Your data matches 26 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000422
(load all 2 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [.,.]
 => ([],1)
 => 0
{{1,2}}
 => [2,1] => [[.,.],.]
 => ([(0,1)],2)
 => 2
{{1},{2}}
 => [1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => 2
{{1,3,5,6},{2},{4}}
 => [3,2,5,4,6,1] => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
{{1,3,6},{2},{4},{5}}
 => [3,2,6,4,5,1] => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
{{1,5,6},{2,3,4}}
 => [5,3,4,2,6,1] => [[[[.,.],[.,.]],[.,.]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
{{1,6},{2,3,4},{5}}
 => [6,3,4,2,5,1] => [[[[.,.],[.,.]],[.,.]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
{{1},{2,3},{4,5},{6}}
 => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
{{1},{2,4,5},{3},{6}}
 => [1,4,3,5,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
{{1,6},{2,4},{3,5}}
 => [6,4,5,2,3,1] => [[[[.,.],[.,.]],[.,.]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
{{1,5,6},{2},{3,4}}
 => [5,2,4,3,6,1] => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
{{1},{2,5},{3,4,6}}
 => [1,5,4,6,2,3] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
{{1,6},{2},{3,4},{5}}
 => [6,2,4,3,5,1] => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
{{1},{2,5},{3},{4,6}}
 => [1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
{{1},{2,5},{3},{4},{6}}
 => [1,5,3,4,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
{{1,2,5,6,7},{3,4}}
 => [2,5,4,3,6,7,1] => [[.,[[[.,.],.],[.,[.,.]]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,2,5,7},{3,4},{6}}
 => [2,5,4,3,7,6,1] => [[.,[[[.,.],.],[[.,.],.]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,2},{3,4,5},{6,7}}
 => [2,1,4,5,3,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,2},{3,4,5},{6},{7}}
 => [2,1,4,5,3,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,2,6,7},{3,4},{5}}
 => [2,6,4,3,5,7,1] => [[.,[[[.,.],.],[.,[.,.]]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,2,7},{3,4},{5,6}}
 => [2,7,4,3,6,5,1] => [[.,[[[.,.],.],[[.,.],.]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,2,7},{3,4},{5},{6}}
 => [2,7,4,3,5,6,1] => [[.,[[[.,.],.],[.,[.,.]]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,2},{3,5},{4,6,7}}
 => [2,1,5,6,3,7,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,2},{3,5},{4,6},{7}}
 => [2,1,5,6,3,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,2},{3,5},{4,7},{6}}
 => [2,1,5,7,3,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,2},{3,5},{4},{6,7}}
 => [2,1,5,4,3,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,2},{3,5},{4},{6},{7}}
 => [2,1,5,4,3,6,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,4,5,6},{2,3},{7}}
 => [4,3,2,5,6,1,7] => [[[[.,.],.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,4,6},{2,3},{5,7}}
 => [4,3,2,6,7,1,5] => [[[[.,.],.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,4,6},{2,3},{5},{7}}
 => [4,3,2,6,5,1,7] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,5,6},{2,3},{4,7}}
 => [5,3,2,7,6,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,5,6},{2,3},{4},{7}}
 => [5,3,2,4,6,1,7] => [[[[.,.],.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,6},{2,3},{4,5,7}}
 => [6,3,2,5,7,1,4] => [[[[.,.],.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,6},{2,3},{4,5},{7}}
 => [6,3,2,5,4,1,7] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,6},{2,3},{4,7},{5}}
 => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,6},{2,3},{4},{5,7}}
 => [6,3,2,4,7,1,5] => [[[[.,.],.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,6},{2,3},{4},{5},{7}}
 => [6,3,2,4,5,1,7] => [[[[.,.],.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,4,5,6},{2,7},{3}}
 => [4,7,3,5,6,1,2] => [[[.,[.,.]],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,4,5,7},{2,6},{3}}
 => [4,6,3,5,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,4,6},{2,5,7},{3}}
 => [4,5,3,6,7,1,2] => [[[.,[.,.]],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,4,7},{2,5,6},{3}}
 => [4,5,3,7,6,2,1] => [[[[.,[.,.]],[[.,.],.]],.],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,4,6},{2,7},{3},{5}}
 => [4,7,3,6,5,1,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,4,7},{2,6},{3},{5}}
 => [4,6,3,7,5,2,1] => [[[[.,[.,.]],[[.,.],.]],.],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,5,6},{2,4,7},{3}}
 => [5,4,3,7,6,1,2] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,5,7},{2,4,6},{3}}
 => [5,4,3,6,7,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,6},{2,4,5,7},{3}}
 => [6,4,3,5,7,1,2] => [[[[.,.],.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,7},{2,4,5,6},{3}}
 => [7,4,3,5,6,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,6},{2,4,7},{3},{5}}
 => [6,4,3,7,5,1,2] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1,7},{2,4,6},{3},{5}}
 => [7,4,3,6,5,2,1] => [[[[[.,.],.],[[.,.],.]],.],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
{{1},{2,5,6,7},{3,4}}
 => [1,5,4,3,6,7,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8
Description
The energy of a graph, if it is integral. The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3]. The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St000235
(load all 2 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000235: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 75%
Values
{{1}}
 => [1] => [1] => [1] => 0
{{1,2}}
 => [2,1] => [1,2] => [1,2] => 2
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 2
{{1,3,5,6},{2},{4}}
 => [3,2,5,4,6,1] => [1,3,5,6,2,4] => [5,6,3,1,2,4] => 6
{{1,3,6},{2},{4},{5}}
 => [3,2,6,4,5,1] => [1,3,6,2,4,5] => [6,1,3,2,4,5] => 6
{{1,5,6},{2,3,4}}
 => [5,3,4,2,6,1] => [1,5,6,2,3,4] => [1,5,2,6,3,4] => 6
{{1,6},{2,3,4},{5}}
 => [6,3,4,2,5,1] => [1,6,2,3,4,5] => [1,2,3,6,4,5] => 6
{{1},{2,3},{4,5},{6}}
 => [1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
{{1},{2,4,5},{3},{6}}
 => [1,4,3,5,2,6] => [1,2,4,5,3,6] => [4,5,1,2,3,6] => 6
{{1,6},{2,4},{3,5}}
 => [6,4,5,2,3,1] => [1,6,2,4,3,5] => [6,1,2,4,3,5] => 6
{{1,5,6},{2},{3,4}}
 => [5,2,4,3,6,1] => [1,5,6,2,3,4] => [1,5,2,6,3,4] => 6
{{1},{2,5},{3,4,6}}
 => [1,5,4,6,2,3] => [1,2,5,3,4,6] => [1,5,2,3,4,6] => 6
{{1,6},{2},{3,4},{5}}
 => [6,2,4,3,5,1] => [1,6,2,3,4,5] => [1,2,3,6,4,5] => 6
{{1},{2,5},{3},{4,6}}
 => [1,5,3,6,2,4] => [1,2,5,3,4,6] => [1,5,2,3,4,6] => 6
{{1},{2,5},{3},{4},{6}}
 => [1,5,3,4,2,6] => [1,2,5,3,4,6] => [1,5,2,3,4,6] => 6
{{1,2,5,6,7},{3,4}}
 => [2,5,4,3,6,7,1] => [1,2,5,6,7,3,4] => [5,6,1,7,2,3,4] => ? = 8
{{1,2,5,7},{3,4},{6}}
 => [2,5,4,3,7,6,1] => [1,2,5,7,3,4,6] => [1,7,5,2,3,4,6] => ? = 8
{{1,2},{3,4,5},{6,7}}
 => [2,1,4,5,3,7,6] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 8
{{1,2},{3,4,5},{6},{7}}
 => [2,1,4,5,3,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 8
{{1,2,6,7},{3,4},{5}}
 => [2,6,4,3,5,7,1] => [1,2,6,7,3,4,5] => [1,6,2,7,3,4,5] => ? = 8
{{1,2,7},{3,4},{5,6}}
 => [2,7,4,3,6,5,1] => [1,2,7,3,4,5,6] => [1,2,3,7,4,5,6] => ? = 8
{{1,2,7},{3,4},{5},{6}}
 => [2,7,4,3,5,6,1] => [1,2,7,3,4,5,6] => [1,2,3,7,4,5,6] => ? = 8
{{1,2},{3,5},{4,6,7}}
 => [2,1,5,6,3,7,4] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 8
{{1,2},{3,5},{4,6},{7}}
 => [2,1,5,6,3,4,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 8
{{1,2},{3,5},{4,7},{6}}
 => [2,1,5,7,3,6,4] => [1,2,3,5,4,7,6] => [5,7,1,2,3,4,6] => ? = 8
{{1,2},{3,5},{4},{6,7}}
 => [2,1,5,4,3,7,6] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 8
{{1,2},{3,5},{4},{6},{7}}
 => [2,1,5,4,3,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 8
{{1,4,5,6},{2,3},{7}}
 => [4,3,2,5,6,1,7] => [1,4,5,6,2,3,7] => [4,5,1,6,2,3,7] => ? = 8
{{1,4,6},{2,3},{5,7}}
 => [4,3,2,6,7,1,5] => [1,4,6,2,3,5,7] => [1,6,4,2,3,5,7] => ? = 8
{{1,4,6},{2,3},{5},{7}}
 => [4,3,2,6,5,1,7] => [1,4,6,2,3,5,7] => [1,6,4,2,3,5,7] => ? = 8
{{1,5,6},{2,3},{4,7}}
 => [5,3,2,7,6,1,4] => [1,5,6,2,3,4,7] => [1,5,2,6,3,4,7] => ? = 8
{{1,5,6},{2,3},{4},{7}}
 => [5,3,2,4,6,1,7] => [1,5,6,2,3,4,7] => [1,5,2,6,3,4,7] => ? = 8
{{1,6},{2,3},{4,5,7}}
 => [6,3,2,5,7,1,4] => [1,6,2,3,4,5,7] => [1,2,3,6,4,5,7] => ? = 8
{{1,6},{2,3},{4,5},{7}}
 => [6,3,2,5,4,1,7] => [1,6,2,3,4,5,7] => [1,2,3,6,4,5,7] => ? = 8
{{1,6},{2,3},{4,7},{5}}
 => [6,3,2,7,5,1,4] => [1,6,2,3,4,7,5] => [1,2,3,6,7,4,5] => ? = 8
{{1,6},{2,3},{4},{5,7}}
 => [6,3,2,4,7,1,5] => [1,6,2,3,4,5,7] => [1,2,3,6,4,5,7] => ? = 8
{{1,6},{2,3},{4},{5},{7}}
 => [6,3,2,4,5,1,7] => [1,6,2,3,4,5,7] => [1,2,3,6,4,5,7] => ? = 8
{{1,4,5,6},{2,7},{3}}
 => [4,7,3,5,6,1,2] => [1,4,5,6,2,7,3] => [4,5,1,6,7,2,3] => ? = 8
{{1,4,5,7},{2,6},{3}}
 => [4,6,3,5,7,2,1] => [1,4,5,7,2,6,3] => [4,5,7,1,6,2,3] => ? = 8
{{1,4,6},{2,5,7},{3}}
 => [4,5,3,6,7,1,2] => [1,4,6,2,5,7,3] => [4,6,1,5,7,2,3] => ? = 8
{{1,4,7},{2,5,6},{3}}
 => [4,5,3,7,6,2,1] => [1,4,7,2,5,6,3] => [7,4,1,5,6,2,3] => ? = 8
{{1,4,6},{2,7},{3},{5}}
 => [4,7,3,6,5,1,2] => [1,4,6,2,7,3,5] => [6,4,1,7,2,3,5] => ? = 8
{{1,4,7},{2,6},{3},{5}}
 => [4,6,3,7,5,2,1] => [1,4,7,2,6,3,5] => [4,1,7,6,2,3,5] => ? = 8
{{1,5,6},{2,4,7},{3}}
 => [5,4,3,7,6,1,2] => [1,5,6,2,4,7,3] => [1,5,6,4,7,2,3] => ? = 8
{{1,5,7},{2,4,6},{3}}
 => [5,4,3,6,7,2,1] => [1,5,7,2,4,6,3] => [5,1,7,4,6,2,3] => ? = 8
{{1,6},{2,4,5,7},{3}}
 => [6,4,3,5,7,1,2] => [1,6,2,4,5,7,3] => [1,6,4,5,7,2,3] => ? = 8
{{1,7},{2,4,5,6},{3}}
 => [7,4,3,5,6,2,1] => [1,7,2,4,5,6,3] => [1,4,7,5,6,2,3] => ? = 8
{{1,6},{2,4,7},{3},{5}}
 => [6,4,3,7,5,1,2] => [1,6,2,4,7,3,5] => [6,7,1,2,4,3,5] => ? = 8
{{1,7},{2,4,6},{3},{5}}
 => [7,4,3,6,5,2,1] => [1,7,2,4,6,3,5] => [7,6,1,2,4,3,5] => ? = 8
{{1},{2,5,6,7},{3,4}}
 => [1,5,4,3,6,7,2] => [1,2,5,6,7,3,4] => [5,6,1,7,2,3,4] => ? = 8
{{1},{2,5,7},{3,4},{6}}
 => [1,5,4,3,7,6,2] => [1,2,5,7,3,4,6] => [1,7,5,2,3,4,6] => ? = 8
{{1},{2},{3,4,5},{6,7}}
 => [1,2,4,5,3,7,6] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 8
{{1},{2},{3,4,5},{6},{7}}
 => [1,2,4,5,3,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 8
{{1},{2,6,7},{3,4},{5}}
 => [1,6,4,3,5,7,2] => [1,2,6,7,3,4,5] => [1,6,2,7,3,4,5] => ? = 8
{{1},{2,7},{3,4},{5,6}}
 => [1,7,4,3,6,5,2] => [1,2,7,3,4,5,6] => [1,2,3,7,4,5,6] => ? = 8
{{1},{2,7},{3,4},{5},{6}}
 => [1,7,4,3,5,6,2] => [1,2,7,3,4,5,6] => [1,2,3,7,4,5,6] => ? = 8
{{1,5,6},{2,7},{3},{4}}
 => [5,7,3,4,6,1,2] => [1,5,6,2,7,3,4] => [1,5,6,2,7,3,4] => ? = 8
{{1,5,7},{2,6},{3},{4}}
 => [5,6,3,4,7,2,1] => [1,5,7,2,6,3,4] => [5,1,7,2,6,3,4] => ? = 8
{{1,6},{2,5,7},{3},{4}}
 => [6,5,3,4,7,1,2] => [1,6,2,5,7,3,4] => [6,5,1,2,7,3,4] => ? = 8
{{1,7},{2,5,6},{3},{4}}
 => [7,5,3,4,6,2,1] => [1,7,2,5,6,3,4] => [7,1,5,2,6,3,4] => ? = 8
{{1},{2},{3,5},{4,6,7}}
 => [1,2,5,6,3,7,4] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 8
{{1},{2},{3,5},{4,6},{7}}
 => [1,2,5,6,3,4,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 8
{{1},{2},{3,5},{4,7},{6}}
 => [1,2,5,7,3,6,4] => [1,2,3,5,4,7,6] => [5,7,1,2,3,4,6] => ? = 8
{{1},{2},{3,5},{4},{6,7}}
 => [1,2,5,4,3,7,6] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 8
{{1},{2},{3,5},{4},{6},{7}}
 => [1,2,5,4,3,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 8
Description
The number of indices that are not cyclical small weak excedances. A cyclical small weak excedance is an index $i < n$ such that $\pi_i = i+1$, or the index $i = n$ if $\pi_n = 1$.
