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Your data matches 202 different statistics following compositions of up to 3 maps.
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(click to perform a complete search on your data)
Matching statistic: St000344
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> 1
([],2)
=> 1
([(0,1)],2)
=> 0
([],3)
=> 1
([(1,2)],3)
=> 0
([(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(1,2)],3)
=> 1
([],4)
=> 1
([(2,3)],4)
=> 0
([(1,3),(2,3)],4)
=> 0
([(0,3),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2)],4)
=> 0
([(0,3),(1,2),(2,3)],4)
=> 0
([(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
([],5)
=> 1
([(3,4)],5)
=> 0
([(2,4),(3,4)],5)
=> 0
([(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(1,4),(2,3)],5)
=> 0
([(1,4),(2,3),(3,4)],5)
=> 0
([(0,1),(2,4),(3,4)],5)
=> 0
([(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 6
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 14
Description
The number of strongly connected outdegree sequences of a graph.
This is the evaluation of the Tutte polynomial at $x=0$ and $y=1$. According to [1,2], the set of strongly connected outdegree sequences is in bijection with strongly connected minimal orientations and also with external spanning trees.
Matching statistic: St000205
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 5% ●values known / values provided: 53%●distinct values known / distinct values provided: 5%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 5% ●values known / values provided: 53%●distinct values known / distinct values provided: 5%
Values
([],1)
=> [1] => [[1],[]]
=> []
=> ? = 1
([],2)
=> [2] => [[2],[]]
=> []
=> ? ∊ {0,1}
([(0,1)],2)
=> [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1}
([],3)
=> [3] => [[3],[]]
=> []
=> ? ∊ {0,1,1}
([(1,2)],3)
=> [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,1,1}
([(0,2),(1,2)],3)
=> [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1}
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [[2,2],[1]]
=> [1]
=> 0
([],4)
=> [4] => [[4],[]]
=> []
=> ? ∊ {1,1,1,2,6}
([(2,3)],4)
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {1,1,1,2,6}
([(1,3),(2,3)],4)
=> [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,6}
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
([(0,3),(1,2)],4)
=> [2,2] => [[3,2],[1]]
=> [1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,6}
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [[3,2],[1]]
=> [1]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,6}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [[3,3],[2]]
=> [2]
=> 0
([],5)
=> [5] => [[5],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(3,4)],5)
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(2,4),(3,4)],5)
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
([(1,4),(2,3)],5)
=> [2,3] => [[4,2],[1]]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [[4,2],[1]]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [[4,3],[2]]
=> [2]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => [[4,4],[3]]
=> [3]
=> 0
([],6)
=> [6] => [[6],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(4,5)],6)
=> [1,5] => [[5,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(3,5),(4,5)],6)
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 0
([(2,5),(3,4)],6)
=> [2,4] => [[5,2],[1]]
=> [1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,2),(3,5),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(3,4),(3,5),(4,5)],6)
=> [2,4] => [[5,2],[1]]
=> [1]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [[5,3],[2]]
=> [2]
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
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
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 5% ●values known / values provided: 53%●distinct values known / distinct values provided: 5%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 5% ●values known / values provided: 53%●distinct values known / distinct values provided: 5%
Values
([],1)
=> [1] => [[1],[]]
=> []
=> ? = 1
([],2)
=> [2] => [[2],[]]
=> []
=> ? ∊ {0,1}
([(0,1)],2)
=> [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1}
([],3)
=> [3] => [[3],[]]
=> []
=> ? ∊ {0,1,1}
([(1,2)],3)
=> [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,1,1}
([(0,2),(1,2)],3)
=> [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1}
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [[2,2],[1]]
=> [1]
=> 0
([],4)
=> [4] => [[4],[]]
=> []
=> ? ∊ {1,1,1,2,6}
([(2,3)],4)
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {1,1,1,2,6}
([(1,3),(2,3)],4)
=> [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,6}
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
([(0,3),(1,2)],4)
=> [2,2] => [[3,2],[1]]
=> [1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,6}
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [[3,2],[1]]
=> [1]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,6}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [[3,3],[2]]
=> [2]
=> 0
([],5)
=> [5] => [[5],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(3,4)],5)
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(2,4),(3,4)],5)
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
([(1,4),(2,3)],5)
=> [2,3] => [[4,2],[1]]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [[4,2],[1]]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [[4,3],[2]]
=> [2]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => [[4,4],[3]]
=> [3]
=> 0
([],6)
=> [6] => [[6],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(4,5)],6)
=> [1,5] => [[5,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(3,5),(4,5)],6)
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 0
([(2,5),(3,4)],6)
=> [2,4] => [[5,2],[1]]
=> [1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,2),(3,5),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(3,4),(3,5),(4,5)],6)
=> [2,4] => [[5,2],[1]]
=> [1]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [[5,3],[2]]
=> [2]
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
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: St000944
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000944: Integer partitions ⟶ ℤResult quality: 5% ●values known / values provided: 52%●distinct values known / distinct values provided: 5%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000944: Integer partitions ⟶ ℤResult quality: 5% ●values known / values provided: 52%●distinct values known / distinct values provided: 5%
Values
([],1)
=> []
=> ?
