searching the database
Your data matches 12 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000148
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [1]
=> []
=> 0
[1,2,3] => [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2] => [2,1]
=> [1]
=> []
=> 0
[2,1,3] => [2,1]
=> [1]
=> []
=> 0
[3,2,1] => [2,1]
=> [1]
=> []
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> []
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> []
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> []
=> 0
[2,3,1,4] => [3,1]
=> [1]
=> []
=> 0
[2,4,3,1] => [3,1]
=> [1]
=> []
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> []
=> 0
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> []
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> []
=> 0
[4,1,3,2] => [3,1]
=> [1]
=> []
=> 0
[4,2,1,3] => [3,1]
=> [1]
=> []
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> []
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> 0
[1,3,5,2,4] => [4,1]
=> [1]
=> []
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> []
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> []
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,5,4,2,3] => [4,1]
=> [1]
=> []
=> 0
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1]
=> 1
Description
The number of odd parts of a partition.
Matching statistic: St001330
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 64%
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 64%
Values
[1,2] => {{1},{2}}
=> [1,1] => ([(0,1)],2)
=> 2 = 0 + 2
[1,2,3] => {{1},{2},{3}}
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,3,2] => {{1},{2,3}}
=> [1,2] => ([(1,2)],3)
=> 2 = 0 + 2
[2,1,3] => {{1,2},{3}}
=> [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[3,2,1] => {{1,3},{2}}
=> [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3] => ([(2,3)],4)
=> 2 = 0 + 2
[1,4,2,3] => {{1},{2,3,4}}
=> [1,3] => ([(2,3)],4)
=> 2 = 0 + 2
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,3,1,4] => {{1,2,3},{4}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,1,2,4] => {{1,2,3},{4}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[3,2,4,1] => {{1,3,4},{2}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[4,1,3,2] => {{1,2,4},{3}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[4,2,1,3] => {{1,3,4},{2}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[4,3,2,1] => {{1,4},{2,3}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 3 + 2
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,4] => ([(3,4)],5)
=> 2 = 0 + 2
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> [1,4] => ([(3,4)],5)
=> 2 = 0 + 2
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> [1,4] => ([(3,4)],5)
=> 2 = 0 + 2
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,4,5,3,2] => {{1},{2,3,4,5}}
=> [1,4] => ([(3,4)],5)
=> 2 = 0 + 2
[1,5,2,3,4] => {{1},{2,3,4,5}}
=> [1,4] => ([(3,4)],5)
=> 2 = 0 + 2
[1,5,2,4,3] => {{1},{2,3,5},{4}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,5,3,2,4] => {{1},{2,4,5},{3}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,5,4,2,3] => {{1},{2,3,4,5}}
=> [1,4] => ([(3,4)],5)
=> 2 = 0 + 2
[1,5,4,3,2] => {{1},{2,5},{3,4}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[2,1,3,5,4] => {{1,2},{3},{4,5}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,1,4,3,5] => {{1,2},{3,4},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,1,4,5,3] => {{1,2},{3,4,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,5,3,4] => {{1,2},{3,4,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,5,4,3] => {{1,2},{3,5},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,3,1,4,5] => {{1,2,3},{4},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,3,1,5,4] => {{1,2,3},{4,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,3,4,1,5] => {{1,2,3,4},{5}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,3,5,4,1] => {{1,2,3,5},{4}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,4,1,3,5] => {{1,2,3,4},{5}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,4,3,1,5] => {{1,2,4},{3},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,4,3,5,1] => {{1,2,4,5},{3}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,4,5,1,3] => {{1,2,4},{3,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,5,1,4,3] => {{1,2,3,5},{4}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,5,3,1,4] => {{1,2,4,5},{3}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,5,3,4,1] => {{1,2,5},{3},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,5,4,3,1] => {{1,2,5},{3,4}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,1,2,4,5] => {{1,2,3},{4},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[3,1,2,5,4] => {{1,2,3},{4,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,1,4,2,5] => {{1,2,3,4},{5}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,1,5,4,2] => {{1,2,3,5},{4}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[3,2,1,5,4] => {{1,3},{2},{4,5}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[3,2,4,1,5] => {{1,3,4},{2},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[3,2,4,5,1] => {{1,3,4,5},{2}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,2,5,1,4] => {{1,3,4,5},{2}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,2,5,4,1] => {{1,3,5},{2},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[3,4,1,2,5] => {{1,3},{2,4},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[3,4,1,5,2] => {{1,3},{2,4,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,4,2,1,5] => {{1,2,3,4},{5}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,4,5,2,1] => {{1,3,5},{2,4}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,5,1,2,4] => {{1,3},{2,4,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,5,1,4,2] => {{1,3},{2,5},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[3,5,2,4,1] => {{1,2,3,5},{4}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,5,4,1,2] => {{1,3,4},{2,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[4,1,3,2,5] => {{1,2,4},{3},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[4,2,1,3,5] => {{1,3,4},{2},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[4,2,3,5,1] => {{1,4,5},{2},{3}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[4,2,5,1,3] => {{1,4},{2},{3,5}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[4,3,2,1,5] => {{1,4},{2,3},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[4,5,3,1,2] => {{1,4},{2,5},{3}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[5,1,3,4,2] => {{1,2,5},{3},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[5,2,1,4,3] => {{1,3,5},{2},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[5,2,3,1,4] => {{1,4,5},{2},{3}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[5,2,4,3,1] => {{1,5},{2},{3,4}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[5,3,2,4,1] => {{1,5},{2,3},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[5,4,3,2,1] => {{1,5},{2,4},{3}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,2,3,5,4,6] => {{1},{2},{3},{4,5},{6}}
=> [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 2
[1,2,3,6,5,4] => {{1},{2},{3},{4,6},{5}}
=> [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 2
[1,2,4,3,5,6] => {{1},{2},{3,4},{5},{6}}
=> [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of q possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number HG(G) of a graph G is the largest integer q such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of q possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St001232
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 55%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 55%
Values