Matching statistic: St000311
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000311: Graphs ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 75%
Values
{{1}}
 => [1] => [.,.]
 => ([],1)
 => 0
{{1,2}}
 => [2,1] => [[.,.],.]
 => ([(0,1)],2)
 => 2
{{1},{2}}
 => [1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => 2
{{1,3,5,6},{2},{4}}
 => [3,2,5,4,6,1] => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
{{1,3,6},{2},{4},{5}}
 => [3,2,6,4,5,1] => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
{{1,5,6},{2,3,4}}
 => [5,3,4,2,6,1] => [[[[.,.],[.,.]],[.,.]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
{{1,6},{2,3,4},{5}}
 => [6,3,4,2,5,1] => [[[[.,.],[.,.]],[.,.]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
{{1},{2,3},{4,5},{6}}
 => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
{{1},{2,4,5},{3},{6}}
 => [1,4,3,5,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
{{1,6},{2,4},{3,5}}
 => [6,4,5,2,3,1] => [[[[.,.],[.,.]],[.,.]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
{{1,5,6},{2},{3,4}}
 => [5,2,4,3,6,1] => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
{{1},{2,5},{3,4,6}}
 => [1,5,4,6,2,3] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
{{1,6},{2},{3,4},{5}}
 => [6,2,4,3,5,1] => [[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
{{1},{2,5},{3},{4,6}}
 => [1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
{{1},{2,5},{3},{4},{6}}
 => [1,5,3,4,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
{{1,2,5,6,7},{3,4}}
 => [2,5,4,3,6,7,1] => [[.,[[[.,.],.],[.,[.,.]]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,2,5,7},{3,4},{6}}
 => [2,5,4,3,7,6,1] => [[.,[[[.,.],.],[[.,.],.]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,2},{3,4,5},{6,7}}
 => [2,1,4,5,3,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,2},{3,4,5},{6},{7}}
 => [2,1,4,5,3,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,2,6,7},{3,4},{5}}
 => [2,6,4,3,5,7,1] => [[.,[[[.,.],.],[.,[.,.]]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,2,7},{3,4},{5,6}}
 => [2,7,4,3,6,5,1] => [[.,[[[.,.],.],[[.,.],.]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,2,7},{3,4},{5},{6}}
 => [2,7,4,3,5,6,1] => [[.,[[[.,.],.],[.,[.,.]]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,2},{3,5},{4,6,7}}
 => [2,1,5,6,3,7,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,2},{3,5},{4,6},{7}}
 => [2,1,5,6,3,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,2},{3,5},{4,7},{6}}
 => [2,1,5,7,3,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,2},{3,5},{4},{6,7}}
 => [2,1,5,4,3,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,2},{3,5},{4},{6},{7}}
 => [2,1,5,4,3,6,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,4,5,6},{2,3},{7}}
 => [4,3,2,5,6,1,7] => [[[[.,.],.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,4,6},{2,3},{5,7}}
 => [4,3,2,6,7,1,5] => [[[[.,.],.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,4,6},{2,3},{5},{7}}
 => [4,3,2,6,5,1,7] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,5,6},{2,3},{4,7}}
 => [5,3,2,7,6,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,5,6},{2,3},{4},{7}}
 => [5,3,2,4,6,1,7] => [[[[.,.],.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,6},{2,3},{4,5,7}}
 => [6,3,2,5,7,1,4] => [[[[.,.],.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,6},{2,3},{4,5},{7}}
 => [6,3,2,5,4,1,7] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,6},{2,3},{4,7},{5}}
 => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,6},{2,3},{4},{5,7}}
 => [6,3,2,4,7,1,5] => [[[[.,.],.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,6},{2,3},{4},{5},{7}}
 => [6,3,2,4,5,1,7] => [[[[.,.],.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,4,5,6},{2,7},{3}}
 => [4,7,3,5,6,1,2] => [[[.,[.,.]],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,4,5,7},{2,6},{3}}
 => [4,6,3,5,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,4,6},{2,5,7},{3}}
 => [4,5,3,6,7,1,2] => [[[.,[.,.]],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,4,7},{2,5,6},{3}}
 => [4,5,3,7,6,2,1] => [[[[.,[.,.]],[[.,.],.]],.],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,4,6},{2,7},{3},{5}}
 => [4,7,3,6,5,1,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,4,7},{2,6},{3},{5}}
 => [4,6,3,7,5,2,1] => [[[[.,[.,.]],[[.,.],.]],.],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,5,6},{2,4,7},{3}}
 => [5,4,3,7,6,1,2] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,5,7},{2,4,6},{3}}
 => [5,4,3,6,7,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,6},{2,4,5,7},{3}}
 => [6,4,3,5,7,1,2] => [[[[.,.],.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,7},{2,4,5,6},{3}}
 => [7,4,3,5,6,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,6},{2,4,7},{3},{5}}
 => [6,4,3,7,5,1,2] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,7},{2,4,6},{3},{5}}
 => [7,4,3,6,5,2,1] => [[[[[.,.],.],[[.,.],.]],.],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1},{2,5,6,7},{3,4}}
 => [1,5,4,3,6,7,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1},{2,5,7},{3,4},{6}}
 => [1,5,4,3,7,6,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1},{2},{3,4,5},{6,7}}
 => [1,2,4,5,3,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1},{2},{3,4,5},{6},{7}}
 => [1,2,4,5,3,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1},{2,6,7},{3,4},{5}}
 => [1,6,4,3,5,7,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1},{2,7},{3,4},{5,6}}
 => [1,7,4,3,6,5,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1},{2,7},{3,4},{5},{6}}
 => [1,7,4,3,5,6,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,5,6},{2,7},{3},{4}}
 => [5,7,3,4,6,1,2] => [[[.,[.,.]],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,5,7},{2,6},{3},{4}}
 => [5,6,3,4,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,6},{2,5,7},{3},{4}}
 => [6,5,3,4,7,1,2] => [[[[.,.],.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1,7},{2,5,6},{3},{4}}
 => [7,5,3,4,6,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1},{2},{3,5},{4,6,7}}
 => [1,2,5,6,3,7,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1},{2},{3,5},{4,6},{7}}
 => [1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1},{2},{3,5},{4,7},{6}}
 => [1,2,5,7,3,6,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1},{2},{3,5},{4},{6,7}}
 => [1,2,5,4,3,7,6] => [.,[.,[[[.,.],.],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
{{1},{2},{3,5},{4},{6},{7}}
 => [1,2,5,4,3,6,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
Description
The number of vertices of odd degree in a graph.
