,1,3] => [6,2,1,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,5,3,4,1,6] => [6,2,4,5,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,5,3,6,1,4] => [6,2,4,1,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,5,4,3,1,6] => [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,5,4,6,1,3] => [6,2,1,4,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,5,6,3,1,4] => [6,2,5,1,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[5,2,3,1,4,6] => [5,3,4,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 0
[5,2,3,1,6,4] => [5,3,4,1,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 0
[5,2,4,1,3,6] => [5,3,6,4,2,1] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 0
[5,2,4,1,6,3] => [5,3,1,4,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 0
[5,2,6,1,3,4] => [5,3,6,1,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 0
[5,2,6,1,4,3] => [5,3,1,6,2,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 0
[5,3,4,1,2,6] => [5,6,3,4,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 0
[5,3,4,1,6,2] => [5,1,3,4,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 0
[5,3,6,1,2,4] => [5,6,3,1,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 0
[5,3,6,1,4,2] => [5,1,3,6,2,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 0
[5,4,6,1,2,3] => [5,6,1,3,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 0
[5,4,6,1,3,2] => [5,1,6,3,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 0
[5,6,2,1,3,4] => [5,4,6,1,2,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 0
[5,6,2,1,4,3] => [5,4,1,6,2,3] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 0
[5,6,4,1,2,3] => [5,6,1,4,2,3] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 0
[5,6,4,1,3,2] => [5,1,6,4,2,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 0
[6,4,5,1,2,3] => [5,6,1,3,4,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 0
[6,4,5,1,3,2] => [5,1,6,3,4,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 0
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St000206
Mp00088: Permutations Kreweras complementPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00027: Dyck paths to partitionInteger partitions
St000206: Integer partitions ⟶ ℤResult quality: 12% values known / values provided: 12%distinct values known / distinct values provided: 33%
Values
[1,2] => [2,1] => [1,1,0,0]
=> []
=> ? = 1
[2,1,3] => [3,2,1] => [1,1,1,0,0,0]
=> []
=> ? = 1
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> []
=> ? = 2
[2,3,1,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 2
[2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 2
[3,2,1,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 0
[3,4,1,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 2
[4,2,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 0
[4,3,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 0
[2,3,4,1,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[2,3,5,1,4] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[2,4,3,1,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[2,4,5,1,3] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[2,5,3,1,4] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[2,5,4,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[3,2,4,1,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[3,2,5,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[3,4,2,1,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[3,4,5,1,2] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[3,5,2,1,4] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[3,5,4,1,2] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[4,2,3,1,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 2
[4,2,5,1,3] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 2
[4,3,2,1,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[4,3,5,1,2] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[4,5,1,2,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 0
[4,5,1,3,2] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 0
[4,5,2,1,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[4,5,3,1,2] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[5,2,3,1,4] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[5,2,4,1,3] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[5,3,2,1,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[5,3,4,1,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[5,4,2,1,3] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[5,4,3,1,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[2,3,4,5,1,6] => [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,3,4,6,1,5] => [6,2,3,4,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,3,5,4,1,6] => [6,2,3,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,3,5,6,1,4] => [6,2,3,1,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,3,6,4,1,5] => [6,2,3,5,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,3,6,5,1,4] => [6,2,3,1,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,4,3,5,1,6] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,4,3,6,1,5] => [6,2,4,3,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,4,5,3,1,6] => [6,2,5,3,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,4,5,6,1,3] => [6,2,1,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,4,6,3,1,5] => [6,2,5,3,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,4,6,5,1,3] => [6,2,1,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,5,3,4,1,6] => [6,2,4,5,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,5,3,6,1,4] => [6,2,4,1,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,5,4,3,1,6] => [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,5,4,6,1,3] => [6,2,1,4,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,5,6,3,1,4] => [6,2,5,1,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[5,2,3,1,4,6] => [5,3,4,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 0
[5,2,3,1,6,4] => [5,3,4,1,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 0
[5,2,4,1,3,6] => [5,3,6,4,2,1] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 0
[5,2,4,1,6,3] => [5,3,1,4,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 0
[5,2,6,1,3,4] => [5,3,6,1,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 0
[5,2,6,1,4,3] => [5,3,1,6,2,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 0
[5,3,4,1,2,6] => [5,6,3,4,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 0
[5,3,4,1,6,2] => [5,1,3,4,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 0
[5,3,6,1,2,4] => [5,6,3,1,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 0
[5,3,6,1,4,2] => [5,1,3,6,2,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 0
[5,4,6,1,2,3] => [5,6,1,3,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 0
[5,4,6,1,3,2] => [5,1,6,3,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 0
[5,6,2,1,3,4] => [5,4,6,1,2,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 0
[5,6,2,1,4,3] => [5,4,1,6,2,3] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 0
[5,6,4,1,2,3] => [5,6,1,4,2,3] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 0
[5,6,4,1,3,2] => [5,1,6,4,2,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 0
[6,4,5,1,2,3] => [5,6,1,3,4,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 0
[6,4,5,1,3,2] => [5,1,6,3,4,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 0
Description
Number of non-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 non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex. See also [[St000205]]. Each value in this statistic is greater than or equal to corresponding value in [[St000205]].