[1,2] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[2,4,3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[3,1,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,2,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[4,1,3,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[4,2,1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,3,5,2,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,4,5,3,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,5,2,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[1,5,4,2,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,5,4,3,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[2,1,3,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[2,1,4,3,5] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[2,1,4,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[2,1,5,3,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[2,1,5,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[2,3,1,4,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[2,3,1,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[2,3,4,1,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[2,3,5,4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[2,4,1,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[2,4,3,5,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[2,4,5,1,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[2,5,1,4,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[2,5,3,1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[2,5,4,3,1] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[3,1,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[3,2,1,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[3,4,1,2,5] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[3,4,1,5,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[3,4,5,2,1] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[3,5,1,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[3,5,1,4,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[3,5,4,1,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[4,1,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[4,2,5,1,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[4,3,2,1,5] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[4,3,2,5,1] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[4,3,5,1,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[4,5,1,3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[4,5,2,1,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[4,5,3,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[5,1,4,3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[5,2,4,3,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[5,3,2,1,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[5,3,2,4,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[5,3,4,2,1] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[5,4,1,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[5,4,2,3,1] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[5,4,3,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,2,4,3,6,5] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,2,5,6,3,4] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,2,6,5,4,3] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,3,2,4,6,5] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,3,2,5,4,6] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,3,2,5,6,4] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,3,2,6,4,5] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,3,2,6,5,4] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,3,4,2,6,5] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,3,5,6,2,4] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,3,6,5,4,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,4,2,3,6,5] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001960
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 36%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 36%
Values
[1,2] => [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[2,1,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[3,2,1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 2
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,4,2,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[2,3,1,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[2,4,3,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[3,1,2,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[3,2,4,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[4,1,3,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[4,2,1,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 3
[1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 2
[1,2,4,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 2
[1,2,4,5,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,2,5,3,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,2,5,4,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 2
[1,3,2,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 2
[1,3,2,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[1,3,4,2,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,3,4,5,2] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,3,5,2,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,3,5,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,2,3,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,2,5,3] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,4,3,2,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 2
[1,4,3,5,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,5,2,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[1,4,5,3,2] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,5,2,3,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,5,2,4,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,5,3,2,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,5,3,4,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 2
[1,5,4,2,3] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,5,4,3,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[2,1,3,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 2
[2,1,3,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[2,1,4,3,5] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ? = 4
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ? = 3
[1,2,3,5,4,6] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ? = 3
[1,2,3,5,6,4] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 2
[1,2,3,6,4,5] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 2
[1,2,3,6,5,4] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ? = 3
[1,2,4,3,5,6] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ? = 3
[1,2,4,3,6,5] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ? = 2
[1,2,4,5,3,6] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 2
[1,2,4,5,6,3] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[1,2,4,6,3,5] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[1,2,4,6,5,3] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 2
[1,2,5,3,4,6] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 2
[1,2,5,3,6,4] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[1,2,5,4,3,6] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ? = 3
[1,2,5,4,6,3] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 2
[1,2,5,6,3,4] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ? = 2
[1,2,5,6,4,3] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[1,2,6,3,4,5] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[1,2,6,3,5,4] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 2
[1,2,6,4,3,5] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 2
[1,2,6,4,5,3] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ? = 3
[1,2,6,5,3,4] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[1,2,6,5,4,3] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ? = 2