Matching statistic: St001388
(load all 5 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St001388: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Values
{{1}}
 => [1] => [1] => ? = 0 - 2
{{1,2}}
 => [2,1] => [2,1] => 0 = 2 - 2
{{1},{2}}
 => [1,2] => [1,2] => 0 = 2 - 2
{{1,3,5,6},{2},{4}}
 => [3,2,5,4,6,1] => [5,3,2,4,6,1] => 4 = 6 - 2
{{1,3,6},{2},{4},{5}}
 => [3,2,6,4,5,1] => [3,6,2,4,5,1] => 4 = 6 - 2
{{1,5,6},{2,3,4}}
 => [5,3,4,2,6,1] => [3,5,4,2,6,1] => 4 = 6 - 2
{{1,6},{2,3,4},{5}}
 => [6,3,4,2,5,1] => [3,4,6,2,5,1] => 4 = 6 - 2
{{1},{2,3},{4,5},{6}}
 => [1,3,2,5,4,6] => [5,3,1,2,4,6] => 4 = 6 - 2
{{1},{2,4,5},{3},{6}}
 => [1,4,3,5,2,6] => [4,3,1,5,2,6] => 4 = 6 - 2
{{1,6},{2,4},{3,5}}
 => [6,4,5,2,3,1] => [2,4,6,5,3,1] => 4 = 6 - 2
{{1,5,6},{2},{3,4}}
 => [5,2,4,3,6,1] => [5,2,4,3,6,1] => 4 = 6 - 2
{{1},{2,5},{3,4,6}}
 => [1,5,4,6,2,3] => [1,5,4,2,6,3] => 4 = 6 - 2
{{1,6},{2},{3,4},{5}}
 => [6,2,4,3,5,1] => [2,6,4,3,5,1] => 4 = 6 - 2
{{1},{2,5},{3},{4,6}}
 => [1,5,3,6,2,4] => [3,5,1,2,6,4] => 4 = 6 - 2
{{1},{2,5},{3},{4},{6}}
 => [1,5,3,4,2,6] => [5,1,3,4,2,6] => 4 = 6 - 2
{{1,2,5,6,7},{3,4}}
 => [2,5,4,3,6,7,1] => [5,4,2,3,6,7,1] => ? = 8 - 2
{{1,2,5,7},{3,4},{6}}
 => [2,5,4,3,7,6,1] => [7,5,4,2,3,6,1] => ? = 8 - 2
{{1,2},{3,4,5},{6,7}}
 => [2,1,4,5,3,7,6] => [7,4,2,1,5,3,6] => ? = 8 - 2
{{1,2},{3,4,5},{6},{7}}
 => [2,1,4,5,3,6,7] => [4,2,1,5,3,6,7] => ? = 8 - 2
{{1,2,6,7},{3,4},{5}}
 => [2,6,4,3,5,7,1] => [4,6,2,3,5,7,1] => ? = 8 - 2
{{1,2,7},{3,4},{5,6}}
 => [2,7,4,3,6,5,1] => [7,4,6,2,3,5,1] => ? = 8 - 2
{{1,2,7},{3,4},{5},{6}}
 => [2,7,4,3,5,6,1] => [4,2,7,3,5,6,1] => ? = 8 - 2
{{1,2},{3,5},{4,6,7}}
 => [2,1,5,6,3,7,4] => [5,6,2,1,7,3,4] => ? = 8 - 2
{{1,2},{3,5},{4,6},{7}}
 => [2,1,5,6,3,4,7] => [2,5,1,3,6,4,7] => ? = 8 - 2
{{1,2},{3,5},{4,7},{6}}
 => [2,1,5,7,3,6,4] => [5,7,2,1,3,6,4] => ? = 8 - 2
{{1,2},{3,5},{4},{6,7}}
 => [2,1,5,4,3,7,6] => [7,5,4,2,1,3,6] => ? = 8 - 2
{{1,2},{3,5},{4},{6},{7}}
 => [2,1,5,4,3,6,7] => [5,4,2,1,3,6,7] => ? = 8 - 2
{{1,4,5,6},{2,3},{7}}
 => [4,3,2,5,6,1,7] => [4,3,2,5,6,1,7] => ? = 8 - 2
{{1,4,6},{2,3},{5,7}}
 => [4,3,2,6,7,1,5] => [4,3,2,1,6,7,5] => ? = 8 - 2
{{1,4,6},{2,3},{5},{7}}
 => [4,3,2,6,5,1,7] => [6,4,3,2,5,1,7] => ? = 8 - 2
{{1,5,6},{2,3},{4,7}}
 => [5,3,2,7,6,1,4] => [3,7,5,2,1,6,4] => ? = 8 - 2
{{1,5,6},{2,3},{4},{7}}
 => [5,3,2,4,6,1,7] => [3,5,2,4,6,1,7] => ? = 8 - 2
{{1,6},{2,3},{4,5,7}}
 => [6,3,2,5,7,1,4] => [3,2,6,1,5,7,4] => ? = 8 - 2
{{1,6},{2,3},{4,5},{7}}
 => [6,3,2,5,4,1,7] => [6,3,5,2,4,1,7] => ? = 8 - 2
{{1,6},{2,3},{4,7},{5}}
 => [6,3,2,7,5,1,4] => [3,6,7,2,1,5,4] => ? = 8 - 2
{{1,6},{2,3},{4},{5,7}}
 => [6,3,2,4,7,1,5] => [3,2,6,4,1,7,5] => ? = 8 - 2
{{1,6},{2,3},{4},{5},{7}}
 => [6,3,2,4,5,1,7] => [3,2,6,4,5,1,7] => ? = 8 - 2
{{1,4,5,6},{2,7},{3}}
 => [4,7,3,5,6,1,2] => [1,4,3,5,7,6,2] => ? = 8 - 2
{{1,4,5,7},{2,6},{3}}
 => [4,6,3,5,7,2,1] => [4,3,6,5,7,2,1] => ? = 8 - 2
{{1,4,6},{2,5,7},{3}}
 => [4,5,3,6,7,1,2] => [1,4,5,3,6,7,2] => ? = 8 - 2
{{1,4,7},{2,5,6},{3}}
 => [4,5,3,7,6,2,1] => [7,4,5,3,6,2,1] => ? = 8 - 2
{{1,4,6},{2,7},{3},{5}}
 => [4,7,3,6,5,1,2] => [1,7,4,3,6,5,2] => ? = 8 - 2
{{1,4,7},{2,6},{3},{5}}
 => [4,6,3,7,5,2,1] => [6,4,7,3,5,2,1] => ? = 8 - 2
{{1,5,6},{2,4,7},{3}}
 => [5,4,3,7,6,1,2] => [1,7,5,4,3,6,2] => ? = 8 - 2
{{1,5,7},{2,4,6},{3}}
 => [5,4,3,6,7,2,1] => [5,4,3,6,7,2,1] => ? = 8 - 2
{{1,6},{2,4,5,7},{3}}
 => [6,4,3,5,7,1,2] => [1,4,6,3,5,7,2] => ? = 8 - 2
{{1,7},{2,4,5,6},{3}}
 => [7,4,3,5,6,2,1] => [4,3,7,5,6,2,1] => ? = 8 - 2
{{1,6},{2,4,7},{3},{5}}
 => [6,4,3,7,5,1,2] => [1,6,7,4,3,5,2] => ? = 8 - 2
{{1,7},{2,4,6},{3},{5}}
 => [7,4,3,6,5,2,1] => [7,4,6,3,5,2,1] => ? = 8 - 2
{{1},{2,5,6,7},{3,4}}
 => [1,5,4,3,6,7,2] => [5,4,3,1,6,7,2] => ? = 8 - 2
{{1},{2,5,7},{3,4},{6}}
 => [1,5,4,3,7,6,2] => [7,5,4,3,1,6,2] => ? = 8 - 2
{{1},{2},{3,4,5},{6,7}}
 => [1,2,4,5,3,7,6] => [7,4,1,2,5,3,6] => ? = 8 - 2
{{1},{2},{3,4,5},{6},{7}}
 => [1,2,4,5,3,6,7] => [4,1,2,5,3,6,7] => ? = 8 - 2
{{1},{2,6,7},{3,4},{5}}
 => [1,6,4,3,5,7,2] => [4,6,3,1,5,7,2] => ? = 8 - 2
{{1},{2,7},{3,4},{5,6}}
 => [1,7,4,3,6,5,2] => [7,4,6,3,1,5,2] => ? = 8 - 2
{{1},{2,7},{3,4},{5},{6}}
 => [1,7,4,3,5,6,2] => [4,7,1,3,5,6,2] => ? = 8 - 2
{{1,5,6},{2,7},{3},{4}}
 => [5,7,3,4,6,1,2] => [1,3,5,4,7,6,2] => ? = 8 - 2
{{1,5,7},{2,6},{3},{4}}
 => [5,6,3,4,7,2,1] => [3,5,6,4,7,2,1] => ? = 8 - 2
{{1,6},{2,5,7},{3},{4}}
 => [6,5,3,4,7,1,2] => [1,3,6,5,4,7,2] => ? = 8 - 2
{{1,7},{2,5,6},{3},{4}}
 => [7,5,3,4,6,2,1] => [3,5,7,4,6,2,1] => ? = 8 - 2
{{1},{2},{3,5},{4,6,7}}
 => [1,2,5,6,3,7,4] => [5,6,1,2,7,3,4] => ? = 8 - 2
{{1},{2},{3,5},{4,6},{7}}
 => [1,2,5,6,3,4,7] => [1,5,2,3,6,4,7] => ? = 8 - 2
{{1},{2},{3,5},{4,7},{6}}
 => [1,2,5,7,3,6,4] => [5,7,1,2,3,6,4] => ? = 8 - 2
{{1},{2},{3,5},{4},{6,7}}
 => [1,2,5,4,3,7,6] => [7,5,4,1,2,3,6] => ? = 8 - 2
Description
The number of non-attacking neighbors of a permutation. For a permutation $\sigma$, the indices $i$ and $i+1$ are attacking if $|\sigma(i)-\sigma(i+1)| = 1$. Visually, this is, for $\sigma$ considered as a placement of kings on a chessboard, if the kings placed in columns $i$ and $i+1$ are non-attacking.
Matching statistic: St000724
(load all 2 compositions to match this statistic)
Mapping
Mp00221: Set partitions —conjugate⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000724: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Values
{{1}}
 => {{1}}
 => [1] => [1] => ? = 0
{{1,2}}
 => {{1},{2}}
 => [1,2] => [1,2] => 2
{{1},{2}}
 => {{1,2}}
 => [2,1] => [2,1] => 2
{{1,3,5,6},{2},{4}}
 => {{1},{2},{3,4},{5,6}}
 => [1,2,4,3,6,5] => [1,6,5,4,3,2] => 6
{{1,3,6},{2},{4},{5}}
 => {{1},{2,3,4},{5,6}}
 => [1,3,4,2,6,5] => [1,6,5,4,3,2] => 6
{{1,5,6},{2,3,4}}
 => {{1},{2},{3,6},{4},{5}}
 => [1,2,6,4,5,3] => [1,6,5,4,3,2] => 6
{{1,6},{2,3,4},{5}}
 => {{1},{2,3,6},{4},{5}}
 => [1,3,6,4,5,2] => [1,6,5,4,3,2] => 6
{{1},{2,3},{4,5},{6}}
 => {{1,2,4,6},{3},{5}}
 => [2,4,3,6,5,1] => [2,6,5,4,3,1] => 6
{{1},{2,4,5},{3},{6}}
 => {{1,2,6},{3},{4,5}}
 => [2,6,3,5,4,1] => [2,6,5,4,3,1] => 6
{{1,6},{2,4},{3,5}}
 => {{1},{2,4,6},{3,5}}
 => [1,4,5,6,3,2] => [1,6,5,4,3,2] => 6
{{1,5,6},{2},{3,4}}
 => {{1},{2},{3,5,6},{4}}
 => [1,2,5,4,6,3] => [1,6,5,4,3,2] => 6
{{1},{2,5},{3,4,6}}
 => {{1,3,6},{2,5},{4}}
 => [3,5,6,4,2,1] => [3,6,5,4,2,1] => 6
{{1,6},{2},{3,4},{5}}
 => {{1},{2,3,5,6},{4}}
 => [1,3,5,4,6,2] => [1,6,5,4,3,2] => 6
{{1},{2,5},{3},{4,6}}
 => {{1,3,6},{2,4,5}}
 => [3,4,6,5,2,1] => [3,6,5,4,2,1] => 6
{{1},{2,5},{3},{4},{6}}
 => {{1,2,6},{3,4,5}}
 => [2,6,4,5,3,1] => [2,6,5,4,3,1] => 6
{{1,2,5,6,7},{3,4}}
 => {{1},{2},{3},{4,6},{5},{7}}
 => [1,2,3,6,5,4,7] => [1,7,6,5,4,3,2] => ? = 8
{{1,2,5,7},{3,4},{6}}
 => {{1},{2,3},{4,6},{5},{7}}
 => [1,3,2,6,5,4,7] => [1,7,6,5,4,3,2] => ? = 8
{{1,2},{3,4,5},{6,7}}
 => {{1,3,6},{2},{4},{5},{7}}
 => [3,2,6,4,5,1,7] => [3,2,7,6,5,1,4] => ? = 8
{{1,2},{3,4,5},{6},{7}}
 => {{1,2,3,6},{4},{5},{7}}
 => [2,3,6,4,5,1,7] => [2,7,6,5,4,1,3] => ? = 8
{{1,2,6,7},{3,4},{5}}
 => {{1},{2},{3,4,6},{5},{7}}
 => [1,2,4,6,5,3,7] => [1,7,6,5,4,3,2] => ? = 8
{{1,2,7},{3,4},{5,6}}
 => {{1},{2,4,6},{3},{5},{7}}
 => [1,4,3,6,5,2,7] => [1,7,6,5,4,3,2] => ? = 8
{{1,2,7},{3,4},{5},{6}}
 => {{1},{2,3,4,6},{5},{7}}
 => [1,3,4,6,5,2,7] => [1,7,6,5,4,3,2] => ? = 8
{{1,2},{3,5},{4,6,7}}
 => {{1,4,6},{2},{3,5},{7}}
 => [4,2,5,6,3,1,7] => [4,2,7,6,5,1,3] => ? = 8
{{1,2},{3,5},{4,6},{7}}
 => {{1,2,4,6},{3,5},{7}}
 => [2,4,5,6,3,1,7] => [2,7,6,5,4,1,3] => ? = 8
{{1,2},{3,5},{4,7},{6}}
 => {{1,4,6},{2,3,5},{7}}
 => [4,3,5,6,2,1,7] => [4,3,7,6,2,1,5] => ? = 8
{{1,2},{3,5},{4},{6,7}}
 => {{1,3,6},{2},{4,5},{7}}
 => [3,2,6,5,4,1,7] => [3,2,7,6,5,1,4] => ? = 8
{{1,2},{3,5},{4},{6},{7}}
 => {{1,2,3,6},{4,5},{7}}
 => [2,3,6,5,4,1,7] => [2,7,6,5,4,1,3] => ? = 8
{{1,4,5,6},{2,3},{7}}
 => {{1,2},{3},{4},{5,7},{6}}
 => [2,1,3,4,7,6,5] => [2,1,7,6,5,4,3] => ? = 8
{{1,4,6},{2,3},{5,7}}
 => {{1,3},{2,4},{5,7},{6}}
 => [3,4,1,2,7,6,5] => [3,7,1,6,5,4,2] => ? = 8
{{1,4,6},{2,3},{5},{7}}
 => {{1,2},{3,4},{5,7},{6}}
 => [2,1,4,3,7,6,5] => [2,1,7,6,5,4,3] => ? = 8