Matching statistic: St000225
Mp00088: Permutations Kreweras complementPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00027: Dyck paths to partitionInteger partitions
St000225: Integer partitions ⟶ ℤResult quality: 12% values known / values provided: 12%distinct values known / distinct values provided: 33%
Values
[1,2] => [2,1] => [1,1,0,0]
=> []
=> ? = 1
[2,1,3] => [3,2,1] => [1,1,1,0,0,0]
=> []
=> ? = 1
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> []
=> ? = 2
[2,3,1,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 2
[2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 2
[3,2,1,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 0
[3,4,1,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 2
[4,2,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 0
[4,3,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 0
[2,3,4,1,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[2,3,5,1,4] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[2,4,3,1,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[2,4,5,1,3] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[2,5,3,1,4] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[2,5,4,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[3,2,4,1,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[3,2,5,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[3,4,2,1,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[3,4,5,1,2] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[3,5,2,1,4] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[3,5,4,1,2] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[4,2,3,1,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 2
[4,2,5,1,3] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 2
[4,3,2,1,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[4,3,5,1,2] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[4,5,1,2,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 0
[4,5,1,3,2] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 0
[4,5,2,1,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[4,5,3,1,2] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[5,2,3,1,4] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[5,2,4,1,3] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[5,3,2,1,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[5,3,4,1,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[5,4,2,1,3] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[5,4,3,1,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[2,3,4,5,1,6] => [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,3,4,6,1,5] => [6,2,3,4,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,3,5,4,1,6] => [6,2,3,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,3,5,6,1,4] => [6,2,3,1,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,3,6,4,1,5] => [6,2,3,5,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,3,6,5,1,4] => [6,2,3,1,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,4,3,5,1,6] => [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,4,3,6,1,5] => [6,2,4,3,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,4,5,3,1,6] => [6,2,5,3,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,4,5,6,1,3] => [6,2,1,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,4,6,3,1,5] => [6,2,5,3,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,4,6,5,1,3] => [6,2,1,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,5,3,4,1,6] => [6,2,4,5,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,5,3,6,1,4] => [6,2,4,1,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,5,4,3,1,6] => [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,5,4,6,1,3] => [6,2,1,4,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[2,5,6,3,1,4] => [6,2,5,1,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[5,2,3,1,4,6] => [5,3,4,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 0
[5,2,3,1,6,4] => [5,3,4,1,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 0
[5,2,4,1,3,6] => [5,3,6,4,2,1] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 0
[5,2,4,1,6,3] => [5,3,1,4,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 0
[5,2,6,1,3,4] => [5,3,6,1,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 0
[5,2,6,1,4,3] => [5,3,1,6,2,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 0
[5,3,4,1,2,6] => [5,6,3,4,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 0
[5,3,4,1,6,2] => [5,1,3,4,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 0
[5,3,6,1,2,4] => [5,6,3,1,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 0
[5,3,6,1,4,2] => [5,1,3,6,2,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 0
[5,4,6,1,2,3] => [5,6,1,3,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 0
[5,4,6,1,3,2] => [5,1,6,3,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 0
[5,6,2,1,3,4] => [5,4,6,1,2,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 0
[5,6,2,1,4,3] => [5,4,1,6,2,3] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 0
[5,6,4,1,2,3] => [5,6,1,4,2,3] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 0
[5,6,4,1,3,2] => [5,1,6,4,2,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 0
[6,4,5,1,2,3] => [5,6,1,3,4,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 0
[6,4,5,1,3,2] => [5,1,6,3,4,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 0
Description
Difference between largest and smallest parts in a partition.
The following 34 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. 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. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001586The number of odd parts smaller than the largest even part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001961The sum of the greatest common divisors of all pairs of parts. St000260The radius of a connected graph. St000618The number of self-evacuating tableaux of given shape. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000781The number of proper colouring schemes of a Ferrers diagram. 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. 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. St001593This is the number of standard Young tableaux of the given shifted shape. St001780The order of promotion on the set of standard tableaux of given shape. 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. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000259The diameter of a connected graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. 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. St001645The pebbling number of a connected graph.