=> ? = 1
([],2)
=> []
=> ?
=> ? ∊ {0,1}
([(0,1)],2)
=> [1]
=> []
=> ? ∊ {0,1}
([],3)
=> []
=> ?
=> ? ∊ {0,1,1}
([(1,2)],3)
=> [1]
=> []
=> ? ∊ {0,1,1}
([(0,2),(1,2)],3)
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,1,1}
([],4)
=> []
=> ?
=> ? ∊ {0,1,1,1,2,6}
([(2,3)],4)
=> [1]
=> []
=> ? ∊ {0,1,1,1,2,6}
([(1,3),(2,3)],4)
=> [1,1]
=> [1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> [1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> []
=> ? ∊ {0,1,1,1,2,6}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {0,1,1,1,2,6}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> []
=> ? ∊ {0,1,1,1,2,6}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> []
=> ? ∊ {0,1,1,1,2,6}
([],5)
=> []
=> ?
=> ? ∊ {0,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(3,4)],5)
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(2,4),(3,4)],5)
=> [1,1]
=> [1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [3]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([],6)
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(4,5)],6)
=> [1]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(3,5),(4,5)],6)
=> [1,1]
=> [1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(2,5),(3,4)],6)
=> [1,1]
=> [1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6,1]
=> [1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7,1]
=> [1]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [8]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [7]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [8]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [9]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [8]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
Description
The 3-degree of an integer partition.
For an integer partition $\lambda$, this is given by the exponent of 3 in the Gram determinant of the integal Specht module of the symmetric group indexed by $\lambda$.
This stupid comment should not be accepted as an edit!
Matching statistic: St001008
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001008: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 47%●distinct values known / distinct values provided: 10%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001008: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 47%●distinct values known / distinct values provided: 10%
Values
([],1)
=> []
=> []
=> []
=> ? = 1
([],2)
=> []
=> []
=> []
=> ? = 1
([(0,1)],2)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
([],3)
=> []
=> []
=> []
=> ? = 1
([(1,2)],3)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
([(0,2),(1,2)],3)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
([],4)
=> []
=> []
=> []
=> ? ∊ {2,6}
([(2,3)],4)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
([(1,3),(2,3)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {2,6}
([],5)
=> []
=> []
=> []
=> ? ∊ {0,1,2,2,2,4,6,6,10,14,26,51}
([(3,4)],5)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
([(2,4),(3,4)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,4,6,6,10,14,26,51}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,4,6,6,10,14,26,51}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,4,6,6,10,14,26,51}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? ∊ {0,1,2,2,2,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,4,6,6,10,14,26,51}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,4,6,6,10,14,26,51}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,4,6,6,10,14,26,51}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,4,6,6,10,14,26,51}
([],6)
=> []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(4,5)],6)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
([(3,5),(4,5)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0
([(2,5),(3,4)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
Description
Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000379
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ? = 1
([],2)
=> ([],2)
=> ([],1)
=> ? ∊ {0,1}
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1}
([],3)
=> ([],3)
=> ([],1)
=> ? ∊ {0,1,1}
([(1,2)],3)
=> ([],2)
=> ([],1)
=> ? ∊ {0,1,1}
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1}
([],4)
=> ([],4)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([(2,3)],4)
=> ([],3)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([],5)
=> ([],5)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(3,4)],5)
=> ([],4)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([],6)
=> ([],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(4,5)],6)
=> ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 0
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 0
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(1,2),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(3,4),(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([],3)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],2)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([],2)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],2)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
Description
The number of Hamiltonian cycles in a graph.
A Hamiltonian cycle in a graph $G$ is a subgraph (this is, a subset of the edges) that is a cycle which contains every vertex of $G$.
Since it is unclear whether the graph on one vertex is Hamiltonian, the statistic is undefined for this graph.
Matching statistic: St000699
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? = 1
([],2)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1}
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1}
([],3)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1}
([(1,2)],3)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1}
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1}
([],4)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([(2,3)],4)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([],5)
=> ([],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(3,4)],5)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ?
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([],6)
=> ([],6)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(4,5)],6)
=> ([],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,2),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(3,4),(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
Description
The toughness times the least common multiple of 1,...,n-1 of a non-complete graph.
A graph $G$ is $t$-tough if $G$ cannot be split into $k$ different connected components by the removal of fewer than $tk$ vertices for all integers $k>1$.
The toughness of $G$ is the maximal number $t$ such that $G$ is $t$-tough. It is a rational number except for the complete graph, where it is infinity. The toughness of a disconnected graph is zero.
This statistic is the toughness multiplied by the least common multiple of $1,\dots,n-1$, where $n$ is the number of vertices of $G$.
Matching statistic: St001281
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? = 1
([],2)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1}
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1}
([],3)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1}
([(1,2)],3)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1}
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1}
([],4)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([(2,3)],4)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([],5)
=> ([],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(3,4)],5)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ?