{{1,5,6},{2,3},{4,7}}
 => {{1,4},{2,5,7},{3},{6}}
 => [4,5,3,1,7,6,2] => [4,7,3,1,6,5,2] => ? = 8
{{1,5,6},{2,3},{4},{7}}
 => {{1,2},{3},{4,5,7},{6}}
 => [2,1,3,5,7,6,4] => [2,1,7,6,5,4,3] => ? = 8
{{1,6},{2,3},{4,5,7}}
 => {{1,3},{2,5,7},{4},{6}}
 => [3,5,1,4,7,6,2] => [3,7,1,6,5,4,2] => ? = 8
{{1,6},{2,3},{4,5},{7}}
 => {{1,2},{3,5,7},{4},{6}}
 => [2,1,5,4,7,6,3] => [2,1,7,6,5,4,3] => ? = 8
{{1,6},{2,3},{4,7},{5}}
 => {{1,3,4},{2,5,7},{6}}
 => [3,5,4,1,7,6,2] => [3,7,6,1,5,4,2] => ? = 8
{{1,6},{2,3},{4},{5,7}}
 => {{1,3},{2,4,5,7},{6}}
 => [3,4,1,5,7,6,2] => [3,7,1,6,5,4,2] => ? = 8
{{1,6},{2,3},{4},{5},{7}}
 => {{1,2},{3,4,5,7},{6}}
 => [2,1,4,5,7,6,3] => [2,1,7,6,5,4,3] => ? = 8
{{1,4,5,6},{2,7},{3}}
 => {{1,5,6},{2,7},{3},{4}}
 => [5,7,3,4,6,1,2] => [5,7,3,6,4,1,2] => ? = 8
{{1,4,5,7},{2,6},{3}}
 => {{1},{2,5,6},{3,7},{4}}
 => [1,5,7,4,6,2,3] => [1,7,6,5,4,3,2] => ? = 8
{{1,4,6},{2,5,7},{3}}
 => {{1,3,5,6},{2,4,7}}
 => [3,4,5,7,6,1,2] => [3,7,6,5,4,1,2] => ? = 8
{{1,4,7},{2,5,6},{3}}
 => {{1},{2,5,6},{3},{4,7}}
 => [1,5,3,7,6,2,4] => [1,7,6,5,4,3,2] => ? = 8
{{1,4,6},{2,7},{3},{5}}
 => {{1,5,6},{2,7},{3,4}}
 => [5,7,4,3,6,1,2] => [5,7,4,3,6,1,2] => ? = 8
{{1,4,7},{2,6},{3},{5}}
 => {{1},{2,5,6},{3,4,7}}
 => [1,5,4,7,6,2,3] => [1,7,6,5,4,3,2] => ? = 8
{{1,5,6},{2,4,7},{3}}
 => {{1,4},{2,7},{3},{5,6}}
 => [4,7,3,1,6,5,2] => [4,7,3,1,6,5,2] => ? = 8
{{1,5,7},{2,4,6},{3}}
 => {{1},{2,4},{3,7},{5,6}}
 => [1,4,7,2,6,5,3] => [1,7,6,5,4,3,2] => ? = 8
{{1,6},{2,4,5,7},{3}}
 => {{1,3},{2,7},{4},{5,6}}
 => [3,7,1,4,6,5,2] => [3,7,1,6,5,4,2] => ? = 8
{{1,7},{2,4,5,6},{3}}
 => {{1},{2,7},{3},{4},{5,6}}
 => [1,7,3,4,6,5,2] => [1,7,6,5,4,3,2] => ? = 8
{{1,6},{2,4,7},{3},{5}}
 => {{1,3,4},{2,7},{5,6}}
 => [3,7,4,1,6,5,2] => [3,7,6,1,5,4,2] => ? = 8
{{1,7},{2,4,6},{3},{5}}
 => {{1},{2,7},{3,4},{5,6}}
 => [1,7,4,3,6,5,2] => [1,7,6,5,4,3,2] => ? = 8
{{1},{2,5,6,7},{3,4}}
 => {{1,7},{2},{3},{4,6},{5}}
 => [7,2,3,6,5,4,1] => [7,2,6,5,4,3,1] => ? = 8
{{1},{2,5,7},{3,4},{6}}
 => {{1,7},{2,3},{4,6},{5}}
 => [7,3,2,6,5,4,1] => [7,3,2,6,5,4,1] => ? = 8
{{1},{2},{3,4,5},{6,7}}
 => {{1,3,6,7},{2},{4},{5}}
 => [3,2,6,4,5,7,1] => [3,2,7,6,5,4,1] => ? = 8
{{1},{2},{3,4,5},{6},{7}}
 => {{1,2,3,6,7},{4},{5}}
 => [2,3,6,4,5,7,1] => [2,7,6,5,4,3,1] => ? = 8
{{1},{2,6,7},{3,4},{5}}
 => {{1,7},{2},{3,4,6},{5}}
 => [7,2,4,6,5,3,1] => [7,2,6,5,4,3,1] => ? = 8
{{1},{2,7},{3,4},{5,6}}
 => {{1,7},{2,4,6},{3},{5}}
 => [7,4,3,6,5,2,1] => [7,4,3,6,5,2,1] => ? = 8
{{1},{2,7},{3,4},{5},{6}}
 => {{1,7},{2,3,4,6},{5}}
 => [7,3,4,6,5,2,1] => [7,3,6,5,4,2,1] => ? = 8
{{1,5,6},{2,7},{3},{4}}
 => {{1,4,5,6},{2,7},{3}}
 => [4,7,3,5,6,1,2] => [4,7,3,6,5,1,2] => ? = 8
{{1,5,7},{2,6},{3},{4}}
 => {{1},{2,4,5,6},{3,7}}
 => [1,4,7,5,6,2,3] => [1,7,6,5,4,3,2] => ? = 8
{{1,6},{2,5,7},{3},{4}}
 => {{1,3},{2,7},{4,5,6}}
 => [3,7,1,5,6,4,2] => [3,7,1,6,5,4,2] => ? = 8
{{1,7},{2,5,6},{3},{4}}
 => {{1},{2,7},{3},{4,5,6}}
 => [1,7,3,5,6,4,2] => [1,7,6,5,4,3,2] => ? = 8
{{1},{2},{3,5},{4,6,7}}
 => {{1,4,6,7},{2},{3,5}}
 => [4,2,5,6,3,7,1] => [4,2,7,6,5,3,1] => ? = 8
{{1},{2},{3,5},{4,6},{7}}
 => {{1,2,4,6,7},{3,5}}
 => [2,4,5,6,3,7,1] => [2,7,6,5,4,3,1] => ? = 8
{{1},{2},{3,5},{4,7},{6}}
 => {{1,4,6,7},{2,3,5}}
 => [4,3,5,6,2,7,1] => [4,3,7,6,2,5,1] => ? = 8
{{1},{2},{3,5},{4},{6,7}}
 => {{1,3,6,7},{2},{4,5}}
 => [3,2,6,5,4,7,1] => [3,2,7,6,5,4,1] => ? = 8
Description
The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. Associate an increasing binary tree to the permutation using [[Mp00061]]. Then follow the path starting at the root which always selects the child with the smaller label. This statistic is the label of the leaf in the path, see [1]. Han [2] showed that this statistic is (up to a shift) equidistributed on zigzag permutations (permutations $\pi$ such that $\pi(1) < \pi(2) > \pi(3) \cdots$) with the greater neighbor of the maximum ([[St000060]]), see also [3].
Matching statistic: St000844
(load all 7 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000844: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Values
{{1}}
 => [1] => [1] => [1] => ? = 0
{{1,2}}
 => [2,1] => [1,2] => [2,1] => 2
{{1},{2}}
 => [1,2] => [1,2] => [2,1] => 2
{{1,3,5,6},{2},{4}}
 => [3,2,5,4,6,1] => [1,3,5,6,2,4] => [4,2,6,5,3,1] => 6
{{1,3,6},{2},{4},{5}}
 => [3,2,6,4,5,1] => [1,3,6,2,4,5] => [5,4,2,6,3,1] => 6
{{1,5,6},{2,3,4}}
 => [5,3,4,2,6,1] => [1,5,6,2,3,4] => [4,3,2,6,5,1] => 6
{{1,6},{2,3,4},{5}}
 => [6,3,4,2,5,1] => [1,6,2,3,4,5] => [5,4,3,2,6,1] => 6
{{1},{2,3},{4,5},{6}}
 => [1,3,2,5,4,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 6
{{1},{2,4,5},{3},{6}}
 => [1,4,3,5,2,6] => [1,2,4,5,3,6] => [6,3,5,4,2,1] => 6
{{1,6},{2,4},{3,5}}
 => [6,4,5,2,3,1] => [1,6,2,4,3,5] => [5,3,4,2,6,1] => 6
{{1,5,6},{2},{3,4}}
 => [5,2,4,3,6,1] => [1,5,6,2,3,4] => [4,3,2,6,5,1] => 6
{{1},{2,5},{3,4,6}}
 => [1,5,4,6,2,3] => [1,2,5,3,4,6] => [6,4,3,5,2,1] => 6
{{1,6},{2},{3,4},{5}}
 => [6,2,4,3,5,1] => [1,6,2,3,4,5] => [5,4,3,2,6,1] => 6
{{1},{2,5},{3},{4,6}}
 => [1,5,3,6,2,4] => [1,2,5,3,4,6] => [6,4,3,5,2,1] => 6
{{1},{2,5},{3},{4},{6}}
 => [1,5,3,4,2,6] => [1,2,5,3,4,6] => [6,4,3,5,2,1] => 6
{{1,2,5,6,7},{3,4}}
 => [2,5,4,3,6,7,1] => [1,2,5,6,7,3,4] => [4,3,7,6,5,2,1] => ? = 8
{{1,2,5,7},{3,4},{6}}
 => [2,5,4,3,7,6,1] => [1,2,5,7,3,4,6] => [6,4,3,7,5,2,1] => ? = 8
{{1,2},{3,4,5},{6,7}}
 => [2,1,4,5,3,7,6] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 8
{{1,2},{3,4,5},{6},{7}}
 => [2,1,4,5,3,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 8
{{1,2,6,7},{3,4},{5}}
 => [2,6,4,3,5,7,1] => [1,2,6,7,3,4,5] => [5,4,3,7,6,2,1] => ? = 8
{{1,2,7},{3,4},{5,6}}
 => [2,7,4,3,6,5,1] => [1,2,7,3,4,5,6] => [6,5,4,3,7,2,1] => ? = 8
{{1,2,7},{3,4},{5},{6}}
 => [2,7,4,3,5,6,1] => [1,2,7,3,4,5,6] => [6,5,4,3,7,2,1] => ? = 8
{{1,2},{3,5},{4,6,7}}
 => [2,1,5,6,3,7,4] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 8
{{1,2},{3,5},{4,6},{7}}
 => [2,1,5,6,3,4,7] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 8
{{1,2},{3,5},{4,7},{6}}
 => [2,1,5,7,3,6,4] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => ? = 8
{{1,2},{3,5},{4},{6,7}}
 => [2,1,5,4,3,7,6] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 8
{{1,2},{3,5},{4},{6},{7}}
 => [2,1,5,4,3,6,7] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 8
{{1,4,5,6},{2,3},{7}}
 => [4,3,2,5,6,1,7] => [1,4,5,6,2,3,7] => [7,3,2,6,5,4,1] => ? = 8
{{1,4,6},{2,3},{5,7}}
 => [4,3,2,6,7,1,5] => [1,4,6,2,3,5,7] => [7,5,3,2,6,4,1] => ? = 8
{{1,4,6},{2,3},{5},{7}}
 => [4,3,2,6,5,1,7] => [1,4,6,2,3,5,7] => [7,5,3,2,6,4,1] => ? = 8
{{1,5,6},{2,3},{4,7}}
 => [5,3,2,7,6,1,4] => [1,5,6,2,3,4,7] => [7,4,3,2,6,5,1] => ? = 8
{{1,5,6},{2,3},{4},{7}}
 => [5,3,2,4,6,1,7] => [1,5,6,2,3,4,7] => [7,4,3,2,6,5,1] => ? = 8
{{1,6},{2,3},{4,5,7}}
 => [6,3,2,5,7,1,4] => [1,6,2,3,4,5,7] => [7,5,4,3,2,6,1] => ? = 8
{{1,6},{2,3},{4,5},{7}}
 => [6,3,2,5,4,1,7] => [1,6,2,3,4,5,7] => [7,5,4,3,2,6,1] => ? = 8
{{1,6},{2,3},{4,7},{5}}
 => [6,3,2,7,5,1,4] => [1,6,2,3,4,7,5] => [5,7,4,3,2,6,1] => ? = 8
{{1,6},{2,3},{4},{5,7}}
 => [6,3,2,4,7,1,5] => [1,6,2,3,4,5,7] => [7,5,4,3,2,6,1] => ? = 8
{{1,6},{2,3},{4},{5},{7}}
 => [6,3,2,4,5,1,7] => [1,6,2,3,4,5,7] => [7,5,4,3,2,6,1] => ? = 8
{{1,4,5,6},{2,7},{3}}
 => [4,7,3,5,6,1,2] => [1,4,5,6,2,7,3] => [3,7,2,6,5,4,1] => ? = 8
{{1,4,5,7},{2,6},{3}}
 => [4,6,3,5,7,2,1] => [1,4,5,7,2,6,3] => [3,6,2,7,5,4,1] => ? = 8
{{1,4,6},{2,5,7},{3}}
 => [4,5,3,6,7,1,2] => [1,4,6,2,5,7,3] => [3,7,5,2,6,4,1] => ? = 8
{{1,4,7},{2,5,6},{3}}
 => [4,5,3,7,6,2,1] => [1,4,7,2,5,6,3] => [3,6,5,2,7,4,1] => ? = 8
{{1,4,6},{2,7},{3},{5}}
 => [4,7,3,6,5,1,2] => [1,4,6,2,7,3,5] => [5,3,7,2,6,4,1] => ? = 8
{{1,4,7},{2,6},{3},{5}}
 => [4,6,3,7,5,2,1] => [1,4,7,2,6,3,5] => [5,3,6,2,7,4,1] => ? = 8
{{1,5,6},{2,4,7},{3}}
 => [5,4,3,7,6,1,2] => [1,5,6,2,4,7,3] => [3,7,4,2,6,5,1] => ? = 8
{{1,5,7},{2,4,6},{3}}
 => [5,4,3,6,7,2,1] => [1,5,7,2,4,6,3] => [3,6,4,2,7,5,1] => ? = 8
{{1,6},{2,4,5,7},{3}}
 => [6,4,3,5,7,1,2] => [1,6,2,4,5,7,3] => [3,7,5,4,2,6,1] => ? = 8
{{1,7},{2,4,5,6},{3}}
 => [7,4,3,5,6,2,1] => [1,7,2,4,5,6,3] => [3,6,5,4,2,7,1] => ? = 8