[1,3,2,4,5,6] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ? = 3
[1,3,2,4,6,5] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ? = 2
[1,3,2,5,4,6] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ? = 2
[1,3,2,5,6,4] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ? = 1
[1,3,2,6,4,5] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ? = 1
[1,3,2,6,5,4] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ? = 2
[1,3,4,2,5,6] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 2
[1,3,4,2,6,5] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ? = 1
[1,3,4,5,2,6] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[1,3,4,5,6,2] => [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 0
[1,3,4,6,2,5] => [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 0
[1,3,4,6,5,2] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[1,3,5,2,4,6] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[1,3,5,2,6,4] => [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 0
[1,3,5,4,2,6] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 2
[1,3,5,4,6,2] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[1,3,5,6,2,4] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ? = 1
[1,3,5,6,4,2] => [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 0
[1,3,6,2,4,5] => [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 0
[1,3,6,2,5,4] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[1,3,6,4,2,5] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 1
[1,3,6,4,5,2] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 2
[1,3,6,5,2,4] => [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 0
[1,3,6,5,4,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ? = 1
[1,4,2,3,5,6] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 2
[1,4,2,3,6,5] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ? = 1
Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Matching statistic: St001169
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001169: Dyck paths ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 36%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001169: Dyck paths ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 36%
Values
[1,2] => [1,2] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 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]
=> 3 = 2 + 1
[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]
=> 2 = 1 + 1
[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]
=> 2 = 1 + 1
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[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]
=> 1 = 0 + 1
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[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]
=> 2 = 1 + 1
[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]
=> 1 = 0 + 1
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[2,4,3,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[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]
=> 1 = 0 + 1
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,2,4,1] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[4,1,3,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[4,2,1,3] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[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]
=> 4 = 3 + 1
[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]
=> 3 = 2 + 1
[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]
=> 3 = 2 + 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[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]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[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]
=> 3 = 2 + 1
[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]
=> 2 = 1 + 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,3,5,4,2] => [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]
=> 2 = 1 + 1
[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]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,4,3,5,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,4,5,3,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,5,2,3,4] => [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]
=> 1 = 0 + 1
[1,5,2,4,3] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,5,3,2,4] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,5,3,4,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,5,4,2,3] => [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]
=> 1 = 0 + 1
[1,5,4,3,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[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]
=> 3 = 2 + 1
[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]
=> 2 = 1 + 1
[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]
=> 2 = 1 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 3 + 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 3 + 1
[1,2,3,5,6,4] => [1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,2,3,6,4,5] => [1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,2,3,6,5,4] => [1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 3 + 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 2 + 1
[1,2,4,5,3,6] => [1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 2 + 1
[1,2,4,5,6,3] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,2,4,6,3,5] => [1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,2,4,6,5,3] => [1,2,5,6,3,4] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 1
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 2 + 1
[1,2,5,3,6,4] => [1,2,6,4,3,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,2,5,4,3,6] => [1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 3 + 1
[1,2,5,4,6,3] => [1,2,4,6,3,5] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 2 + 1
[1,2,5,6,3,4] => [1,2,5,3,6,4] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 2 + 1
[1,2,5,6,4,3] => [1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,2,6,3,4,5] => [1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,2,6,3,5,4] => [1,2,5,6,4,3] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 1
[1,2,6,4,3,5] => [1,2,4,6,5,3] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 2 + 1
[1,2,6,4,5,3] => [1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3 + 1
[1,2,6,5,3,4] => [1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,2,6,5,4,3] => [1,2,5,4,6,3] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 2 + 1
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 2 + 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 2 + 1
[1,3,2,5,6,4] => [1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,3,2,6,4,5] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,3,2,6,5,4] => [1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,3,4,2,5,6] => [1,4,2,3,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 + 1
[1,3,4,2,6,5] => [1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 1 + 1
[1,3,4,5,2,6] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 1
[1,3,4,5,6,2] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,3,4,6,2,5] => [1,6,5,2,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,3,4,6,5,2] => [1,5,6,2,3,4] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 + 1
[1,3,5,2,4,6] => [1,5,4,2,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 1
[1,3,5,2,6,4] => [1,6,4,2,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,3,5,4,2,6] => [1,4,5,2,3,6] => [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 + 1
[1,3,5,4,6,2] => [1,4,6,2,3,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,3,5,6,2,4] => [1,5,2,3,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 1 + 1
