Your data matches 35 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000147
Mp00306: Posets rowmotion cycle typeInteger partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> 2
([],2)
=> [2,2]
=> 2
([(0,1)],2)
=> [3]
=> 3
([],3)
=> [2,2,2,2]
=> 2
([(1,2)],3)
=> [6]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> 3
([(0,2),(2,1)],3)
=> [4]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> 3
([],4)
=> [2,2,2,2,2,2,2,2]
=> 2
([(2,3)],4)
=> [6,6]
=> 6
([(1,2),(1,3)],4)
=> [6,2,2]
=> 6
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> 3
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 4
([(1,2),(2,3)],4)
=> [4,4]
=> 4
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 4
([(1,3),(2,3)],4)
=> [6,2,2]
=> 6
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 4
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> 3
([(0,3),(1,2)],4)
=> [3,3,3]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> 4
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> 4
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> 4
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> 4
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> 4
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> 4
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> 10
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> 5
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> 8
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> 3
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> 10
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> 7
([(1,4),(3,2),(4,3)],5)
=> [10]
=> 10
Description
The largest part of an integer partition.
Mp00306: Posets rowmotion cycle typeInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [1,1]
=> 2
([],2)
=> [2,2]
=> [2,2]
=> 2
([(0,1)],2)
=> [3]
=> [1,1,1]
=> 3
([],3)
=> [2,2,2,2]
=> [4,4]
=> 2
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> [2,2,1]
=> 3
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> 3
([],4)
=> [2,2,2,2,2,2,2,2]
=> [8,8]
=> 2
([(2,3)],4)
=> [6,6]
=> [2,2,2,2,2,2]
=> 6
([(1,2),(1,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> 6
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> 3
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2,2,1,1]
=> 4
([(1,2),(2,3)],4)
=> [4,4]
=> [2,2,2,2]
=> 4
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 4
([(1,3),(2,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> 6
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 4
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> 3
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3,3]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [2,2,2,1,1]
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [3,3,1]
=> 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> 4
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [3,3,3,1]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [3,3,1,1]
=> 4
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [4,4,4,4]
=> 4
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> 4
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> 4
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [3,3,1,1]
=> 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [5,5,1]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [2,2,1,1,1,1,1]
=> 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [3,3,1,1]
=> 4
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> 10
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 5
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 8
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [5,5,1]
=> 3
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> 10
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [2,2,1,1,1,1,1]
=> 7
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> 10
Description
The length of the partition.
Matching statistic: St000734
Mp00306: Posets rowmotion cycle typeInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [[1,2]]
=> 2
([],2)
=> [2,2]
=> [[1,2],[3,4]]
=> 2
([(0,1)],2)
=> [3]
=> [[1,2,3]]
=> 3
([],3)
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> 2
([(1,2)],3)
=> [6]
=> [[1,2,3,4,5,6]]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
([(0,2),(2,1)],3)
=> [4]
=> [[1,2,3,4]]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
([],4)
=> [2,2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
=> 2
([(2,3)],4)
=> [6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> 6
([(1,2),(1,3)],4)
=> [6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> 6
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> 3
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 4
([(1,2),(2,3)],4)
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 4
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 4
([(1,3),(2,3)],4)
=> [6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> 6
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 4
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> 3
([(0,3),(1,2)],4)
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [[1,2,3,4,5]]
=> 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> 4
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 4
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> 4
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> 4
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> 4
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 4
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> 10
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> 5
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> 8
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> 3
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> 10
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> 7
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> 10
Description
The last entry in the first row of a standard tableau.