{{1,6},{2,4,7},{3},{5}}
 => [6,4,3,7,5,1,2] => [1,6,2,4,7,3,5] => [5,3,7,4,2,6,1] => ? = 8
{{1,7},{2,4,6},{3},{5}}
 => [7,4,3,6,5,2,1] => [1,7,2,4,6,3,5] => [5,3,6,4,2,7,1] => ? = 8
{{1},{2,5,6,7},{3,4}}
 => [1,5,4,3,6,7,2] => [1,2,5,6,7,3,4] => [4,3,7,6,5,2,1] => ? = 8
{{1},{2,5,7},{3,4},{6}}
 => [1,5,4,3,7,6,2] => [1,2,5,7,3,4,6] => [6,4,3,7,5,2,1] => ? = 8
{{1},{2},{3,4,5},{6,7}}
 => [1,2,4,5,3,7,6] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 8
{{1},{2},{3,4,5},{6},{7}}
 => [1,2,4,5,3,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 8
{{1},{2,6,7},{3,4},{5}}
 => [1,6,4,3,5,7,2] => [1,2,6,7,3,4,5] => [5,4,3,7,6,2,1] => ? = 8
{{1},{2,7},{3,4},{5,6}}
 => [1,7,4,3,6,5,2] => [1,2,7,3,4,5,6] => [6,5,4,3,7,2,1] => ? = 8
{{1},{2,7},{3,4},{5},{6}}
 => [1,7,4,3,5,6,2] => [1,2,7,3,4,5,6] => [6,5,4,3,7,2,1] => ? = 8
{{1,5,6},{2,7},{3},{4}}
 => [5,7,3,4,6,1,2] => [1,5,6,2,7,3,4] => [4,3,7,2,6,5,1] => ? = 8
{{1,5,7},{2,6},{3},{4}}
 => [5,6,3,4,7,2,1] => [1,5,7,2,6,3,4] => [4,3,6,2,7,5,1] => ? = 8
{{1,6},{2,5,7},{3},{4}}
 => [6,5,3,4,7,1,2] => [1,6,2,5,7,3,4] => [4,3,7,5,2,6,1] => ? = 8
{{1,7},{2,5,6},{3},{4}}
 => [7,5,3,4,6,2,1] => [1,7,2,5,6,3,4] => [4,3,6,5,2,7,1] => ? = 8
{{1},{2},{3,5},{4,6,7}}
 => [1,2,5,6,3,7,4] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 8
{{1},{2},{3,5},{4,6},{7}}
 => [1,2,5,6,3,4,7] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 8
{{1},{2},{3,5},{4,7},{6}}
 => [1,2,5,7,3,6,4] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => ? = 8
{{1},{2},{3,5},{4},{6,7}}
 => [1,2,5,4,3,7,6] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 8
Description
The size of the largest block in the direct sum decomposition of a permutation. A component of a permutation $\pi$ is a set of consecutive numbers $\{a,a+1,\dots, b\}$ such that $a\leq \pi(i) \leq b$ for all $a\leq i\leq b$. This statistic is the size of the largest component which does not properly contain another component.
Matching statistic: St000060
(load all 4 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000060: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Values
{{1}}
 => [1] => [1] => [1] => ? = 0 - 1
{{1,2}}
 => [2,1] => [2,1] => [2,1] => 1 = 2 - 1
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 1 = 2 - 1
{{1,3,5,6},{2},{4}}
 => [3,2,5,4,6,1] => [2,4,3,6,5,1] => [2,6,5,4,3,1] => 5 = 6 - 1
{{1,3,6},{2},{4},{5}}
 => [3,2,6,4,5,1] => [2,4,3,5,6,1] => [2,6,5,4,3,1] => 5 = 6 - 1
{{1,5,6},{2,3,4}}
 => [5,3,4,2,6,1] => [3,6,4,1,5,2] => [3,6,5,1,4,2] => 5 = 6 - 1
{{1,6},{2,3,4},{5}}
 => [6,3,4,2,5,1] => [3,5,4,1,6,2] => [3,6,5,1,4,2] => 5 = 6 - 1
{{1},{2,3},{4,5},{6}}
 => [1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,6,5,4,3,2] => 5 = 6 - 1
{{1},{2,4,5},{3},{6}}
 => [1,4,3,5,2,6] => [1,3,5,4,2,6] => [1,6,5,4,3,2] => 5 = 6 - 1
{{1,6},{2,4},{3,5}}
 => [6,4,5,2,3,1] => [4,6,5,2,1,3] => [4,6,5,2,1,3] => 5 = 6 - 1
{{1,5,6},{2},{3,4}}
 => [5,2,4,3,6,1] => [2,4,6,1,5,3] => [2,6,5,1,4,3] => 5 = 6 - 1
{{1},{2,5},{3,4,6}}
 => [1,5,4,6,2,3] => [1,5,6,4,3,2] => [1,6,5,4,3,2] => 5 = 6 - 1
{{1,6},{2},{3,4},{5}}
 => [6,2,4,3,5,1] => [2,4,5,1,6,3] => [2,6,5,1,4,3] => 5 = 6 - 1
{{1},{2,5},{3},{4,6}}
 => [1,5,3,6,2,4] => [1,3,6,5,4,2] => [1,6,5,4,3,2] => 5 = 6 - 1
{{1},{2,5},{3},{4},{6}}
 => [1,5,3,4,2,6] => [1,3,4,5,2,6] => [1,6,5,4,3,2] => 5 = 6 - 1
{{1,2,5,6,7},{3,4}}
 => [2,5,4,3,6,7,1] => [4,2,7,1,5,6,3] => [4,2,7,1,6,5,3] => ? = 8 - 1
{{1,2,5,7},{3,4},{6}}
 => [2,5,4,3,7,6,1] => [4,2,6,1,5,7,3] => [4,2,7,1,6,5,3] => ? = 8 - 1
{{1,2},{3,4,5},{6,7}}
 => [2,1,4,5,3,7,6] => [2,1,5,4,3,7,6] => [2,1,7,6,5,4,3] => ? = 8 - 1
{{1,2},{3,4,5},{6},{7}}
 => [2,1,4,5,3,6,7] => [2,1,5,4,3,6,7] => [2,1,7,6,5,4,3] => ? = 8 - 1
{{1,2,6,7},{3,4},{5}}
 => [2,6,4,3,5,7,1] => [4,2,5,1,7,6,3] => [4,2,7,1,6,5,3] => ? = 8 - 1
{{1,2,7},{3,4},{5,6}}
 => [2,7,4,3,6,5,1] => [4,2,6,1,7,3,5] => [4,2,7,1,6,5,3] => ? = 8 - 1
{{1,2,7},{3,4},{5},{6}}
 => [2,7,4,3,5,6,1] => [4,2,5,1,6,7,3] => [4,2,7,1,6,5,3] => ? = 8 - 1
{{1,2},{3,5},{4,6,7}}
 => [2,1,5,6,3,7,4] => [2,1,7,5,4,6,3] => [2,1,7,6,5,4,3] => ? = 8 - 1
{{1,2},{3,5},{4,6},{7}}
 => [2,1,5,6,3,4,7] => [2,1,6,5,4,3,7] => [2,1,7,6,5,4,3] => ? = 8 - 1
{{1,2},{3,5},{4,7},{6}}
 => [2,1,5,7,3,6,4] => [2,1,6,5,4,7,3] => [2,1,7,6,5,4,3] => ? = 8 - 1
{{1,2},{3,5},{4},{6,7}}
 => [2,1,5,4,3,7,6] => [2,1,4,5,3,7,6] => [2,1,7,6,5,4,3] => ? = 8 - 1
{{1,2},{3,5},{4},{6},{7}}
 => [2,1,5,4,3,6,7] => [2,1,4,5,3,6,7] => [2,1,7,6,5,4,3] => ? = 8 - 1
{{1,4,5,6},{2,3},{7}}
 => [4,3,2,5,6,1,7] => [3,6,1,4,5,2,7] => [3,7,1,6,5,4,2] => ? = 8 - 1
{{1,4,6},{2,3},{5,7}}
 => [4,3,2,6,7,1,5] => [3,7,1,4,6,5,2] => [3,7,1,6,5,4,2] => ? = 8 - 1
{{1,4,6},{2,3},{5},{7}}
 => [4,3,2,6,5,1,7] => [3,5,1,4,6,2,7] => [3,7,1,6,5,4,2] => ? = 8 - 1
{{1,5,6},{2,3},{4,7}}
 => [5,3,2,7,6,1,4] => [3,7,1,5,6,4,2] => [3,7,1,6,5,4,2] => ? = 8 - 1
{{1,5,6},{2,3},{4},{7}}
 => [5,3,2,4,6,1,7] => [3,4,1,6,5,2,7] => [3,7,1,6,5,4,2] => ? = 8 - 1
{{1,6},{2,3},{4,5,7}}
 => [6,3,2,5,7,1,4] => [3,6,1,7,5,4,2] => [3,7,1,6,5,4,2] => ? = 8 - 1
{{1,6},{2,3},{4,5},{7}}
 => [6,3,2,5,4,1,7] => [3,5,1,6,2,4,7] => [3,7,1,6,5,4,2] => ? = 8 - 1
{{1,6},{2,3},{4,7},{5}}
 => [6,3,2,7,5,1,4] => [3,6,1,5,7,4,2] => [3,7,1,6,5,4,2] => ? = 8 - 1
{{1,6},{2,3},{4},{5,7}}
 => [6,3,2,4,7,1,5] => [3,4,1,7,6,5,2] => [3,7,1,6,5,4,2] => ? = 8 - 1
{{1,6},{2,3},{4},{5},{7}}
 => [6,3,2,4,5,1,7] => [3,4,1,5,6,2,7] => [3,7,1,6,5,4,2] => ? = 8 - 1
{{1,4,5,6},{2,7},{3}}
 => [4,7,3,5,6,1,2] => [4,3,7,5,6,2,1] => [4,3,7,6,5,2,1] => ? = 8 - 1
{{1,4,5,7},{2,6},{3}}
 => [4,6,3,5,7,2,1] => [4,3,7,6,5,1,2] => [4,3,7,6,5,1,2] => ? = 8 - 1
{{1,4,6},{2,5,7},{3}}
 => [4,5,3,6,7,1,2] => [6,3,7,4,5,2,1] => [6,3,7,5,4,2,1] => ? = 8 - 1
{{1,4,7},{2,5,6},{3}}
 => [4,5,3,7,6,2,1] => [5,3,7,4,6,1,2] => [5,3,7,6,4,1,2] => ? = 8 - 1
{{1,4,6},{2,7},{3},{5}}
 => [4,7,3,6,5,1,2] => [4,3,6,5,7,2,1] => [4,3,7,6,5,2,1] => ? = 8 - 1
{{1,4,7},{2,6},{3},{5}}
 => [4,6,3,7,5,2,1] => [5,3,6,4,7,1,2] => [5,3,7,6,4,1,2] => ? = 8 - 1
{{1,5,6},{2,4,7},{3}}
 => [5,4,3,7,6,1,2] => [3,5,7,4,6,2,1] => [3,7,6,5,4,2,1] => ? = 8 - 1
{{1,5,7},{2,4,6},{3}}
 => [5,4,3,6,7,2,1] => [3,6,7,4,5,1,2] => [3,7,6,5,4,1,2] => ? = 8 - 1
{{1,6},{2,4,5,7},{3}}
 => [6,4,3,5,7,1,2] => [3,4,7,6,5,2,1] => [3,7,6,5,4,2,1] => ? = 8 - 1
{{1,7},{2,4,5,6},{3}}
 => [7,4,3,5,6,2,1] => [3,4,7,5,6,1,2] => [3,7,6,5,4,1,2] => ? = 8 - 1
{{1,6},{2,4,7},{3},{5}}
 => [6,4,3,7,5,1,2] => [3,5,6,4,7,2,1] => [3,7,6,5,4,2,1] => ? = 8 - 1
{{1,7},{2,4,6},{3},{5}}
 => [7,4,3,6,5,2,1] => [3,4,6,5,7,1,2] => [3,7,6,5,4,1,2] => ? = 8 - 1
{{1},{2,5,6,7},{3,4}}
 => [1,5,4,3,6,7,2] => [1,4,7,2,5,6,3] => [1,7,6,5,4,3,2] => ? = 8 - 1
{{1},{2,5,7},{3,4},{6}}
 => [1,5,4,3,7,6,2] => [1,4,6,2,5,7,3] => [1,7,6,5,4,3,2] => ? = 8 - 1
{{1},{2},{3,4,5},{6,7}}
 => [1,2,4,5,3,7,6] => [1,2,5,4,3,7,6] => [1,7,6,5,4,3,2] => ? = 8 - 1
{{1},{2},{3,4,5},{6},{7}}
 => [1,2,4,5,3,6,7] => [1,2,5,4,3,6,7] => [1,7,6,5,4,3,2] => ? = 8 - 1
{{1},{2,6,7},{3,4},{5}}
 => [1,6,4,3,5,7,2] => [1,4,5,2,7,6,3] => [1,7,6,5,4,3,2] => ? = 8 - 1
{{1},{2,7},{3,4},{5,6}}
 => [1,7,4,3,6,5,2] => [1,4,6,2,7,3,5] => [1,7,6,5,4,3,2] => ? = 8 - 1
{{1},{2,7},{3,4},{5},{6}}
 => [1,7,4,3,5,6,2] => [1,4,5,2,6,7,3] => [1,7,6,5,4,3,2] => ? = 8 - 1
{{1,5,6},{2,7},{3},{4}}
 => [5,7,3,4,6,1,2] => [4,3,5,7,6,2,1] => [4,3,7,6,5,2,1] => ? = 8 - 1
{{1,5,7},{2,6},{3},{4}}
 => [5,6,3,4,7,2,1] => [4,3,6,7,5,1,2] => [4,3,7,6,5,1,2] => ? = 8 - 1
{{1,6},{2,5,7},{3},{4}}
 => [6,5,3,4,7,1,2] => [3,4,6,7,5,2,1] => [3,7,6,5,4,2,1] => ? = 8 - 1
{{1,7},{2,5,6},{3},{4}}
 => [7,5,3,4,6,2,1] => [3,4,5,7,6,1,2] => [3,7,6,5,4,1,2] => ? = 8 - 1
{{1},{2},{3,5},{4,6,7}}
 => [1,2,5,6,3,7,4] => [1,2,7,5,4,6,3] => [1,7,6,5,4,3,2] => ? = 8 - 1
{{1},{2},{3,5},{4,6},{7}}
 => [1,2,5,6,3,4,7] => [1,2,6,5,4,3,7] => [1,7,6,5,4,3,2] => ? = 8 - 1
{{1},{2},{3,5},{4,7},{6}}
 => [1,2,5,7,3,6,4] => [1,2,6,5,4,7,3] => [1,7,6,5,4,3,2] => ? = 8 - 1
{{1},{2},{3,5},{4},{6,7}}
 => [1,2,5,4,3,7,6] => [1,2,4,5,3,7,6] => [1,7,6,5,4,3,2] => ? = 8 - 1
Description
