- FindStat

Your data matches 55 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000344
(load all 2 compositions to match this statistic)

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
St000344: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => ([],1)
 => 1
[1,2] => ([],2)
 => 1
[2,1] => ([(0,1)],2)
 => 0
[1,2,3] => ([],3)
 => 1
[1,3,2] => ([(1,2)],3)
 => 0
[2,1,3] => ([(1,2)],3)
 => 0
[2,3,1] => ([(0,2),(1,2)],3)
 => 0
[3,1,2] => ([(0,2),(1,2)],3)
 => 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,2,3,4] => ([],4)
 => 1
[1,2,4,3] => ([(2,3)],4)
 => 0
[1,3,2,4] => ([(2,3)],4)
 => 0
[1,3,4,2] => ([(1,3),(2,3)],4)
 => 0
[1,4,2,3] => ([(1,3),(2,3)],4)
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 1
[2,1,3,4] => ([(2,3)],4)
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => 0
[2,3,1,4] => ([(1,3),(2,3)],4)
 => 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 0
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[3,1,2,4] => ([(1,3),(2,3)],4)
 => 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 0
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
[1,2,3,4,5] => ([],5)
 => 1
[1,2,3,5,4] => ([(3,4)],5)
 => 0
[1,2,4,3,5] => ([(3,4)],5)
 => 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,3,2,4,5] => ([(3,4)],5)
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 0
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1

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 4 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 52%●distinct values known / distinct values provided: 6%

Values

[1] => [1,0]
 => [[1],[]]
 => []
 => ? = 1
[1,2] => [1,0,1,0]
 => [[1,1],[]]
 => []
 => ? ∊ {0,1}
[2,1] => [1,1,0,0]
 => [[2],[]]
 => []
 => ? ∊ {0,1}
[1,2,3] => [1,0,1,0,1,0]
 => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1}
[1,3,2] => [1,0,1,1,0,0]
 => [[2,1],[]]
 => []
 => ? ∊ {0,0,0,1,1}
[2,1,3] => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => [1]
 => 0
[2,3,1] => [1,1,0,1,0,0]
 => [[3],[]]
 => []
 => ? ∊ {0,0,0,1,1}
[3,1,2] => [1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => ? ∊ {0,0,0,1,1}
[3,2,1] => [1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => ? ∊ {0,0,0,1,1}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1]
 => 0
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => 0
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => 0
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [[2,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => [1]
 => 0
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => 0
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => 0
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => [2]
 => 0
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => [1]
 => 0
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => [1]
 => 0
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1]
 => 0
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1]
 => 0
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => [1,1]
 => 0
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [2,1]
 => 0
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => [1]
 => 0
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => 0
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => 0
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => [2,2]
 => 0
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => [2]
 => 0
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => [3]
 => 0
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => [[4,4],[2]]
 => [2]
 => 0
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => [[4,4],[2]]
 => [2]
 => 0
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => [2,1]
 => 0
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
 => [[4,3],[1]]
 => [1]
 => 0
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => [2,1]
 => 0
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => [[4,3],[1]]
 => [1]
 => 0
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => [1,1]
 => 0
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => [1,1]
 => 0
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => [1]
 => 0
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => [1]
 => 0
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => [1]
 => 0
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => [1]
 => 0
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => [1]
 => 0
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => [1]
 => 0
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 0
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => 0
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => 0
[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => [1]
 => 0
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => [1]
 => 0
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 0
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => 0
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => 0
[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => [1]
 => 0
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => [1]
 => 0
[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => [[2,2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [[2,2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}

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 4 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 52%●distinct values known / distinct values provided: 6%