[1,3,5,6,4,2] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,3,6,2,4,5] => [1,6,5,4,2,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,3,6,2,5,4] => [1,5,6,4,2,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 + 1
[1,3,6,4,2,5] => [1,4,6,5,2,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,3,6,4,5,2] => [1,4,5,6,2,3] => [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2 + 1
[1,3,6,5,2,4] => [1,6,4,5,2,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,3,6,5,4,2] => [1,5,4,6,2,3] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1 + 1
[1,4,2,3,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 + 1
[1,4,2,3,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 1 + 1
Description
Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St000015
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 36%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 36%
Values
[1,2] => [1,2] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 0 + 2
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 0 + 2
[3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 0 + 2
[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]
=> 4 = 2 + 2
[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]
=> 3 = 1 + 2
[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]
=> 3 = 1 + 2
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[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]
=> 2 = 0 + 2
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[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]
=> 3 = 1 + 2
[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]
=> 2 = 0 + 2
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 0 + 2
[2,4,3,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 0 + 2
[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]
=> 2 = 0 + 2
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
[3,2,4,1] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 0 + 2
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 0 + 2
[4,1,3,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 0 + 2
[4,2,1,3] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 0 + 2
[4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 1 + 2
[4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 0 + 2
[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]
=> 5 = 3 + 2
[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]
=> 4 = 2 + 2
[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]
=> 4 = 2 + 2
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 3 = 1 + 2
[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]
=> 3 = 1 + 2
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[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]
=> 4 = 2 + 2
[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]
=> 3 = 1 + 2
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 3 = 1 + 2
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[1,3,5,2,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[1,3,5,4,2] => [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]
=> 3 = 1 + 2
[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]
=> 3 = 1 + 2
[1,4,2,5,3] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,4,3,5,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 3 = 1 + 2
[1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,4,5,3,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[1,5,2,3,4] => [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]
=> 2 = 0 + 2
[1,5,2,4,3] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 3 = 1 + 2
[1,5,3,2,4] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 3 = 1 + 2
[1,5,3,4,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 4 = 2 + 2
[1,5,4,2,3] => [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]
=> 2 = 0 + 2
[1,5,4,3,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[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]
=> 4 = 2 + 2
[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]
=> 3 = 1 + 2
[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]
=> 3 = 1 + 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 2
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 3 + 2
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 3 + 2
[1,2,3,5,6,4] => [1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 2
[1,2,3,6,4,5] => [1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 2
[1,2,3,6,5,4] => [1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 3 + 2
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 3 + 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 2 + 2
[1,2,4,5,3,6] => [1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 2 + 2
[1,2,4,5,6,3] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,2,4,6,3,5] => [1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,2,4,6,5,3] => [1,2,5,6,3,4] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 2
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 2 + 2
[1,2,5,3,6,4] => [1,2,6,4,3,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,2,5,4,3,6] => [1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 3 + 2
[1,2,5,4,6,3] => [1,2,4,6,3,5] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 2 + 2
[1,2,5,6,3,4] => [1,2,5,3,6,4] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 2 + 2
[1,2,5,6,4,3] => [1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,2,6,3,4,5] => [1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,2,6,3,5,4] => [1,2,5,6,4,3] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 2
[1,2,6,4,3,5] => [1,2,4,6,5,3] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 2 + 2
[1,2,6,4,5,3] => [1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3 + 2
[1,2,6,5,3,4] => [1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,2,6,5,4,3] => [1,2,5,4,6,3] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 2 + 2
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 2
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 2 + 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 2 + 2
[1,3,2,5,6,4] => [1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 1 + 2
[1,3,2,6,4,5] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 1 + 2
[1,3,2,6,5,4] => [1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 2 + 2
[1,3,4,2,5,6] => [1,4,2,3,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 + 2
[1,3,4,2,6,5] => [1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 1 + 2
[1,3,4,5,2,6] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 2
[1,3,4,5,6,2] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,3,4,6,2,5] => [1,6,5,2,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,3,4,6,5,2] => [1,5,6,2,3,4] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 + 2
[1,3,5,2,4,6] => [1,5,4,2,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 2
[1,3,5,2,6,4] => [1,6,4,2,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,3,5,4,2,6] => [1,4,5,2,3,6] => [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 + 2
[1,3,5,4,6,2] => [1,4,6,2,3,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,3,5,6,2,4] => [1,5,2,3,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 1 + 2
[1,3,5,6,4,2] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,3,6,2,4,5] => [1,6,5,4,2,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,3,6,2,5,4] => [1,5,6,4,2,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 + 2
[1,3,6,4,2,5] => [1,4,6,5,2,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,3,6,4,5,2] => [1,4,5,6,2,3] => [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2 + 2
[1,3,6,5,2,4] => [1,6,4,5,2,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,3,6,5,4,2] => [1,5,4,6,2,3] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1 + 2
[1,4,2,3,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 + 2
[1,4,2,3,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 1 + 2