The greater neighbor of the maximum. Han [2] showed that this statistic is (up to a shift) equidistributed on zigzag permutations (permutations $\pi$ such that $\pi(1) < \pi(2) > \pi(3) \cdots$) with the smallest path leaf label of the binary tree associated to a permutation ([[St000724]]), see also [3].
Matching statistic: St000197
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000197: Alternating sign matrices ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Values
{{1}}
 => [1] => [] => []
 => ? = 0 - 1
{{1,2}}
 => [2,1] => [1] => [[1]]
 => 1 = 2 - 1
{{1},{2}}
 => [1,2] => [1] => [[1]]
 => 1 = 2 - 1
{{1,3,5,6},{2},{4}}
 => [3,2,5,4,6,1] => [3,2,5,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
 => 5 = 6 - 1
{{1,3,6},{2},{4},{5}}
 => [3,2,6,4,5,1] => [3,2,4,5,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 5 = 6 - 1
{{1,5,6},{2,3,4}}
 => [5,3,4,2,6,1] => [5,3,4,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
 => 5 = 6 - 1
{{1,6},{2,3,4},{5}}
 => [6,3,4,2,5,1] => [3,4,2,5,1] => [[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
 => 5 = 6 - 1
{{1},{2,3},{4,5},{6}}
 => [1,3,2,5,4,6] => [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 5 = 6 - 1
{{1},{2,4,5},{3},{6}}
 => [1,4,3,5,2,6] => [1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
 => 5 = 6 - 1
{{1,6},{2,4},{3,5}}
 => [6,4,5,2,3,1] => [4,5,2,3,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
 => 5 = 6 - 1
{{1,5,6},{2},{3,4}}
 => [5,2,4,3,6,1] => [5,2,4,3,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
 => 5 = 6 - 1
{{1},{2,5},{3,4,6}}
 => [1,5,4,6,2,3] => [1,5,4,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
 => 5 = 6 - 1
{{1,6},{2},{3,4},{5}}
 => [6,2,4,3,5,1] => [2,4,3,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
 => 5 = 6 - 1
{{1},{2,5},{3},{4,6}}
 => [1,5,3,6,2,4] => [1,5,3,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
 => 5 = 6 - 1
{{1},{2,5},{3},{4},{6}}
 => [1,5,3,4,2,6] => [1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
 => 5 = 6 - 1
{{1,2,5,6,7},{3,4}}
 => [2,5,4,3,6,7,1] => [2,5,4,3,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
 => ? = 8 - 1
{{1,2,5,7},{3,4},{6}}
 => [2,5,4,3,7,6,1] => [2,5,4,3,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
 => ? = 8 - 1
{{1,2},{3,4,5},{6,7}}
 => [2,1,4,5,3,7,6] => [2,1,4,5,3,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => ? = 8 - 1
{{1,2},{3,4,5},{6},{7}}
 => [2,1,4,5,3,6,7] => [2,1,4,5,3,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => ? = 8 - 1
{{1,2,6,7},{3,4},{5}}
 => [2,6,4,3,5,7,1] => [2,6,4,3,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
 => ? = 8 - 1
{{1,2,7},{3,4},{5,6}}
 => [2,7,4,3,6,5,1] => [2,4,3,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 8 - 1
{{1,2,7},{3,4},{5},{6}}
 => [2,7,4,3,5,6,1] => [2,4,3,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 8 - 1
{{1,2},{3,5},{4,6,7}}
 => [2,1,5,6,3,7,4] => [2,1,5,6,3,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
 => ? = 8 - 1
{{1,2},{3,5},{4,6},{7}}
 => [2,1,5,6,3,4,7] => [2,1,5,6,3,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
 => ? = 8 - 1
{{1,2},{3,5},{4,7},{6}}
 => [2,1,5,7,3,6,4] => [2,1,5,3,6,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 8 - 1
{{1,2},{3,5},{4},{6,7}}
 => [2,1,5,4,3,7,6] => [2,1,5,4,3,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
 => ? = 8 - 1
{{1,2},{3,5},{4},{6},{7}}
 => [2,1,5,4,3,6,7] => [2,1,5,4,3,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
 => ? = 8 - 1
{{1,4,5,6},{2,3},{7}}
 => [4,3,2,5,6,1,7] => [4,3,2,5,6,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 8 - 1
{{1,4,6},{2,3},{5,7}}
 => [4,3,2,6,7,1,5] => [4,3,2,6,1,5] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
 => ? = 8 - 1
{{1,4,6},{2,3},{5},{7}}
 => [4,3,2,6,5,1,7] => [4,3,2,6,5,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 8 - 1
{{1,5,6},{2,3},{4,7}}
 => [5,3,2,7,6,1,4] => [5,3,2,6,1,4] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 8 - 1
{{1,5,6},{2,3},{4},{7}}
 => [5,3,2,4,6,1,7] => [5,3,2,4,6,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
 => ? = 8 - 1
{{1,6},{2,3},{4,5,7}}
 => [6,3,2,5,7,1,4] => [6,3,2,5,1,4] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0]]
 => ? = 8 - 1
{{1,6},{2,3},{4,5},{7}}
 => [6,3,2,5,4,1,7] => [6,3,2,5,4,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
 => ? = 8 - 1
{{1,6},{2,3},{4,7},{5}}
 => [6,3,2,7,5,1,4] => [6,3,2,5,1,4] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0]]
 => ? = 8 - 1
{{1,6},{2,3},{4},{5,7}}
 => [6,3,2,4,7,1,5] => [6,3,2,4,1,5] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
 => ? = 8 - 1
{{1,6},{2,3},{4},{5},{7}}
 => [6,3,2,4,5,1,7] => [6,3,2,4,5,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0]]
 => ? = 8 - 1
{{1,4,5,6},{2,7},{3}}
 => [4,7,3,5,6,1,2] => [4,3,5,6,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
 => ? = 8 - 1
{{1,4,5,7},{2,6},{3}}
 => [4,6,3,5,7,2,1] => [4,6,3,5,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => ? = 8 - 1
{{1,4,6},{2,5,7},{3}}
 => [4,5,3,6,7,1,2] => [4,5,3,6,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 8 - 1
{{1,4,7},{2,5,6},{3}}
 => [4,5,3,7,6,2,1] => [4,5,3,6,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 8 - 1
{{1,4,6},{2,7},{3},{5}}
 => [4,7,3,6,5,1,2] => [4,3,6,5,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => ? = 8 - 1
{{1,4,7},{2,6},{3},{5}}
 => [4,6,3,7,5,2,1] => [4,6,3,5,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => ? = 8 - 1
{{1,5,6},{2,4,7},{3}}
 => [5,4,3,7,6,1,2] => [5,4,3,6,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 8 - 1
{{1,5,7},{2,4,6},{3}}
 => [5,4,3,6,7,2,1] => [5,4,3,6,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 8 - 1
{{1,6},{2,4,5,7},{3}}
 => [6,4,3,5,7,1,2] => [6,4,3,5,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
 => ? = 8 - 1
{{1,7},{2,4,5,6},{3}}
 => [7,4,3,5,6,2,1] => [4,3,5,6,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
 => ? = 8 - 1
{{1,6},{2,4,7},{3},{5}}
 => [6,4,3,7,5,1,2] => [6,4,3,5,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
 => ? = 8 - 1
{{1,7},{2,4,6},{3},{5}}
 => [7,4,3,6,5,2,1] => [4,3,6,5,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => ? = 8 - 1
{{1},{2,5,6,7},{3,4}}
 => [1,5,4,3,6,7,2] => [1,5,4,3,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
 => ? = 8 - 1
{{1},{2,5,7},{3,4},{6}}
 => [1,5,4,3,7,6,2] => [1,5,4,3,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
 => ? = 8 - 1
{{1},{2},{3,4,5},{6,7}}
 => [1,2,4,5,3,7,6] => [1,2,4,5,3,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => ? = 8 - 1
{{1},{2},{3,4,5},{6},{7}}
 => [1,2,4,5,3,6,7] => [1,2,4,5,3,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => ? = 8 - 1
{{1},{2,6,7},{3,4},{5}}
 => [1,6,4,3,5,7,2] => [1,6,4,3,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
 => ? = 8 - 1
{{1},{2,7},{3,4},{5,6}}
 => [1,7,4,3,6,5,2] => [1,4,3,6,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 8 - 1
{{1},{2,7},{3,4},{5},{6}}
 => [1,7,4,3,5,6,2] => [1,4,3,5,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 8 - 1
{{1,5,6},{2,7},{3},{4}}
 => [5,7,3,4,6,1,2] => [5,3,4,6,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 8 - 1
{{1,5,7},{2,6},{3},{4}}
 => [5,6,3,4,7,2,1] => [5,6,3,4,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0]]
 => ? = 8 - 1
{{1,6},{2,5,7},{3},{4}}
 => [6,5,3,4,7,1,2] => [6,5,3,4,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 8 - 1
{{1,7},{2,5,6},{3},{4}}
 => [7,5,3,4,6,2,1] => [5,3,4,6,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 8 - 1
{{1},{2},{3,5},{4,6,7}}
 => [1,2,5,6,3,7,4] => [1,2,5,6,3,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
 => ? = 8 - 1
{{1},{2},{3,5},{4,6},{7}}
 => [1,2,5,6,3,4,7] => [1,2,5,6,3,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
 => ? = 8 - 1
{{1},{2},{3,5},{4,7},{6}}
 => [1,2,5,7,3,6,4] => [1,2,5,3,6,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 8 - 1
{{1},{2},{3,5},{4},{6,7}}
 => [1,2,5,4,3,7,6] => [1,2,5,4,3,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
 => ? = 8 - 1
Description
The number of entries equal to positive one in the alternating sign matrix.