Values

[1] => [1,0]
 => [[1],[]]
 => []
 => ? = 1
[1,2] => [1,0,1,0]
 => [[1,1],[]]
 => []
 => ? ∊ {0,1}
[2,1] => [1,1,0,0]
 => [[2],[]]
 => []
 => ? ∊ {0,1}
[1,2,3] => [1,0,1,0,1,0]
 => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1}
[1,3,2] => [1,0,1,1,0,0]
 => [[2,1],[]]
 => []
 => ? ∊ {0,0,0,1,1}
[2,1,3] => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => [1]
 => 0
[2,3,1] => [1,1,0,1,0,0]
 => [[3],[]]
 => []
 => ? ∊ {0,0,0,1,1}
[3,1,2] => [1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => ? ∊ {0,0,0,1,1}
[3,2,1] => [1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => ? ∊ {0,0,0,1,1}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1]
 => 0
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => 0
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => 0
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [[2,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => [1]
 => 0
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => 0
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => 0
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => [2]
 => 0
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => [1]
 => 0
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => [1]
 => 0
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1]
 => 0
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1]
 => 0
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => [1,1]
 => 0
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [2,1]
 => 0
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => [1]
 => 0
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => 0
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => 0
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => [2,2]
 => 0
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => [2]
 => 0
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => [3]
 => 0
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => [[4,4],[2]]
 => [2]
 => 0
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => [[4,4],[2]]
 => [2]
 => 0
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => [2,1]
 => 0
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
 => [[4,3],[1]]
 => [1]
 => 0
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => [2,1]
 => 0
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => [[4,3],[1]]
 => [1]
 => 0
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => [1,1]
 => 0
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => [1,1]
 => 0
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => [1]
 => 0
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => [1]
 => 0
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => [1]
 => 0
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => [1]
 => 0
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => [1]
 => 0
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => [1]
 => 0
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 0
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => 0
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => 0
[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => [1]
 => 0
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => [1]
 => 0
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 0
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => 0
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => 0
[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => [1]
 => 0
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => [1]
 => 0
[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => [[2,2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [[2,2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}

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: St000699
(load all 2 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
St000699: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 50%●distinct values known / distinct values provided: 3%

Values

[1] => [1] => ([],1)
 => ([],1)
 => ? = 1
[1,2] => [2] => ([],2)
 => ([],2)
 => 0
[2,1] => [1,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 1
[1,2,3] => [3] => ([],3)
 => ([],3)
 => 0
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ? ∊ {0,1,1}
[2,1,3] => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ? ∊ {0,1,1}
[3,1,2] => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ? ∊ {0,1,1}
[1,2,3,4] => [4] => ([],4)
 => ([],4)
 => 0
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,6}
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,6}
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,6}
[2,1,3,4] => [1,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,6}
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,6}
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,6}
[3,1,2,4] => [1,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,6}
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,6}
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,6}
[4,1,2,3] => [1,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,6}
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,6}
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
[4,3,2,1] => [1,1,1,1] => ([(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)
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,6}
[1,2,3,4,5] => [5] => ([],5)
 => ([],5)
 => 0
[1,2,3,5,4] => [4,1] => ([(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,2,4,5,3] => [4,1] => ([(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 0
[1,3,2,5,4] => [2,2,1] => ([(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,3,4,5,2] => [4,1] => ([(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 0
[1,4,2,5,3] => [2,2,1] => ([(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,4,3,5,2] => [2,2,1] => ([(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 0
[1,5,2,4,3] => [2,2,1] => ([(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,5,3,4,2] => [2,2,1] => ([(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,5,4,3,2] => [2,1,1,1] => ([(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,1,3,4,5] => [1,4] => ([(3,4)],5)
 => ([(3,4)],5)
 => 0
[2,1,3,5,4] => [1,3,1] => ([(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,1,4,3,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[2,1,4,5,3] => [1,3,1] => ([(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,1,5,3,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[2,1,5,4,3] => [1,2,1,1] => ([(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 0
[2,3,1,5,4] => [2,2,1] => ([(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[2,3,4,5,1] => [4,1] => ([(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 0
[2,4,1,5,3] => [2,2,1] => ([(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,4,3,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[2,4,3,5,1] => [2,2,1] => ([(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 0
[2,5,1,4,3] => [2,2,1] => ([(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,5,3,1,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[2,5,3,4,1] => [2,2,1] => ([(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,5,4,1,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[2,5,4,3,1] => [2,1,1,1] => ([(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,1,2,4,5] => [1,4] => ([(3,4)],5)
 => ([(3,4)],5)
 => 0
[3,1,2,5,4] => [1,3,1] => ([(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,1,4,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[3,1,4,5,2] => [1,3,1] => ([(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,1,5,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[3,1,5,4,2] => [1,2,1,1] => ([(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,2,1,4,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 0
[3,2,1,5,4] => [1,1,2,1] => ([(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,2,4,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[3,2,4,5,1] => [1,3,1] => ([(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,2,5,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[3,2,5,4,1] => [1,2,1,1] => ([(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 0
[3,4,1,5,2] => [2,2,1] => ([(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,4,2,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[3,4,2,5,1] => [2,2,1] => ([(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,4,5,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 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: St001498