Description
The number of peaks of a Dyck path.
Matching statistic: St000702
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 36%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 36%
Values
[1,2] => [1,2] => [1,0,1,0]
=> [3,1,2] => 2 = 0 + 2
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 1 + 2
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> [4,3,1,2] => 2 = 0 + 2
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 4 = 2 + 2
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 3 = 1 + 2
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 3 = 1 + 2
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 2 = 0 + 2
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 2 = 0 + 2
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 3 = 1 + 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 3 = 1 + 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2 = 0 + 2
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 0 + 2
[2,4,3,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 2 = 0 + 2
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 0 + 2
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 3 = 1 + 2
[3,2,4,1] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 2 = 0 + 2
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 2 = 0 + 2
[4,1,3,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 2 = 0 + 2
[4,2,1,3] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 2 = 0 + 2
[4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 3 = 1 + 2
[4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 2 = 0 + 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 5 = 3 + 2
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 4 = 2 + 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 4 = 2 + 2
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 3 = 1 + 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 3 = 1 + 2
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 4 = 2 + 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 4 = 2 + 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 3 = 1 + 2
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 3 = 1 + 2
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 2 = 0 + 2
[1,3,5,2,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 2 = 0 + 2
[1,3,5,4,2] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 3 = 1 + 2
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 3 = 1 + 2
[1,4,2,5,3] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 2 = 0 + 2
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => 4 = 2 + 2
[1,4,3,5,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 3 = 1 + 2
[1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 3 = 1 + 2
[1,4,5,3,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 2 = 0 + 2
[1,5,2,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 2 = 0 + 2
[1,5,2,4,3] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 3 = 1 + 2
[1,5,3,2,4] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 3 = 1 + 2
[1,5,3,4,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 4 = 2 + 2
[1,5,4,2,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 2 = 0 + 2
[1,5,4,3,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 3 = 1 + 2
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 4 = 2 + 2
[2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 3 = 1 + 2
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 3 = 1 + 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 4 + 2
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 3 + 2
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => ? = 3 + 2
[1,2,3,5,6,4] => [1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => ? = 2 + 2
[1,2,3,6,4,5] => [1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => ? = 2 + 2
[1,2,3,6,5,4] => [1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 3 + 2
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 3 + 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ? = 2 + 2
[1,2,4,5,3,6] => [1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 2 + 2
[1,2,4,5,6,3] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ? = 1 + 2
[1,2,4,6,3,5] => [1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ? = 1 + 2
[1,2,4,6,5,3] => [1,2,5,6,3,4] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => ? = 2 + 2
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 2 + 2
[1,2,5,3,6,4] => [1,2,6,4,3,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ? = 1 + 2
[1,2,5,4,3,6] => [1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => ? = 3 + 2
[1,2,5,4,6,3] => [1,2,4,6,3,5] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => ? = 2 + 2
[1,2,5,6,3,4] => [1,2,5,3,6,4] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => ? = 2 + 2
[1,2,5,6,4,3] => [1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ? = 1 + 2
[1,2,6,3,4,5] => [1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ? = 1 + 2
[1,2,6,3,5,4] => [1,2,5,6,4,3] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => ? = 2 + 2
[1,2,6,4,3,5] => [1,2,4,6,5,3] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => ? = 2 + 2
[1,2,6,4,5,3] => [1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 3 + 2
[1,2,6,5,3,4] => [1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ? = 1 + 2
[1,2,6,5,4,3] => [1,2,5,4,6,3] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => ? = 2 + 2
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ? = 3 + 2
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ? = 2 + 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ? = 2 + 2
[1,3,2,5,6,4] => [1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 1 + 2
[1,3,2,6,4,5] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 1 + 2
[1,3,2,6,5,4] => [1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => ? = 2 + 2
[1,3,4,2,5,6] => [1,4,2,3,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 2 + 2
[1,3,4,2,6,5] => [1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 1 + 2
[1,3,4,5,2,6] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 1 + 2
[1,3,4,5,6,2] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 0 + 2
[1,3,4,6,2,5] => [1,6,5,2,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 0 + 2
[1,3,4,6,5,2] => [1,5,6,2,3,4] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 1 + 2
[1,3,5,2,4,6] => [1,5,4,2,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 1 + 2
[1,3,5,2,6,4] => [1,6,4,2,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 0 + 2
[1,3,5,4,2,6] => [1,4,5,2,3,6] => [1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => ? = 2 + 2
[1,3,5,4,6,2] => [1,4,6,2,3,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 1 + 2
[1,3,5,6,2,4] => [1,5,2,3,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ? = 1 + 2
[1,3,5,6,4,2] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 0 + 2
[1,3,6,2,4,5] => [1,6,5,4,2,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 0 + 2
[1,3,6,2,5,4] => [1,5,6,4,2,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 1 + 2
[1,3,6,4,2,5] => [1,4,6,5,2,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 1 + 2
[1,3,6,4,5,2] => [1,4,5,6,2,3] => [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => ? = 2 + 2
[1,3,6,5,2,4] => [1,6,4,5,2,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 0 + 2
[1,3,6,5,4,2] => [1,5,4,6,2,3] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 1 + 2
[1,4,2,3,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 2 + 2
[1,4,2,3,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 1 + 2
Description
The number of weak deficiencies of a permutation.
This is defined as
wdec(σ)=#{i:σ(i)≤i}.
The number of weak exceedances is [[St000213]], the number of deficiencies is [[St000703]].