Mp00306: Posets rowmotion cycle typeInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
Mp00095: Integer partitions to binary wordBinary words
St000288: Binary words ⟶ ℤResult quality: 90% values known / values provided: 90%distinct values known / distinct values provided: 90%
Values
([],1)
=> [2]
=> [1,1]
=> 110 => 2
([],2)
=> [2,2]
=> [2,2]
=> 1100 => 2
([(0,1)],2)
=> [3]
=> [1,1,1]
=> 1110 => 3
([],3)
=> [2,2,2,2]
=> [4,4]
=> 110000 => 2
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1]
=> 1111110 => 6
([(0,1),(0,2)],3)
=> [3,2]
=> [2,2,1]
=> 11010 => 3
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 11110 => 4
([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> 11010 => 3
([],4)
=> [2,2,2,2,2,2,2,2]
=> [8,8]
=> 1100000000 => ? = 2
([(2,3)],4)
=> [6,6]
=> [2,2,2,2,2,2]
=> 11111100 => 6
([(1,2),(1,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> 110011110 => 6
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> 1100010 => 3
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 11111110 => 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2,2,1,1]
=> 110110 => 4
([(1,2),(2,3)],4)
=> [4,4]
=> [2,2,2,2]
=> 111100 => 4
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 110110 => 4
([(1,3),(2,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> 110011110 => 6
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 110110 => 4
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> 1100010 => 3
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3,3]
=> 111000 => 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [2,2,2,1,1]
=> 1110110 => 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [3,3,1]
=> 110010 => 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 111110 => 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 11111110 => 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> 11000110 => 4
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> 11001100 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 1101110 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [3,3,3,1]
=> 1110010 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1111010 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [3,3,1,1]
=> 1100110 => 4
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [4,4,4,4]
=> 11110000 => 4
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> 11001100 => 4
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> 11000110 => 4
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> 11001100 => 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [3,3,1,1]
=> 1100110 => 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> 11000110 => 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> 110010110 => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [5,5,1]
=> 11000010 => 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 1101110 => 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1111010 => 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [2,2,1,1,1,1,1]
=> 110111110 => 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [3,3,1,1]
=> 1100110 => 4
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => 10
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1111010 => 5
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 111111110 => 8
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> 110010110 => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [5,5,1]
=> 11000010 => 3
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => 10
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [2,2,1,1,1,1,1]
=> 110111110 => 7
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => 10
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 1101110 => 5
([(0,3),(1,4),(4,2)],5)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5),(4,6),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
([(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(6,1)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
([(0,5),(2,6),(4,1),(4,6),(5,2),(5,4),(6,3)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
([(0,5),(1,4),(3,6),(4,3),(4,5),(5,6),(6,2)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
([(0,6),(1,3),(1,6),(2,5),(3,5),(5,4),(6,2)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
([(0,4),(1,5),(2,5),(2,6),(3,1),(3,6),(4,2),(4,3)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
([(0,3),(0,6),(1,5),(1,6),(3,5),(4,2),(5,4),(6,4)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
([(0,3),(1,4),(1,6),(2,5),(3,4),(3,6),(4,2),(6,5)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
([(0,6),(3,5),(4,3),(5,1),(6,2),(6,4)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
([(0,5),(2,6),(3,4),(4,1),(4,6),(5,2),(5,3)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
([(0,6),(1,3),(3,6),(5,2),(6,4),(6,5)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Matching statistic: St000474
Mp00306: Posets rowmotion cycle typeInteger partitions
St000474: Integer partitions ⟶ ℤResult quality: 89% values known / values provided: 89%distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> 2
([],2)
=> [2,2]
=> 2
([(0,1)],2)
=> [3]
=> 3
([],3)
=> [2,2,2,2]
=> 2
([(1,2)],3)
=> [6]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> 3
([(0,2),(2,1)],3)
=> [4]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> 3
([],4)
=> [2,2,2,2,2,2,2,2]
=> ? = 2
([(2,3)],4)
=> [6,6]
=> 6
([(1,2),(1,3)],4)
=> [6,2,2]
=> 6
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> 3
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 4
([(1,2),(2,3)],4)
=> [4,4]
=> 4
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 4
([(1,3),(2,3)],4)
=> [6,2,2]
=> 6
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 4
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> 3
([(0,3),(1,2)],4)
=> [3,3,3]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> 4
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> 4
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> ? = 4