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 3% ●values known / values provided: 50%●distinct values known / distinct values provided: 3%

Values

[1] => [1] => [1,0]
 => [1,1,0,0]
 => ? = 1
[1,2] => [1,2] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 0
[2,1] => [2,1] => [1,1,0,0]
 => [1,1,1,0,0,0]
 => ? = 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 0
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 0
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1}
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1}
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1}
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 0
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 0
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 0
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 0
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 0
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 0
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,6}
[2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 0
[2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,6}
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0
[3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,6}
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0
[3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,6}
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,6}
[3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,6}
[4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,6}
[4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,6}
[4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,6}
[4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,6}
[4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,6}
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,6}
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 0
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 0
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 0
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 0
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 0
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 0
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[1,3,5,2,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 0
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 0
[1,4,2,5,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 0
[1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[1,5,2,3,4] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[1,5,2,4,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[1,5,3,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[1,5,4,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 0
[2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 0
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 0
[2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 0
[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 0
[2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 0
[2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 0
[2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 0
[2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 0
[2,3,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,3,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 0
[2,3,5,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,4,3,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,4,5,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,5,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,5,4,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,1,4,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,1,5,4,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,2,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,2,5,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,4,1,5,2] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,4,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,4,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,4,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,5,1,4,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,5,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,5,4,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,5,4,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[4,1,2,5,3] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[4,1,3,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[4,1,5,2,3] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[4,1,5,3,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[4,2,1,5,3] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[4,2,3,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[4,2,5,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[4,2,5,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[4,3,1,5,2] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[4,3,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[4,3,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[4,3,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[4,5,1,2,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[4,5,1,3,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[4,5,2,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}

Description

The normalised height of a Nakayama algebra with magnitude 1. We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.

Matching statistic: St000455
(load all 6 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 48%●distinct values known / distinct values provided: 6%