Matching statistic: St000991
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 36%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 36%
Values
[1,2] => [1,2] => [1,0,1,0]
=> [3,1,2] => 2 = 0 + 2
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 1 + 2
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> [4,3,1,2] => 2 = 0 + 2
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 4 = 2 + 2
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 3 = 1 + 2
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 3 = 1 + 2
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 2 = 0 + 2
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 2 = 0 + 2
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 3 = 1 + 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 3 = 1 + 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2 = 0 + 2
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 0 + 2
[2,4,3,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 2 = 0 + 2
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 0 + 2
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 3 = 1 + 2
[3,2,4,1] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 2 = 0 + 2
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 2 = 0 + 2
[4,1,3,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 2 = 0 + 2
[4,2,1,3] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 2 = 0 + 2
[4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 3 = 1 + 2
[4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 2 = 0 + 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 5 = 3 + 2
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 4 = 2 + 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 4 = 2 + 2
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 3 = 1 + 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 3 = 1 + 2
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 4 = 2 + 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 4 = 2 + 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 3 = 1 + 2
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 3 = 1 + 2
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 2 = 0 + 2
[1,3,5,2,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 2 = 0 + 2
[1,3,5,4,2] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 3 = 1 + 2
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 3 = 1 + 2
[1,4,2,5,3] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 2 = 0 + 2
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => 4 = 2 + 2
[1,4,3,5,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 3 = 1 + 2
[1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 3 = 1 + 2
[1,4,5,3,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 2 = 0 + 2
[1,5,2,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 2 = 0 + 2
[1,5,2,4,3] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 3 = 1 + 2
[1,5,3,2,4] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 3 = 1 + 2
[1,5,3,4,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 4 = 2 + 2
[1,5,4,2,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 2 = 0 + 2
[1,5,4,3,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 3 = 1 + 2
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 4 = 2 + 2
[2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 3 = 1 + 2
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 3 = 1 + 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 4 + 2
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 3 + 2
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => ? = 3 + 2
[1,2,3,5,6,4] => [1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => ? = 2 + 2
[1,2,3,6,4,5] => [1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => ? = 2 + 2
[1,2,3,6,5,4] => [1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 3 + 2
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 3 + 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ? = 2 + 2
[1,2,4,5,3,6] => [1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 2 + 2
[1,2,4,5,6,3] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ? = 1 + 2
[1,2,4,6,3,5] => [1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ? = 1 + 2
[1,2,4,6,5,3] => [1,2,5,6,3,4] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => ? = 2 + 2
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 2 + 2
[1,2,5,3,6,4] => [1,2,6,4,3,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ? = 1 + 2
[1,2,5,4,3,6] => [1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => ? = 3 + 2
[1,2,5,4,6,3] => [1,2,4,6,3,5] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => ? = 2 + 2
[1,2,5,6,3,4] => [1,2,5,3,6,4] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => ? = 2 + 2
[1,2,5,6,4,3] => [1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ? = 1 + 2
[1,2,6,3,4,5] => [1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ? = 1 + 2
[1,2,6,3,5,4] => [1,2,5,6,4,3] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => ? = 2 + 2
[1,2,6,4,3,5] => [1,2,4,6,5,3] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => ? = 2 + 2
[1,2,6,4,5,3] => [1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 3 + 2
[1,2,6,5,3,4] => [1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ? = 1 + 2
[1,2,6,5,4,3] => [1,2,5,4,6,3] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => ? = 2 + 2
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ? = 3 + 2
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ? = 2 + 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ? = 2 + 2
[1,3,2,5,6,4] => [1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 1 + 2
[1,3,2,6,4,5] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 1 + 2
[1,3,2,6,5,4] => [1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => ? = 2 + 2
[1,3,4,2,5,6] => [1,4,2,3,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 2 + 2
[1,3,4,2,6,5] => [1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 1 + 2
[1,3,4,5,2,6] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 1 + 2
[1,3,4,5,6,2] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 0 + 2
[1,3,4,6,2,5] => [1,6,5,2,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 0 + 2
[1,3,4,6,5,2] => [1,5,6,2,3,4] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 1 + 2
[1,3,5,2,4,6] => [1,5,4,2,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 1 + 2
[1,3,5,2,6,4] => [1,6,4,2,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 0 + 2
[1,3,5,4,2,6] => [1,4,5,2,3,6] => [1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => ? = 2 + 2
[1,3,5,4,6,2] => [1,4,6,2,3,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 1 + 2
[1,3,5,6,2,4] => [1,5,2,3,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ? = 1 + 2
[1,3,5,6,4,2] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 0 + 2
[1,3,6,2,4,5] => [1,6,5,4,2,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 0 + 2
[1,3,6,2,5,4] => [1,5,6,4,2,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 1 + 2
[1,3,6,4,2,5] => [1,4,6,5,2,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 1 + 2
[1,3,6,4,5,2] => [1,4,5,6,2,3] => [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => ? = 2 + 2
[1,3,6,5,2,4] => [1,6,4,5,2,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 0 + 2
[1,3,6,5,4,2] => [1,5,4,6,2,3] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 1 + 2
[1,4,2,3,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 2 + 2
[1,4,2,3,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 1 + 2
Description
The number of right-to-left minima of a permutation.
For the number of left-to-right maxima, see [[St000314]].
Matching statistic: St001769