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> 4
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> 4
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> 4
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> 10
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> 5
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> 8
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> 3
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> 10
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> 7
([(1,4),(3,2),(4,3)],5)
=> [10]
=> 10
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> 5
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 8
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,4,2,2]
=> ? = 5
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> ? = 5
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> [5,4,2,2]
=> ? = 5
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [5,4,2,2]
=> ? = 5
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
=> [5,4,2,2]
=> ? = 5
([(0,5),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [5,4,2,2]
=> ? = 5
([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> ? = 5
([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
=> [5,4,2,2]
=> ? = 5
([(0,4),(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [5,4,2,2]
=> ? = 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [5,4,2,2]
=> ? = 5
([(0,5),(1,4),(4,2),(5,3)],6)
=> [4,4,4,4]
=> ? = 4
([(0,3),(0,4),(3,5),(3,6),(4,5),(4,6),(5,2),(6,1)],7)
=> [5,3,3,2]
=> ? = 5
([(0,6),(1,6),(4,3),(5,2),(6,4),(6,5)],7)
=> [5,3,3,2]
=> ? = 5
([(0,5),(0,6),(1,5),(1,6),(2,4),(3,4),(5,3),(6,2)],7)
=> [5,3,3,2]
=> ? = 5
([(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1)],7)
=> [5,3,3,2]
=> ? = 5
([(0,5),(1,4),(4,6),(5,6),(6,2),(6,3)],7)
=> [5,3,3,2]
=> ? = 5
([(0,3),(1,2),(2,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
=> [5,3,3,2]
=> ? = 5
Description
Dyson's crank of a partition. Let $\lambda$ be a partition and let $o(\lambda)$ be the number of parts that are equal to 1 ([[St000475]]), and let $\mu(\lambda)$ be the number of parts that are strictly larger than $o(\lambda)$ ([[St000473]]). Dyson's crank is then defined as $$crank(\lambda) = \begin{cases} \text{ largest part of }\lambda & o(\lambda) = 0\\ \mu(\lambda) - o(\lambda) & o(\lambda) > 0. \end{cases}$$
Mp00306: Posets rowmotion cycle typeInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 87% values known / values provided: 87%distinct values known / distinct values provided: 90%
Values
([],1)
=> [2]
=> [1,1]
=> [[1],[2]]
=> 2
([],2)
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
([(0,1)],2)
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
([],3)
=> [2,2,2,2]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 2
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
([],4)
=> [2,2,2,2,2,2,2,2]
=> [8,8]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14,15,16]]
=> ? = 2
([(2,3)],4)
=> [6,6]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> 6
([(1,2),(1,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10]]
=> 6
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> 3
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 4
([(1,2),(2,3)],4)
=> [4,4]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> 4
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 4
([(1,3),(2,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10]]
=> 6
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 4
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> 3
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> 4
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 4
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> ? = 4
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> 4
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> 4
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11]]
=> ? = 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 4
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> 10
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> 5
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 8
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11]]
=> ? = 3
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> 10
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> 7
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> 10
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> 5
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 8
([(0,3),(1,4),(4,2)],5)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 8
([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> 10
([(0,5),(1,4),(4,2),(5,3)],6)
=> [4,4,4,4]
=> [4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> ? = 4
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5),(4,6),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
([(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(6,1)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
([(0,5),(2,6),(4,1),(4,6),(5,2),(5,4),(6,3)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
([(0,5),(1,4),(3,6),(4,3),(4,5),(5,6),(6,2)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
([(0,6),(1,3),(1,6),(2,5),(3,5),(5,4),(6,2)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
([(0,4),(1,5),(2,5),(2,6),(3,1),(3,6),(4,2),(4,3)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
([(0,3),(0,6),(1,5),(1,6),(3,5),(4,2),(5,4),(6,4)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
([(0,3),(1,4),(1,6),(2,5),(3,4),(3,6),(4,2),(6,5)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
([(0,6),(3,5),(4,3),(5,1),(6,2),(6,4)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
([(0,5),(2,6),(3,4),(4,1),(4,6),(5,2),(5,3)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
([(0,6),(1,3),(3,6),(5,2),(6,4),(6,5)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
Description
The row containing the largest entry of a standard tableau.