Values

[1] => [1] => [1] => ([],1)
 => ? = 1
[1,2] => [1,2] => [1,2] => ([],2)
 => ? ∊ {0,1}
[2,1] => [1,2] => [1,2] => ([],2)
 => ? ∊ {0,1}
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,0,1,1}
[1,3,2] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 0
[2,1,3] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 0
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,0,1,1}
[3,1,2] => [1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,0,1,1}
[3,2,1] => [1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,0,1,1}
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,1,1,1,1,2,2,2,6}
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 0
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 0
[1,3,4,2] => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,0,1,1,1,1,2,2,2,6}
[1,4,2,3] => [1,4,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 0
[1,4,3,2] => [1,4,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 0
[2,1,3,4] => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,0,1,1,1,1,2,2,2,6}
[2,1,4,3] => [1,4,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 0
[2,3,1,4] => [1,4,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,1,1,1,1,2,2,2,6}
[2,4,1,3] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 0
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 0
[3,1,2,4] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 0
[3,1,4,2] => [1,4,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 0
[3,2,1,4] => [1,4,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 0
[3,2,4,1] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,1,1,1,1,2,2,2,6}
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,1,1,1,1,2,2,2,6}
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,1,1,1,1,2,2,2,6}
[4,1,3,2] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 0
[4,2,1,3] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,1,1,1,1,2,2,2,6}
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,1,1,1,1,2,2,2,6}
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,1,1,1,1,2,2,2,6}
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,2,4,5,3] => [1,2,4,5,3] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,2,5,3,4] => [1,2,5,3,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 0
[1,2,5,4,3] => [1,2,5,3,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 0
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 0
[1,3,2,5,4] => [1,3,2,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,3,4,2,5] => [1,3,4,2,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,3,4,5,2] => [1,3,4,5,2] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,3,5,2,4] => [1,3,5,2,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,3,5,4,2] => [1,3,5,2,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 0
[1,4,2,5,3] => [1,4,2,5,3] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,3,2,5] => [1,4,2,5,3] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,3,5,2] => [1,4,2,3,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 0
[1,4,5,2,3] => [1,4,5,2,3] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,5,3,2] => [1,4,5,2,3] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,2,3,4] => [1,5,2,3,4] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 0
[1,5,2,4,3] => [1,5,2,4,3] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 0
[1,5,3,2,4] => [1,5,2,4,3] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 0
[1,5,3,4,2] => [1,5,2,3,4] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 0
[1,5,4,2,3] => [1,5,2,3,4] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 0
[1,5,4,3,2] => [1,5,2,3,4] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 0
[2,1,3,4,5] => [1,3,4,5,2] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,1,3,5,4] => [1,3,5,2,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,1,4,3,5] => [1,4,2,3,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 0
[2,1,4,5,3] => [1,4,5,2,3] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,1,5,3,4] => [1,5,2,3,4] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 0
[2,1,5,4,3] => [1,5,2,3,4] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 0
[2,3,1,4,5] => [1,4,5,2,3] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,3,1,5,4] => [1,5,2,3,4] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 0
[2,3,4,1,5] => [1,5,2,3,4] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,3,5,1,4] => [1,4,2,3,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 0
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
[2,4,1,3,5] => [1,3,5,2,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,4,1,5,3] => [1,5,2,4,3] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 0
[2,4,3,1,5] => [1,5,2,4,3] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 0
[2,4,3,5,1] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => 0
[2,4,5,1,3] => [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 0
[2,4,5,3,1] => [1,2,4,5,3] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,5,1,3,4] => [1,3,4,2,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,5,1,4,3] => [1,4,2,5,3] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,5,3,1,4] => [1,4,2,5,3] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,5,3,4,1] => [1,2,5,3,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 0
[2,5,4,1,3] => [1,3,2,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,5,4,3,1] => [1,2,5,3,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 0
[3,1,2,4,5] => [1,2,4,5,3] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,1,2,5,4] => [1,2,5,3,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 0
[3,1,4,2,5] => [1,4,2,5,3] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,1,4,5,2] => [1,4,5,2,3] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,1,5,2,4] => [1,5,2,4,3] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 0
[3,1,5,4,2] => [1,5,2,3,4] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 0
[3,2,1,4,5] => [1,4,5,2,3] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,2,1,5,4] => [1,5,2,3,4] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 0
[3,2,4,1,5] => [1,5,2,4,3] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 0
[3,2,4,5,1] => [1,2,4,5,3] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,2,5,1,4] => [1,4,2,5,3] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,2,5,4,1] => [1,2,5,3,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 0
[3,4,1,2,5] => [1,2,5,3,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 0
[3,4,1,5,2] => [1,5,2,3,4] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 0
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[4,1,3,2,5] => [1,3,2,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[4,1,3,5,2] => [1,3,5,2,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[4,2,1,3,5] => [1,3,5,2,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}

Description

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.

Matching statistic: St000936
(load all 5 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000936: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 48%●distinct values known / distinct values provided: 6%