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00277: Permutations —catalanization⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001769: Signed permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 27%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001769: Signed permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 27%
Values
[1,2] => [1,2] => [2,1] => [2,1] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [2,3,1] => [2,3,1] => 2 = 1 + 1
[1,3,2] => [1,3,2] => [3,2,1] => [3,2,1] => 1 = 0 + 1
[2,1,3] => [2,1,3] => [1,3,2] => [1,3,2] => 1 = 0 + 1
[3,2,1] => [3,2,1] => [2,1,3] => [2,1,3] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 3 = 2 + 1
[1,2,4,3] => [1,2,4,3] => [2,4,3,1] => [2,4,3,1] => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [3,2,4,1] => [3,2,4,1] => 2 = 1 + 1
[1,3,4,2] => [1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 1 = 0 + 1
[1,4,2,3] => [1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 1 = 0 + 1
[1,4,3,2] => [1,4,3,2] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[2,1,3,4] => [2,1,3,4] => [1,3,4,2] => [1,3,4,2] => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [1,4,3,2] => [1,4,3,2] => 1 = 0 + 1
[2,3,1,4] => [2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 1 = 0 + 1
[2,4,3,1] => [2,4,3,1] => [1,3,2,4] => [1,3,2,4] => 1 = 0 + 1
[3,1,2,4] => [2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 1 = 0 + 1
[3,2,1,4] => [3,2,1,4] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[3,2,4,1] => [3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 1 = 0 + 1
[3,4,1,2] => [4,3,2,1] => [3,2,1,4] => [3,2,1,4] => 1 = 0 + 1
[4,1,3,2] => [2,4,3,1] => [1,3,2,4] => [1,3,2,4] => 1 = 0 + 1
[4,2,1,3] => [3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 1 = 0 + 1
[4,2,3,1] => [3,4,2,1] => [3,1,2,4] => [3,1,2,4] => 2 = 1 + 1
[4,3,2,1] => [4,3,2,1] => [3,2,1,4] => [3,2,1,4] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 3 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 2 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 2 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 1 + 1
[1,2,5,3,4] => [1,2,4,5,3] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 1 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 2 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 1 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 1 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 0 + 1
[1,3,5,2,4] => [1,5,4,2,3] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 0 + 1
[1,3,5,4,2] => [1,3,5,4,2] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 1 + 1
[1,4,2,3,5] => [1,3,4,2,5] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 1 + 1
[1,4,2,5,3] => [1,3,4,5,2] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 0 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,2,5,1] => [4,3,2,5,1] => ? = 2 + 1
[1,4,3,5,2] => [1,4,3,5,2] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 1 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1 + 1
[1,4,5,3,2] => [1,4,5,3,2] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 0 + 1
[1,5,2,3,4] => [1,3,4,5,2] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 0 + 1
[1,5,2,4,3] => [1,3,5,4,2] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 1 + 1
[1,5,3,2,4] => [1,4,3,5,2] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 1 + 1
[1,5,3,4,2] => [1,4,5,3,2] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 2 + 1
[1,5,4,2,3] => [1,4,5,3,2] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 0 + 1
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,2] => [1,3,4,5,2] => 3 = 2 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,2] => [1,3,5,4,2] => 2 = 1 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,2] => [1,4,3,5,2] => 2 = 1 + 1
[2,1,4,5,3] => [2,1,4,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 1 = 0 + 1
[2,1,5,3,4] => [2,1,4,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 1 = 0 + 1
[2,1,5,4,3] => [2,1,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2 = 1 + 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,2,4,5,3] => [1,2,4,5,3] => 2 = 1 + 1
[2,3,1,5,4] => [2,3,1,5,4] => [1,2,5,4,3] => [1,2,5,4,3] => 1 = 0 + 1
[2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => 1 = 0 + 1
[2,3,5,4,1] => [2,3,5,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 1 = 0 + 1
[2,4,1,3,5] => [4,3,1,2,5] => [4,2,1,5,3] => [4,2,1,5,3] => ? = 0 + 1
[2,4,3,1,5] => [2,4,3,1,5] => [1,3,2,5,4] => [1,3,2,5,4] => 2 = 1 + 1
[2,4,3,5,1] => [2,4,3,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 1 = 0 + 1
[2,4,5,1,3] => [3,5,4,1,2] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 0 + 1
[2,5,1,4,3] => [5,3,1,4,2] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 0 + 1
[2,5,3,1,4] => [5,4,3,1,2] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 0 + 1
[2,5,3,4,1] => [2,4,5,3,1] => [1,4,2,3,5] => [1,4,2,3,5] => 2 = 1 + 1
[2,5,4,3,1] => [2,5,4,3,1] => [1,4,3,2,5] => [1,4,3,2,5] => 1 = 0 + 1
[3,1,2,4,5] => [2,3,1,4,5] => [1,2,4,5,3] => [1,2,4,5,3] => 2 = 1 + 1
[3,1,2,5,4] => [2,3,1,5,4] => [1,2,5,4,3] => [1,2,5,4,3] => 1 = 0 + 1
[3,1,4,2,5] => [2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => 1 = 0 + 1
[3,1,5,4,2] => [2,3,5,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 1 = 0 + 1
[3,2,1,4,5] => [3,2,1,4,5] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 2 + 1
[3,2,1,5,4] => [3,2,1,5,4] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1 + 1
[3,2,4,1,5] => [3,2,4,1,5] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 1 + 1
[3,2,4,5,1] => [3,2,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 0 + 1
[3,2,5,1,4] => [5,2,4,1,3] => [2,5,3,1,4] => [2,5,3,1,4] => ? = 0 + 1
[3,2,5,4,1] => [3,2,5,4,1] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 1 + 1
[3,4,1,2,5] => [4,3,2,1,5] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1 + 1
[3,4,1,5,2] => [4,3,2,5,1] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0 + 1
[3,4,2,1,5] => [3,4,2,1,5] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0 + 1
[3,4,5,2,1] => [3,4,5,2,1] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 0 + 1
[3,5,1,2,4] => [4,3,5,1,2] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 0 + 1
[3,5,1,4,2] => [5,3,2,4,1] => [3,2,4,1,5] => [3,2,4,1,5] => ? = 1 + 1
[3,5,2,4,1] => [5,4,2,3,1] => [3,4,2,1,5] => [3,4,2,1,5] => ? = 0 + 1
[3,5,4,1,2] => [3,4,5,2,1] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 0 + 1
[4,1,2,3,5] => [2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => 1 = 0 + 1
[4,1,3,2,5] => [2,4,3,1,5] => [1,3,2,5,4] => [1,3,2,5,4] => 2 = 1 + 1
[4,1,3,5,2] => [2,4,3,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 1 = 0 + 1
[4,1,5,2,3] => [2,5,4,3,1] => [1,4,3,2,5] => [1,4,3,2,5] => 1 = 0 + 1
[4,2,1,3,5] => [3,2,4,1,5] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 1 + 1
[4,2,1,5,3] => [3,2,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 0 + 1
[4,2,3,1,5] => [3,4,2,1,5] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 2 + 1
[4,2,3,5,1] => [3,4,2,5,1] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 1 + 1
[4,2,5,1,3] => [5,2,4,3,1] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 1 + 1
[4,2,5,3,1] => [3,4,5,2,1] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 0 + 1
[4,3,1,2,5] => [3,4,2,1,5] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0 + 1
[4,3,2,1,5] => [4,3,2,1,5] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1 + 1
[5,1,2,4,3] => [2,3,5,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 1 = 0 + 1
[5,1,3,2,4] => [2,4,3,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 1 = 0 + 1
[5,1,3,4,2] => [2,4,5,3,1] => [1,4,2,3,5] => [1,4,2,3,5] => 2 = 1 + 1
[5,1,4,3,2] => [2,5,4,3,1] => [1,4,3,2,5] => [1,4,3,2,5] => 1 = 0 + 1
Description
The reflection length of a signed permutation.