Matching statistic: St000530
Mapping
Mp00221: Set partitions —conjugate⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000530: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Values
{{1}}
 => {{1}}
 => [1] => [1] => ? = 0 - 1
{{1,2}}
 => {{1},{2}}
 => [1,2] => [1,2] => 1 = 2 - 1
{{1},{2}}
 => {{1,2}}
 => [2,1] => [2,1] => 1 = 2 - 1
{{1,3,5,6},{2},{4}}
 => {{1},{2},{3,4},{5,6}}
 => [1,2,4,3,6,5] => [1,6,5,4,3,2] => 5 = 6 - 1
{{1,3,6},{2},{4},{5}}
 => {{1},{2,3,4},{5,6}}
 => [1,3,4,2,6,5] => [1,6,5,4,3,2] => 5 = 6 - 1
{{1,5,6},{2,3,4}}
 => {{1},{2},{3,6},{4},{5}}
 => [1,2,6,4,5,3] => [1,6,5,4,3,2] => 5 = 6 - 1
{{1,6},{2,3,4},{5}}
 => {{1},{2,3,6},{4},{5}}
 => [1,3,6,4,5,2] => [1,6,5,4,3,2] => 5 = 6 - 1
{{1},{2,3},{4,5},{6}}
 => {{1,2,4,6},{3},{5}}
 => [2,4,3,6,5,1] => [2,6,5,4,3,1] => 5 = 6 - 1
{{1},{2,4,5},{3},{6}}
 => {{1,2,6},{3},{4,5}}
 => [2,6,3,5,4,1] => [2,6,5,4,3,1] => 5 = 6 - 1
{{1,6},{2,4},{3,5}}
 => {{1},{2,4,6},{3,5}}
 => [1,4,5,6,3,2] => [1,6,5,4,3,2] => 5 = 6 - 1
{{1,5,6},{2},{3,4}}
 => {{1},{2},{3,5,6},{4}}
 => [1,2,5,4,6,3] => [1,6,5,4,3,2] => 5 = 6 - 1
{{1},{2,5},{3,4,6}}
 => {{1,3,6},{2,5},{4}}
 => [3,5,6,4,2,1] => [3,6,5,4,2,1] => 5 = 6 - 1
{{1,6},{2},{3,4},{5}}
 => {{1},{2,3,5,6},{4}}
 => [1,3,5,4,6,2] => [1,6,5,4,3,2] => 5 = 6 - 1
{{1},{2,5},{3},{4,6}}
 => {{1,3,6},{2,4,5}}
 => [3,4,6,5,2,1] => [3,6,5,4,2,1] => 5 = 6 - 1
{{1},{2,5},{3},{4},{6}}
 => {{1,2,6},{3,4,5}}
 => [2,6,4,5,3,1] => [2,6,5,4,3,1] => 5 = 6 - 1
{{1,2,5,6,7},{3,4}}
 => {{1},{2},{3},{4,6},{5},{7}}
 => [1,2,3,6,5,4,7] => [1,7,6,5,4,3,2] => ? = 8 - 1
{{1,2,5,7},{3,4},{6}}
 => {{1},{2,3},{4,6},{5},{7}}
 => [1,3,2,6,5,4,7] => [1,7,6,5,4,3,2] => ? = 8 - 1
{{1,2},{3,4,5},{6,7}}
 => {{1,3,6},{2},{4},{5},{7}}
 => [3,2,6,4,5,1,7] => [3,2,7,6,5,1,4] => ? = 8 - 1
{{1,2},{3,4,5},{6},{7}}
 => {{1,2,3,6},{4},{5},{7}}
 => [2,3,6,4,5,1,7] => [2,7,6,5,4,1,3] => ? = 8 - 1
{{1,2,6,7},{3,4},{5}}
 => {{1},{2},{3,4,6},{5},{7}}
 => [1,2,4,6,5,3,7] => [1,7,6,5,4,3,2] => ? = 8 - 1
{{1,2,7},{3,4},{5,6}}
 => {{1},{2,4,6},{3},{5},{7}}
 => [1,4,3,6,5,2,7] => [1,7,6,5,4,3,2] => ? = 8 - 1
{{1,2,7},{3,4},{5},{6}}
 => {{1},{2,3,4,6},{5},{7}}
 => [1,3,4,6,5,2,7] => [1,7,6,5,4,3,2] => ? = 8 - 1
{{1,2},{3,5},{4,6,7}}
 => {{1,4,6},{2},{3,5},{7}}
 => [4,2,5,6,3,1,7] => [4,2,7,6,5,1,3] => ? = 8 - 1
{{1,2},{3,5},{4,6},{7}}
 => {{1,2,4,6},{3,5},{7}}
 => [2,4,5,6,3,1,7] => [2,7,6,5,4,1,3] => ? = 8 - 1
{{1,2},{3,5},{4,7},{6}}
 => {{1,4,6},{2,3,5},{7}}
 => [4,3,5,6,2,1,7] => [4,3,7,6,2,1,5] => ? = 8 - 1
{{1,2},{3,5},{4},{6,7}}
 => {{1,3,6},{2},{4,5},{7}}
 => [3,2,6,5,4,1,7] => [3,2,7,6,5,1,4] => ? = 8 - 1
{{1,2},{3,5},{4},{6},{7}}
 => {{1,2,3,6},{4,5},{7}}
 => [2,3,6,5,4,1,7] => [2,7,6,5,4,1,3] => ? = 8 - 1
{{1,4,5,6},{2,3},{7}}
 => {{1,2},{3},{4},{5,7},{6}}
 => [2,1,3,4,7,6,5] => [2,1,7,6,5,4,3] => ? = 8 - 1
{{1,4,6},{2,3},{5,7}}
 => {{1,3},{2,4},{5,7},{6}}
 => [3,4,1,2,7,6,5] => [3,7,1,6,5,4,2] => ? = 8 - 1
{{1,4,6},{2,3},{5},{7}}
 => {{1,2},{3,4},{5,7},{6}}
 => [2,1,4,3,7,6,5] => [2,1,7,6,5,4,3] => ? = 8 - 1
{{1,5,6},{2,3},{4,7}}
 => {{1,4},{2,5,7},{3},{6}}
 => [4,5,3,1,7,6,2] => [4,7,3,1,6,5,2] => ? = 8 - 1
{{1,5,6},{2,3},{4},{7}}
 => {{1,2},{3},{4,5,7},{6}}
 => [2,1,3,5,7,6,4] => [2,1,7,6,5,4,3] => ? = 8 - 1
{{1,6},{2,3},{4,5,7}}
 => {{1,3},{2,5,7},{4},{6}}
 => [3,5,1,4,7,6,2] => [3,7,1,6,5,4,2] => ? = 8 - 1
{{1,6},{2,3},{4,5},{7}}
 => {{1,2},{3,5,7},{4},{6}}
 => [2,1,5,4,7,6,3] => [2,1,7,6,5,4,3] => ? = 8 - 1
{{1,6},{2,3},{4,7},{5}}
 => {{1,3,4},{2,5,7},{6}}
 => [3,5,4,1,7,6,2] => [3,7,6,1,5,4,2] => ? = 8 - 1
{{1,6},{2,3},{4},{5,7}}
 => {{1,3},{2,4,5,7},{6}}
 => [3,4,1,5,7,6,2] => [3,7,1,6,5,4,2] => ? = 8 - 1
{{1,6},{2,3},{4},{5},{7}}
 => {{1,2},{3,4,5,7},{6}}
 => [2,1,4,5,7,6,3] => [2,1,7,6,5,4,3] => ? = 8 - 1
{{1,4,5,6},{2,7},{3}}
 => {{1,5,6},{2,7},{3},{4}}
 => [5,7,3,4,6,1,2] => [5,7,3,6,4,1,2] => ? = 8 - 1
{{1,4,5,7},{2,6},{3}}
 => {{1},{2,5,6},{3,7},{4}}
 => [1,5,7,4,6,2,3] => [1,7,6,5,4,3,2] => ? = 8 - 1
{{1,4,6},{2,5,7},{3}}
 => {{1,3,5,6},{2,4,7}}
 => [3,4,5,7,6,1,2] => [3,7,6,5,4,1,2] => ? = 8 - 1
{{1,4,7},{2,5,6},{3}}
 => {{1},{2,5,6},{3},{4,7}}
 => [1,5,3,7,6,2,4] => [1,7,6,5,4,3,2] => ? = 8 - 1
{{1,4,6},{2,7},{3},{5}}
 => {{1,5,6},{2,7},{3,4}}
 => [5,7,4,3,6,1,2] => [5,7,4,3,6,1,2] => ? = 8 - 1
{{1,4,7},{2,6},{3},{5}}
 => {{1},{2,5,6},{3,4,7}}
 => [1,5,4,7,6,2,3] => [1,7,6,5,4,3,2] => ? = 8 - 1
{{1,5,6},{2,4,7},{3}}
 => {{1,4},{2,7},{3},{5,6}}
 => [4,7,3,1,6,5,2] => [4,7,3,1,6,5,2] => ? = 8 - 1
{{1,5,7},{2,4,6},{3}}
 => {{1},{2,4},{3,7},{5,6}}
 => [1,4,7,2,6,5,3] => [1,7,6,5,4,3,2] => ? = 8 - 1
{{1,6},{2,4,5,7},{3}}
 => {{1,3},{2,7},{4},{5,6}}
 => [3,7,1,4,6,5,2] => [3,7,1,6,5,4,2] => ? = 8 - 1
{{1,7},{2,4,5,6},{3}}
 => {{1},{2,7},{3},{4},{5,6}}
 => [1,7,3,4,6,5,2] => [1,7,6,5,4,3,2] => ? = 8 - 1
{{1,6},{2,4,7},{3},{5}}
 => {{1,3,4},{2,7},{5,6}}
 => [3,7,4,1,6,5,2] => [3,7,6,1,5,4,2] => ? = 8 - 1
{{1,7},{2,4,6},{3},{5}}
 => {{1},{2,7},{3,4},{5,6}}
 => [1,7,4,3,6,5,2] => [1,7,6,5,4,3,2] => ? = 8 - 1
{{1},{2,5,6,7},{3,4}}
 => {{1,7},{2},{3},{4,6},{5}}
 => [7,2,3,6,5,4,1] => [7,2,6,5,4,3,1] => ? = 8 - 1
{{1},{2,5,7},{3,4},{6}}
 => {{1,7},{2,3},{4,6},{5}}
 => [7,3,2,6,5,4,1] => [7,3,2,6,5,4,1] => ? = 8 - 1
{{1},{2},{3,4,5},{6,7}}
 => {{1,3,6,7},{2},{4},{5}}
 => [3,2,6,4,5,7,1] => [3,2,7,6,5,4,1] => ? = 8 - 1
{{1},{2},{3,4,5},{6},{7}}
 => {{1,2,3,6,7},{4},{5}}
 => [2,3,6,4,5,7,1] => [2,7,6,5,4,3,1] => ? = 8 - 1
{{1},{2,6,7},{3,4},{5}}
 => {{1,7},{2},{3,4,6},{5}}
 => [7,2,4,6,5,3,1] => [7,2,6,5,4,3,1] => ? = 8 - 1
{{1},{2,7},{3,4},{5,6}}
 => {{1,7},{2,4,6},{3},{5}}
 => [7,4,3,6,5,2,1] => [7,4,3,6,5,2,1] => ? = 8 - 1
{{1},{2,7},{3,4},{5},{6}}
 => {{1,7},{2,3,4,6},{5}}
 => [7,3,4,6,5,2,1] => [7,3,6,5,4,2,1] => ? = 8 - 1
{{1,5,6},{2,7},{3},{4}}
 => {{1,4,5,6},{2,7},{3}}
 => [4,7,3,5,6,1,2] => [4,7,3,6,5,1,2] => ? = 8 - 1
{{1,5,7},{2,6},{3},{4}}
 => {{1},{2,4,5,6},{3,7}}
 => [1,4,7,5,6,2,3] => [1,7,6,5,4,3,2] => ? = 8 - 1
{{1,6},{2,5,7},{3},{4}}
 => {{1,3},{2,7},{4,5,6}}
 => [3,7,1,5,6,4,2] => [3,7,1,6,5,4,2] => ? = 8 - 1
{{1,7},{2,5,6},{3},{4}}
 => {{1},{2,7},{3},{4,5,6}}
 => [1,7,3,5,6,4,2] => [1,7,6,5,4,3,2] => ? = 8 - 1
{{1},{2},{3,5},{4,6,7}}
 => {{1,4,6,7},{2},{3,5}}
 => [4,2,5,6,3,7,1] => [4,2,7,6,5,3,1] => ? = 8 - 1
{{1},{2},{3,5},{4,6},{7}}
 => {{1,2,4,6,7},{3,5}}
 => [2,4,5,6,3,7,1] => [2,7,6,5,4,3,1] => ? = 8 - 1
{{1},{2},{3,5},{4,7},{6}}
 => {{1,4,6,7},{2,3,5}}
 => [4,3,5,6,2,7,1] => [4,3,7,6,2,5,1] => ? = 8 - 1
{{1},{2},{3,5},{4},{6,7}}
 => {{1,3,6,7},{2},{4,5}}
 => [3,2,6,5,4,7,1] => [3,2,7,6,5,4,1] => ? = 8 - 1
Description
The number of permutations with the same descent word as the given permutation. The descent word of a permutation is the binary word given by [[Mp00109]]. For a given permutation, this statistic is the number of permutations with the same descent word, so the number of elements in the fiber of the map [[Mp00109]] containing a given permutation. This statistic appears as ''up-down analysis'' in statistical applications in genetics, see [1] and the references therein.