Mp00306: Posets rowmotion cycle typeInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 87% values known / values provided: 87%distinct values known / distinct values provided: 90%
Values
([],1)
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
([],2)
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 2 - 1
([(0,1)],2)
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 2 = 3 - 1
([],3)
=> [2,2,2,2]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 1 = 2 - 1
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 5 = 6 - 1
([(0,1),(0,2)],3)
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2 = 3 - 1
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3 = 4 - 1
([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2 = 3 - 1
([],4)
=> [2,2,2,2,2,2,2,2]
=> [8,8]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14,15,16]]
=> ? = 2 - 1
([(2,3)],4)
=> [6,6]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> 5 = 6 - 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10]]
=> 5 = 6 - 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> 2 = 3 - 1
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 6 = 7 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 3 = 4 - 1
([(1,2),(2,3)],4)
=> [4,4]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> 3 = 4 - 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 3 = 4 - 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10]]
=> 5 = 6 - 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 3 = 4 - 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> 2 = 3 - 1
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> 2 = 3 - 1
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> 4 = 5 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 4 = 5 - 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 6 = 7 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> 3 = 4 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> 3 = 4 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> 4 = 5 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> 3 = 4 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> 4 = 5 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 3 = 4 - 1
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> ? = 4 - 1
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> 3 = 4 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> 3 = 4 - 1
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> 3 = 4 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 3 = 4 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> 3 = 4 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> 4 = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11]]
=> ? = 3 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> 4 = 5 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> 4 = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> 6 = 7 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 3 = 4 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> 9 = 10 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> 4 = 5 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 7 = 8 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> 4 = 5 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11]]
=> ? = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> 9 = 10 - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> 6 = 7 - 1
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> 9 = 10 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> 4 = 5 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 7 = 8 - 1
([(0,3),(1,4),(4,2)],5)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 7 = 8 - 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> 9 = 10 - 1
([(0,5),(1,4),(4,2),(5,3)],6)
=> [4,4,4,4]
=> [4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> ? = 4 - 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12 - 1
([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12 - 1
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5),(4,6),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12 - 1
([(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(6,1)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12 - 1
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12 - 1
([(0,5),(2,6),(4,1),(4,6),(5,2),(5,4),(6,3)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12 - 1
([(0,5),(1,4),(3,6),(4,3),(4,5),(5,6),(6,2)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12 - 1
([(0,6),(1,3),(1,6),(2,5),(3,5),(5,4),(6,2)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12 - 1
([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12 - 1
([(0,4),(1,5),(2,5),(2,6),(3,1),(3,6),(4,2),(4,3)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12 - 1
([(0,3),(0,6),(1,5),(1,6),(3,5),(4,2),(5,4),(6,4)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12 - 1
([(0,3),(1,4),(1,6),(2,5),(3,4),(3,6),(4,2),(6,5)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12 - 1
([(0,6),(3,5),(4,3),(5,1),(6,2),(6,4)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12 - 1
([(0,5),(2,6),(3,4),(4,1),(4,6),(5,2),(5,3)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12 - 1
([(0,6),(1,3),(3,6),(5,2),(6,4),(6,5)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12 - 1
Description
The number of descents of a standard tableau. Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
St000643: Posets ⟶ ℤResult quality: 80% values known / values provided: 80%distinct values known / distinct values provided: 100%
Values
([],1)
=> ? = 2
([],2)
=> 2
([(0,1)],2)
=> 3
([],3)
=> 2
([(1,2)],3)
=> 6
([(0,1),(0,2)],3)
=> 3
([(0,2),(2,1)],3)
=> 4
([(0,2),(1,2)],3)
=> 3
([],4)
=> 2
([(2,3)],4)
=> 6
([(1,2),(1,3)],4)
=> 6
([(0,1),(0,2),(0,3)],4)
=> 3
([(0,2),(0,3),(3,1)],4)
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(1,2),(2,3)],4)
=> 4
([(0,3),(3,1),(3,2)],4)
=> 4
([(1,3),(2,3)],4)
=> 6
([(0,3),(1,3),(3,2)],4)
=> 4
([(0,3),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2)],4)
=> 3
([(0,3),(1,2),(1,3)],4)
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> 3
([(0,3),(2,1),(3,2)],4)
=> 5
([(0,3),(1,2),(2,3)],4)
=> 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 4
([(1,2),(1,3),(2,4),(3,4)],5)
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
([(2,3),(3,4)],5)
=> 4
([(1,4),(4,2),(4,3)],5)
=> 4
([(0,4),(4,1),(4,2),(4,3)],5)
=> 4
([(1,4),(2,4),(4,3)],5)