This is the minimal numbers of reflections needed to express a signed permutation.
Matching statistic: St000942
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St000942: Parking functions ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 27%
Mp00305: Permutations —parking function⟶ Parking functions
St000942: Parking functions ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 27%
Values
[1,2] => [1,2] => [1,2] => 2 = 0 + 2
[1,2,3] => [1,2,3] => [1,2,3] => 3 = 1 + 2
[1,3,2] => [1,3,2] => [1,3,2] => 2 = 0 + 2
[2,1,3] => [2,1,3] => [2,1,3] => 2 = 0 + 2
[3,2,1] => [2,3,1] => [2,3,1] => 2 = 0 + 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 3 = 1 + 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 3 = 1 + 2
[1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 2 = 0 + 2
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2 = 0 + 2
[1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 3 = 1 + 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 3 = 1 + 2
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2 = 0 + 2
[2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 2 = 0 + 2
[2,4,3,1] => [3,4,1,2] => [3,4,1,2] => 2 = 0 + 2
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => 2 = 0 + 2
[3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 3 = 1 + 2
[3,2,4,1] => [2,4,1,3] => [2,4,1,3] => 2 = 0 + 2
[3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 2 = 0 + 2
[4,1,3,2] => [3,4,2,1] => [3,4,2,1] => 2 = 0 + 2
[4,2,1,3] => [2,4,3,1] => [2,4,3,1] => 2 = 0 + 2
[4,2,3,1] => [2,3,4,1] => [2,3,4,1] => 3 = 1 + 2
[4,3,2,1] => [3,2,4,1] => [3,2,4,1] => 2 = 0 + 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 3 + 2
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 2 + 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 2 + 2
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1 + 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 1 + 2
[1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 2 + 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 2 + 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => ? = 1 + 2
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 1 + 2
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 0 + 2
[1,3,5,2,4] => [1,5,4,2,3] => [1,5,4,2,3] => ? = 0 + 2
[1,3,5,4,2] => [1,4,5,2,3] => [1,4,5,2,3] => ? = 1 + 2
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => ? = 1 + 2
[1,4,2,5,3] => [1,5,3,2,4] => [1,5,3,2,4] => ? = 0 + 2
[1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => ? = 2 + 2
[1,4,3,5,2] => [1,3,5,2,4] => [1,3,5,2,4] => ? = 1 + 2
[1,4,5,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 2
[1,4,5,3,2] => [1,5,2,4,3] => [1,5,2,4,3] => ? = 0 + 2
[1,5,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 0 + 2
[1,5,2,4,3] => [1,4,5,3,2] => [1,4,5,3,2] => ? = 1 + 2
[1,5,3,2,4] => [1,3,5,4,2] => [1,3,5,4,2] => ? = 1 + 2
[1,5,3,4,2] => [1,3,4,5,2] => [1,3,4,5,2] => ? = 2 + 2
[1,5,4,2,3] => [1,5,3,4,2] => [1,5,3,4,2] => ? = 0 + 2
[1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => ? = 1 + 2
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 + 2
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 1 + 2
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 1 + 2
[2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 0 + 2
[2,1,5,3,4] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0 + 2
[2,1,5,4,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 1 + 2
[2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 1 + 2
[2,3,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0 + 2
[2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 0 + 2
[2,3,5,4,1] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 0 + 2
[2,4,1,3,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 0 + 2
[2,4,3,1,5] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 1 + 2
[2,4,3,5,1] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 0 + 2
[2,4,5,1,3] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 0 + 2
[2,5,1,4,3] => [4,5,3,1,2] => [4,5,3,1,2] => ? = 0 + 2
[2,5,3,1,4] => [3,5,4,1,2] => [3,5,4,1,2] => ? = 0 + 2
[2,5,3,4,1] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 + 2
[2,5,4,3,1] => [4,3,5,1,2] => [4,3,5,1,2] => ? = 0 + 2
[3,1,2,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 1 + 2
[3,1,2,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 0 + 2
[3,1,4,2,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 0 + 2
[3,1,5,4,2] => [4,5,2,1,3] => [4,5,2,1,3] => ? = 0 + 2
[3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 2 + 2
[3,2,1,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 1 + 2
[3,2,4,1,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1 + 2
[3,2,4,5,1] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 0 + 2
Description
The number of critical left to right maxima of the parking functions.
An entry p in a parking function is critical, if there are exactly p−1 entries smaller than p and n−p entries larger than p. It is a left to right maximum, if there are no larger entries before it.
This statistic allows the computation of the Tutte polynomial of the complete graph Kn+1, via
\sum_{P} x^{st(P)}y^{\binom{n+1}{2}-\sum P},
where the sum is over all parking functions of length n, see [1, thm.13.5.16].
The following 2 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!