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([],6)
=> ([],6)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(4,5)],6)
=> ([],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,2),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(3,4),(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
Description
The normalized isoperimetric number of a graph.
The isoperimetric number, or Cheeger constant, of a graph $G$ is
$$
i(G) = \min\left\{\frac{|\partial A|}{|A|}\ : \ A\subseteq V(G), 0 < |A|\leq |V(G)|/2\right\},
$$
where
$$
\partial A := \{(x, y)\in E(G)\ : \ x\in A, y\in V(G)\setminus A \}.
$$
This statistic is $i(G)\cdot\lfloor n/2\rfloor$.
Matching statistic: St001592
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? = 1
([],2)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1}
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1}
([],3)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1}
([(1,2)],3)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1}
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1}
([],4)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([(2,3)],4)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1,1,2,6}
([],5)
=> ([],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(3,4)],5)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ?
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([],6)
=> ([],6)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(4,5)],6)
=> ([],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,2),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(3,4),(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
Description
The maximal number of simple paths between any two different vertices of a graph.
Matching statistic: St000455
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? = 1
([],2)
=> ([],2)
=> ([],2)
=> ([],2)
=> ? ∊ {0,1}
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],1)
=> ? ∊ {0,1}
([],3)
=> ([],3)
=> ([],3)
=> ([],3)
=> ? ∊ {0,1,1}
([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([],2)
=> ? ∊ {0,1,1}
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ? ∊ {0,1,1}
([],4)
=> ([],4)
=> ([],4)
=> ([],4)
=> ? ∊ {0,0,0,1,1,2,6}
([(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> ([],3)
=> ? ∊ {0,0,0,1,1,2,6}
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> ? ∊ {0,0,0,1,1,2,6}
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {0,0,0,1,1,2,6}
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {0,0,0,1,1,2,6}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {0,0,0,1,1,2,6}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {0,0,0,1,1,2,6}
([],5)
=> ([],5)
=> ([],5)
=> ([],5)
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(3,4)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> ([],4)
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(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)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(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)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,6,6,10,14,26,51}
([],6)
=> ([],6)
=> ([],6)
=> ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(4,5)],6)
=> ([(4,5)],6)
=> ([(4,5)],6)
=> ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> 0
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> 1
([(1,2),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> 0
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(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)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(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)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,7,8,8,10,10,10,10,10,12,14,14,14,14,16,17,18,20,20,22,24,26,26,26,30,32,34,36,38,42,48,51,51,58,60,66,68,76,92,102,118,133,186,212,342,560}
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 0
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
The following 192 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001570The minimal number of edges to add to make a graph Hamiltonian. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001651The Frankl number of a lattice. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a 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. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. 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. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000012The area of a Dyck path. St000017The number of inversions of a standard tableau. St000117The number of centered tunnels of a Dyck path. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000148The number of odd parts of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000185The weighted size of a partition. St000228The size of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000290The major index of a binary word. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000376The bounce deficit of a Dyck path. St000377The dinv defect of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000459The hook length of the base cell of a partition. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000475The number of parts equal to 1 in a partition. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000547The number of even non-empty partial sums of an integer partition. St000628The balance of a binary word. St000629The defect of a binary word. St000661The number of rises of length 3 of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000691The number of changes of a binary word. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000867The sum of the hook lengths in the first row of an integer partition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000877The depth of the binary word interpreted as a path. St000921The number of internal inversions of a binary word. St000931The number of occurrences of the pattern UUU in a Dyck path. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000995The largest even part of an integer partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001127The sum of the squares of the parts of a partition. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001214The aft of an integer partition. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001371The length of the longest Yamanouchi prefix of a binary word. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001423The number of distinct cubes in a binary word. St001435The number of missing boxes in the first row. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001438The number of missing boxes of a skew partition. St001485The modular major index of a binary word. St001498The normalised height of a Nakayama algebra with magnitude 1. St001524The degree of symmetry of a binary word. St001584The area statistic between a Dyck path and its bounce path. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001596The number of two-by-two squares inside a skew partition. St001657The number of twos in an integer partition. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001721The degree of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001910The height of the middle non-run of a Dyck path. St001961The sum of the greatest common divisors of all pairs of parts. St000478Another weight of a partition according to Alladi. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000929The constant term of the character polynomial of an integer partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000145The Dyson rank of a partition. St000506The number of standard desarrangement tableaux of shape equal to the given 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. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even. St000369The dinv deficit of a Dyck path. St000674The number of hills of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000693The modular (standard) major index of a standard tableau. St000984The number of boxes below precisely one peak. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001139The number of occurrences of hills of size 2 in a Dyck path. St001480The number of simple summands of the module J^2/J^3. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000460The hook length of the last cell along the main diagonal of an integer partition. 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. St001248Sum of the even parts of a partition. St001249Sum of the odd 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. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001525The number of symmetric hooks on the diagonal of a partition. St001587Half of the largest even part of an integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. 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.
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