Values

[1] => [1] => [1]
 => []
 => ? = 1
[1,2] => [1,2] => [2]
 => []
 => ? ∊ {0,1}
[2,1] => [1,2] => [2]
 => []
 => ? ∊ {0,1}
[1,2,3] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,1,1}
[1,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1}
[2,1,3] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1}
[2,3,1] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,1,1}
[3,1,2] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,1,1}
[3,2,1] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,1,1}
[1,2,3,4] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,2,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,3,2,4] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,4,3,2] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,1,3,4] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,1,4,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,3,1,4] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,3,4,1] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,4,1,3] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,4,3,1] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,1,2,4] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,1,4,2] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,2,1,4] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,2,4,1] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,4,1,2] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,4,2,1] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,1,2,3] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,1,3,2] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,2,1,3] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,2,3,1] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,3,1,2] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,3,2,1] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,2,3,4,5] => [1,2,3,4,5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,2,3,5,4] => [1,2,3,5,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,2,4,3,5] => [1,2,4,3,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,2,4,5,3] => [1,2,4,5,3] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,2,5,3,4] => [1,2,5,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,2,5,4,3] => [1,2,5,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,3,2,4,5] => [1,3,2,4,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,3,2,5,4] => [1,3,2,5,4] => [3,2]
 => [2]
 => 0
[1,3,4,2,5] => [1,3,4,2,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,3,4,5,2] => [1,3,4,5,2] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 0
[1,3,5,4,2] => [1,3,5,2,4] => [3,2]
 => [2]
 => 0
[1,4,2,3,5] => [1,4,2,3,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 0
[1,4,3,2,5] => [1,4,2,5,3] => [3,2]
 => [2]
 => 0
[1,4,3,5,2] => [1,4,2,3,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,5,2,3] => [1,4,5,2,3] => [3,2]
 => [2]
 => 0
[1,4,5,3,2] => [1,4,5,2,3] => [3,2]
 => [2]
 => 0
[1,5,2,3,4] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
 => [1,1]
 => 0
[1,5,3,2,4] => [1,5,2,4,3] => [3,1,1]
 => [1,1]
 => 0
[1,5,3,4,2] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,4,2,3] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,4,3,2] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,1,3,4,5] => [1,3,4,5,2] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,1,3,5,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 0
[2,1,4,3,5] => [1,4,2,3,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,1,4,5,3] => [1,4,5,2,3] => [3,2]
 => [2]
 => 0
[2,3,1,4,5] => [1,4,5,2,3] => [3,2]
 => [2]
 => 0
[2,4,1,3,5] => [1,3,5,2,4] => [3,2]
 => [2]
 => 0
[2,4,1,5,3] => [1,5,2,4,3] => [3,1,1]
 => [1,1]
 => 0
[2,4,3,1,5] => [1,5,2,4,3] => [3,1,1]
 => [1,1]
 => 0
[2,5,1,4,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 0
[2,5,3,1,4] => [1,4,2,5,3] => [3,2]
 => [2]
 => 0
[2,5,4,1,3] => [1,3,2,5,4] => [3,2]
 => [2]
 => 0
[3,1,4,2,5] => [1,4,2,5,3] => [3,2]
 => [2]
 => 0
[3,1,4,5,2] => [1,4,5,2,3] => [3,2]
 => [2]
 => 0
[3,1,5,2,4] => [1,5,2,4,3] => [3,1,1]
 => [1,1]
 => 0
[3,2,1,4,5] => [1,4,5,2,3] => [3,2]
 => [2]
 => 0
[3,2,4,1,5] => [1,5,2,4,3] => [3,1,1]
 => [1,1]
 => 0
[3,2,5,1,4] => [1,4,2,5,3] => [3,2]
 => [2]
 => 0
[4,1,3,2,5] => [1,3,2,5,4] => [3,2]
 => [2]
 => 0
[4,1,3,5,2] => [1,3,5,2,4] => [3,2]
 => [2]
 => 0
[4,2,1,3,5] => [1,3,5,2,4] => [3,2]
 => [2]
 => 0
[4,2,5,1,3] => [1,3,2,5,4] => [3,2]
 => [2]
 => 0
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,2]
 => [2]
 => 0
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [4,2]
 => [2]
 => 0
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [4,2]
 => [2]
 => 0
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [4,2]
 => [2]
 => 0
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [4,2]
 => [2]
 => 0
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [4,2]
 => [2]
 => 0
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [4,2]
 => [2]
 => 0
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [4,1,1]
 => [1,1]
 => 0
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [4,1,1]
 => [1,1]
 => 0
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [4,2]
 => [2]
 => 0
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [4,2]
 => [2]
 => 0
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [4,2]
 => [2]
 => 0
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [4,2]
 => [2]
 => 0
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [4,2]
 => [2]
 => 0
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [4,2]
 => [2]
 => 0
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [4,2]
 => [2]
 => 0
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [4,2]
 => [2]
 => 0
[1,3,5,2,4,6] => [1,3,5,2,4,6] => [4,2]
 => [2]
 => 0
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [4,2]
 => [2]
 => 0
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [4,2]
 => [2]
 => 0
[1,3,5,4,6,2] => [1,3,5,2,4,6] => [4,2]
 => [2]
 => 0
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [4,2]
 => [2]
 => 0

Description

The number of even values of the symmetric group character corresponding to the partition. For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugace class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $4$. It is shown in [1] that the sum of the values of the statistic over all partitions of a given size is even.