Matching statistic: St000653
(load all 6 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000653: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Values
{{1}}
 => [1] => [1] => [1] => ? = 0 - 1
{{1,2}}
 => [2,1] => [1,2] => [2,1] => 1 = 2 - 1
{{1},{2}}
 => [1,2] => [1,2] => [2,1] => 1 = 2 - 1
{{1,3,5,6},{2},{4}}
 => [3,2,5,4,6,1] => [1,3,5,6,2,4] => [4,2,6,5,3,1] => 5 = 6 - 1
{{1,3,6},{2},{4},{5}}
 => [3,2,6,4,5,1] => [1,3,6,2,4,5] => [5,4,2,6,3,1] => 5 = 6 - 1
{{1,5,6},{2,3,4}}
 => [5,3,4,2,6,1] => [1,5,6,2,3,4] => [4,3,2,6,5,1] => 5 = 6 - 1
{{1,6},{2,3,4},{5}}
 => [6,3,4,2,5,1] => [1,6,2,3,4,5] => [5,4,3,2,6,1] => 5 = 6 - 1
{{1},{2,3},{4,5},{6}}
 => [1,3,2,5,4,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 5 = 6 - 1
{{1},{2,4,5},{3},{6}}
 => [1,4,3,5,2,6] => [1,2,4,5,3,6] => [6,3,5,4,2,1] => 5 = 6 - 1
{{1,6},{2,4},{3,5}}
 => [6,4,5,2,3,1] => [1,6,2,4,3,5] => [5,3,4,2,6,1] => 5 = 6 - 1
{{1,5,6},{2},{3,4}}
 => [5,2,4,3,6,1] => [1,5,6,2,3,4] => [4,3,2,6,5,1] => 5 = 6 - 1
{{1},{2,5},{3,4,6}}
 => [1,5,4,6,2,3] => [1,2,5,3,4,6] => [6,4,3,5,2,1] => 5 = 6 - 1
{{1,6},{2},{3,4},{5}}
 => [6,2,4,3,5,1] => [1,6,2,3,4,5] => [5,4,3,2,6,1] => 5 = 6 - 1
{{1},{2,5},{3},{4,6}}
 => [1,5,3,6,2,4] => [1,2,5,3,4,6] => [6,4,3,5,2,1] => 5 = 6 - 1
{{1},{2,5},{3},{4},{6}}
 => [1,5,3,4,2,6] => [1,2,5,3,4,6] => [6,4,3,5,2,1] => 5 = 6 - 1
{{1,2,5,6,7},{3,4}}
 => [2,5,4,3,6,7,1] => [1,2,5,6,7,3,4] => [4,3,7,6,5,2,1] => ? = 8 - 1
{{1,2,5,7},{3,4},{6}}
 => [2,5,4,3,7,6,1] => [1,2,5,7,3,4,6] => [6,4,3,7,5,2,1] => ? = 8 - 1
{{1,2},{3,4,5},{6,7}}
 => [2,1,4,5,3,7,6] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 8 - 1
{{1,2},{3,4,5},{6},{7}}
 => [2,1,4,5,3,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 8 - 1
{{1,2,6,7},{3,4},{5}}
 => [2,6,4,3,5,7,1] => [1,2,6,7,3,4,5] => [5,4,3,7,6,2,1] => ? = 8 - 1
{{1,2,7},{3,4},{5,6}}
 => [2,7,4,3,6,5,1] => [1,2,7,3,4,5,6] => [6,5,4,3,7,2,1] => ? = 8 - 1
{{1,2,7},{3,4},{5},{6}}
 => [2,7,4,3,5,6,1] => [1,2,7,3,4,5,6] => [6,5,4,3,7,2,1] => ? = 8 - 1
{{1,2},{3,5},{4,6,7}}
 => [2,1,5,6,3,7,4] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 8 - 1
{{1,2},{3,5},{4,6},{7}}
 => [2,1,5,6,3,4,7] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 8 - 1
{{1,2},{3,5},{4,7},{6}}
 => [2,1,5,7,3,6,4] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => ? = 8 - 1
{{1,2},{3,5},{4},{6,7}}
 => [2,1,5,4,3,7,6] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 8 - 1
{{1,2},{3,5},{4},{6},{7}}
 => [2,1,5,4,3,6,7] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 8 - 1
{{1,4,5,6},{2,3},{7}}
 => [4,3,2,5,6,1,7] => [1,4,5,6,2,3,7] => [7,3,2,6,5,4,1] => ? = 8 - 1
{{1,4,6},{2,3},{5,7}}
 => [4,3,2,6,7,1,5] => [1,4,6,2,3,5,7] => [7,5,3,2,6,4,1] => ? = 8 - 1
{{1,4,6},{2,3},{5},{7}}
 => [4,3,2,6,5,1,7] => [1,4,6,2,3,5,7] => [7,5,3,2,6,4,1] => ? = 8 - 1
{{1,5,6},{2,3},{4,7}}
 => [5,3,2,7,6,1,4] => [1,5,6,2,3,4,7] => [7,4,3,2,6,5,1] => ? = 8 - 1
{{1,5,6},{2,3},{4},{7}}
 => [5,3,2,4,6,1,7] => [1,5,6,2,3,4,7] => [7,4,3,2,6,5,1] => ? = 8 - 1
{{1,6},{2,3},{4,5,7}}
 => [6,3,2,5,7,1,4] => [1,6,2,3,4,5,7] => [7,5,4,3,2,6,1] => ? = 8 - 1
{{1,6},{2,3},{4,5},{7}}
 => [6,3,2,5,4,1,7] => [1,6,2,3,4,5,7] => [7,5,4,3,2,6,1] => ? = 8 - 1
{{1,6},{2,3},{4,7},{5}}
 => [6,3,2,7,5,1,4] => [1,6,2,3,4,7,5] => [5,7,4,3,2,6,1] => ? = 8 - 1
{{1,6},{2,3},{4},{5,7}}
 => [6,3,2,4,7,1,5] => [1,6,2,3,4,5,7] => [7,5,4,3,2,6,1] => ? = 8 - 1
{{1,6},{2,3},{4},{5},{7}}
 => [6,3,2,4,5,1,7] => [1,6,2,3,4,5,7] => [7,5,4,3,2,6,1] => ? = 8 - 1
{{1,4,5,6},{2,7},{3}}
 => [4,7,3,5,6,1,2] => [1,4,5,6,2,7,3] => [3,7,2,6,5,4,1] => ? = 8 - 1
{{1,4,5,7},{2,6},{3}}
 => [4,6,3,5,7,2,1] => [1,4,5,7,2,6,3] => [3,6,2,7,5,4,1] => ? = 8 - 1
{{1,4,6},{2,5,7},{3}}
 => [4,5,3,6,7,1,2] => [1,4,6,2,5,7,3] => [3,7,5,2,6,4,1] => ? = 8 - 1
{{1,4,7},{2,5,6},{3}}
 => [4,5,3,7,6,2,1] => [1,4,7,2,5,6,3] => [3,6,5,2,7,4,1] => ? = 8 - 1
{{1,4,6},{2,7},{3},{5}}
 => [4,7,3,6,5,1,2] => [1,4,6,2,7,3,5] => [5,3,7,2,6,4,1] => ? = 8 - 1
{{1,4,7},{2,6},{3},{5}}
 => [4,6,3,7,5,2,1] => [1,4,7,2,6,3,5] => [5,3,6,2,7,4,1] => ? = 8 - 1
{{1,5,6},{2,4,7},{3}}
 => [5,4,3,7,6,1,2] => [1,5,6,2,4,7,3] => [3,7,4,2,6,5,1] => ? = 8 - 1
{{1,5,7},{2,4,6},{3}}
 => [5,4,3,6,7,2,1] => [1,5,7,2,4,6,3] => [3,6,4,2,7,5,1] => ? = 8 - 1
{{1,6},{2,4,5,7},{3}}
 => [6,4,3,5,7,1,2] => [1,6,2,4,5,7,3] => [3,7,5,4,2,6,1] => ? = 8 - 1
{{1,7},{2,4,5,6},{3}}
 => [7,4,3,5,6,2,1] => [1,7,2,4,5,6,3] => [3,6,5,4,2,7,1] => ? = 8 - 1
{{1,6},{2,4,7},{3},{5}}
 => [6,4,3,7,5,1,2] => [1,6,2,4,7,3,5] => [5,3,7,4,2,6,1] => ? = 8 - 1
{{1,7},{2,4,6},{3},{5}}
 => [7,4,3,6,5,2,1] => [1,7,2,4,6,3,5] => [5,3,6,4,2,7,1] => ? = 8 - 1
{{1},{2,5,6,7},{3,4}}
 => [1,5,4,3,6,7,2] => [1,2,5,6,7,3,4] => [4,3,7,6,5,2,1] => ? = 8 - 1
{{1},{2,5,7},{3,4},{6}}
 => [1,5,4,3,7,6,2] => [1,2,5,7,3,4,6] => [6,4,3,7,5,2,1] => ? = 8 - 1
{{1},{2},{3,4,5},{6,7}}
 => [1,2,4,5,3,7,6] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 8 - 1
{{1},{2},{3,4,5},{6},{7}}
 => [1,2,4,5,3,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 8 - 1
{{1},{2,6,7},{3,4},{5}}
 => [1,6,4,3,5,7,2] => [1,2,6,7,3,4,5] => [5,4,3,7,6,2,1] => ? = 8 - 1
{{1},{2,7},{3,4},{5,6}}
 => [1,7,4,3,6,5,2] => [1,2,7,3,4,5,6] => [6,5,4,3,7,2,1] => ? = 8 - 1
{{1},{2,7},{3,4},{5},{6}}
 => [1,7,4,3,5,6,2] => [1,2,7,3,4,5,6] => [6,5,4,3,7,2,1] => ? = 8 - 1
{{1,5,6},{2,7},{3},{4}}
 => [5,7,3,4,6,1,2] => [1,5,6,2,7,3,4] => [4,3,7,2,6,5,1] => ? = 8 - 1
{{1,5,7},{2,6},{3},{4}}
 => [5,6,3,4,7,2,1] => [1,5,7,2,6,3,4] => [4,3,6,2,7,5,1] => ? = 8 - 1
{{1,6},{2,5,7},{3},{4}}
 => [6,5,3,4,7,1,2] => [1,6,2,5,7,3,4] => [4,3,7,5,2,6,1] => ? = 8 - 1
{{1,7},{2,5,6},{3},{4}}
 => [7,5,3,4,6,2,1] => [1,7,2,5,6,3,4] => [4,3,6,5,2,7,1] => ? = 8 - 1
{{1},{2},{3,5},{4,6,7}}
 => [1,2,5,6,3,7,4] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 8 - 1
{{1},{2},{3,5},{4,6},{7}}
 => [1,2,5,6,3,4,7] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 8 - 1
{{1},{2},{3,5},{4,7},{6}}
 => [1,2,5,7,3,6,4] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => ? = 8 - 1
{{1},{2},{3,5},{4},{6,7}}
 => [1,2,5,4,3,7,6] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 8 - 1
Description
The last descent of a permutation. For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the largest index $0 \leq i < n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(0) = n+1$.
The following 16 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000956The maximal displacement of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St000045The number of linear extensions of a binary tree. St001560The product of the cardinalities of the lower order ideal and upper order ideal generated by a permutation in weak order. St000890The number of nonzero entries in an alternating sign matrix. St001060The distinguishing index of a graph. St001569The maximal modular displacement of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001890The maximum magnitude of the Möbius function of a poset. St001948The number of augmented double ascents of a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000264The girth of a graph, which is not a tree. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001893The flag descent of a signed permutation.