Matching statistic: St000478
(load all 2 compositions to match this statistic)

Mapping

Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000478: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 45%●distinct values known / distinct values provided: 6%

Values

[1] => [1]
 => []
 => ?
 => ? = 1
[1,2] => [1,1]
 => [1]
 => []
 => ? ∊ {0,1}
[2,1] => [2]
 => []
 => ?
 => ? ∊ {0,1}
[1,2,3] => [1,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1}
[1,3,2] => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1}
[2,1,3] => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1}
[2,3,1] => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1}
[3,1,2] => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1}
[3,2,1] => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,1}
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,4,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,3,2,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,3,4,2] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,4,2,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,4,3,2] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,1,3,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,1,4,3] => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,3,1,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,3,4,1] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,4,1,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,4,3,1] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,1,2,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,1,4,2] => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,2,1,4] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,2,4,1] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,4,1,2] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,4,2,1] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,1,2,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,1,3,2] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,2,1,3] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,2,3,1] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,3,1,2] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,3,2,1] => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,4,5,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,5,3,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,5,4,3] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,3,2,5,4] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,3,4,2,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,3,4,5,2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,3,5,2,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,2,3,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,4,2,5,3] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,3,2,5] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,5,2,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,4,5,3,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,2,3,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,5,2,4,3] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,3,2,4] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,3,4,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,4,2,3] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,4,3,2] => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,1,3,5,4] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,1,4,3,5] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,1,4,5,3] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,1,5,3,4] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,1,5,4,3] => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,3,1,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,3,1,5,4] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,3,4,1,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,3,4,5,1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,3,5,1,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,4,1,3,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,4,5,1,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,5,1,3,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[3,1,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[3,4,1,2,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[3,4,5,1,2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[3,5,1,2,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[4,1,2,3,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[4,5,1,2,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[5,1,2,3,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,2,3,4,6,5] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,5,4,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,5,6,4] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,6,4,5] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,6,5,4] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,4,3,5,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,3,6,5] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[1,2,4,5,3,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,5,6,3] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,6,3,5] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,6,5,3] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,5,3,4,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,5,3,6,4] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[1,2,5,4,3,6] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,5,4,6,3] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,5,6,3,4] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,5,6,4,3] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,6,3,4,5] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,6,3,5,4] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,6,4,3,5] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,6,4,5,3] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0

Description

Another weight of a partition according to Alladi. According to Theorem 3.4 (Alladi 2012) in [1] $$ \sum_{\pi\in GG_1(r)} w_1(\pi) $$ equals the number of partitions of $r$ whose odd parts are all distinct. $GG_1(r)$ is the set of partitions of $r$ where consecutive entries differ by at least $2$, and consecutive even entries differ by at least $4$.

Matching statistic: St000566
(load all 2 compositions to match this statistic)

Mapping

Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 45%●distinct values known / distinct values provided: 6%

Values

[1] => [1]
 => []
 => ?
 => ? = 1
[1,2] => [1,1]
 => [1]
 => []
 => ? ∊ {0,1}
[2,1] => [2]
 => []
 => ?
 => ? ∊ {0,1}
[1,2,3] => [1,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1}
[1,3,2] => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1}
[2,1,3] => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1}
[2,3,1] => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1}
[3,1,2] => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1}
[3,2,1] => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,1}
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,4,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,3,2,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,3,4,2] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,4,2,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,4,3,2] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,1,3,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,1,4,3] => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,3,1,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,3,4,1] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,4,1,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,4,3,1] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,1,2,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,1,4,2] => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,2,1,4] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,2,4,1] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,4,1,2] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,4,2,1] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,1,2,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,1,3,2] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,2,1,3] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,2,3,1] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,3,1,2] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,3,2,1] => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,4,5,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,5,3,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,5,4,3] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,3,2,5,4] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,3,4,2,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,3,4,5,2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,3,5,2,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,2,3,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,4,2,5,3] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,3,2,5] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,5,2,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,4,5,3,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,2,3,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,5,2,4,3] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,3,2,4] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,3,4,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,4,2,3] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,4,3,2] => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,1,3,5,4] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,1,4,3,5] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,1,4,5,3] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,1,5,3,4] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,1,5,4,3] => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,3,1,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,3,1,5,4] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,3,4,1,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,3,4,5,1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,3,5,1,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,4,1,3,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,4,5,1,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,5,1,3,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[3,1,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[3,4,1,2,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[3,4,5,1,2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[3,5,1,2,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[4,1,2,3,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[4,5,1,2,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[5,1,2,3,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,2,3,4,6,5] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,5,4,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,5,6,4] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,6,4,5] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,6,5,4] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,4,3,5,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,3,6,5] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[1,2,4,5,3,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,5,6,3] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,6,3,5] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,6,5,3] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,5,3,4,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,5,3,6,4] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[1,2,5,4,3,6] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,5,4,6,3] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,5,6,3,4] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,5,6,4,3] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,6,3,4,5] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,6,3,5,4] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,6,4,3,5] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,6,4,5,3] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0

Description

The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is $$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$

Matching statistic: St000621
(load all 2 compositions to match this statistic)

Mapping

Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 45%●distinct values known / distinct values provided: 6%

Values

[1] => [1]
 => []
 => ?
 => ? = 1
[1,2] => [1,1]
 => [1]
 => []
 => ? ∊ {0,1}
[2,1] => [2]
 => []
 => ?
 => ? ∊ {0,1}
[1,2,3] => [1,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1}
[1,3,2] => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1}
[2,1,3] => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1}
[2,3,1] => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1}
[3,1,2] => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1}
[3,2,1] => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,1}
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,4,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,3,2,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,3,4,2] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,4,2,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,4,3,2] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,1,3,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,1,4,3] => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,3,1,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,3,4,1] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,4,1,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[2,4,3,1] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,1,2,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,1,4,2] => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,2,1,4] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,2,4,1] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,4,1,2] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[3,4,2,1] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,1,2,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,1,3,2] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,2,1,3] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,2,3,1] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,3,1,2] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[4,3,2,1] => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,6}
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,4,5,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,5,3,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,5,4,3] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,3,2,5,4] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,3,4,2,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,3,4,5,2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,3,5,2,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,2,3,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,4,2,5,3] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,3,2,5] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,4,5,2,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,4,5,3,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,2,3,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,5,2,4,3] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,3,2,4] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,3,4,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,4,2,3] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[1,5,4,3,2] => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,1,3,5,4] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,1,4,3,5] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,1,4,5,3] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,1,5,3,4] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,1,5,4,3] => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,3,1,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,3,1,5,4] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,6,6,6,6,6,6,10,10,10,10,10,10,14,14,14,26,26,26,26,51}
[2,3,4,1,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,3,4,5,1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,3,5,1,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,4,1,3,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,4,5,1,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,5,1,3,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[3,1,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[3,4,1,2,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[3,4,5,1,2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[3,5,1,2,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[4,1,2,3,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[4,5,1,2,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[5,1,2,3,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,2,3,4,6,5] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,5,4,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,5,6,4] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,6,4,5] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,6,5,4] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,4,3,5,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,3,6,5] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[1,2,4,5,3,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,5,6,3] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,6,3,5] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,6,5,3] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,5,3,4,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,5,3,6,4] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[1,2,5,4,3,6] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,5,4,6,3] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,5,6,3,4] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,5,6,4,3] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,6,3,4,5] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,6,3,5,4] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,6,4,3,5] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,6,4,5,3] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0

Description

The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is even. This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1]. The case of an odd minimum is [[St000620]].

The following 45 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000934The 2-degree of an integer 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. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000929The constant term of the character polynomial of an integer partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001651The Frankl number of a lattice. 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. St001249Sum of the odd parts of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001383The BG-rank of an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001060The distinguishing index of a graph. St000124The cardinality of the preimage of the Simion-Schmidt map.