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 4
([(0,4),(1,2),(1,4),(4,3)],5)
=> 10
([(0,2),(0,4),(3,1),(4,3)],5)
=> 5
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 8
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 10
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 7
([(1,4),(3,2),(4,3)],5)
=> 10
([(0,3),(3,4),(4,1),(4,2)],5)
=> 5
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5),(4,6),(5,6)],7)
=> ? = 12
([(0,2),(0,4),(1,5),(1,6),(2,5),(2,6),(3,1),(4,3)],7)
=> ? = 6
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 7
([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(6,4)],7)
=> ? = 6
([(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(6,1)],7)
=> ? = 12
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=> ? = 12
([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
=> ? = 6
([(0,5),(2,6),(4,1),(4,6),(5,2),(5,4),(6,3)],7)
=> ? = 12
([(0,2),(1,5),(1,6),(2,3),(3,5),(3,6),(5,4),(6,4)],7)
=> ? = 6
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
=> ? = 6
([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
=> ? = 6
([(0,5),(1,4),(3,6),(4,3),(4,5),(5,6),(6,2)],7)
=> ? = 12
([(0,5),(1,4),(4,6),(5,6),(6,2),(6,3)],7)
=> ? = 5
([(0,3),(1,2),(2,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
=> ? = 5
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 10
([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
=> ? = 6
([(0,6),(1,3),(1,6),(2,5),(3,5),(5,4),(6,2)],7)
=> ? = 12
([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ? = 12
([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> ? = 10
([(0,4),(1,5),(2,5),(2,6),(3,1),(3,6),(4,2),(4,3)],7)
=> ? = 12
([(0,3),(0,6),(1,5),(1,6),(3,5),(4,2),(5,4),(6,4)],7)
=> ? = 12
([(0,3),(1,4),(1,6),(2,5),(3,4),(3,6),(4,2),(6,5)],7)
=> ? = 12
([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
=> ? = 6
([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
=> ? = 7
([(0,6),(3,5),(4,3),(5,1),(6,2),(6,4)],7)
=> ? = 12
([(0,5),(2,6),(3,4),(4,1),(4,6),(5,2),(5,3)],7)
=> ? = 12
([(0,6),(1,3),(3,6),(5,2),(6,4),(6,5)],7)
=> ? = 12
([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
=> ? = 10
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 8
([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 10
([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ? = 10
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 7
([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 7
Description
The size of the largest orbit of antichains under Panyushev complementation.
Matching statistic: St000013
Mp00306: Posets rowmotion cycle typeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00227: Dyck paths Delest-Viennot-inverseDyck paths
St000013: Dyck paths ⟶ ℤResult quality: 80% values known / values provided: 80%distinct values known / distinct values provided: 90%
Values
([],1)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
([],2)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
([(0,1)],2)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
([],3)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
([(1,2)],3)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([(0,2),(2,1)],3)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([],4)
=> [2,2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 2
([(2,3)],4)
=> [6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6
([(1,2),(1,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> 6
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
([(1,2),(2,3)],4)
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
([(1,3),(2,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> 6
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3
([(0,3),(1,2)],4)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 4
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 4
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> 4
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> 4
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 4
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 4
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> 10
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 5
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> 8
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> 10
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> 7
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> 10
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 5
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> 8
([(0,3),(1,4),(4,2)],5)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 5
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 5
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 5
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 5
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 5
([(0,5),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 5
([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 5
([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 5
([(0,4),(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 5
([(0,3),(0,4),(3,5),(3,6),(4,5),(4,6),(5,2),(6,1)],7)
=> [5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> ? = 5
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,6),(1,6),(4,3),(5,2),(6,4),(6,5)],7)
=> [5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> ? = 5
([(0,5),(0,6),(1,5),(1,6),(2,4),(3,4),(5,3),(6,2)],7)
=> [5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> ? = 5
([(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1)],7)
=> [5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> ? = 5
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5),(4,6),(5,6)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(6,1)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,5),(2,6),(4,1),(4,6),(5,2),(5,4),(6,3)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,5),(1,4),(3,6),(4,3),(4,5),(5,6),(6,2)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,5),(1,4),(4,6),(5,6),(6,2),(6,3)],7)
=> [5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> ? = 5
([(0,3),(1,2),(2,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
=> [5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> ? = 5
([(0,6),(1,3),(1,6),(2,5),(3,5),(5,4),(6,2)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,4),(1,5),(2,5),(2,6),(3,1),(3,6),(4,2),(4,3)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,3),(0,6),(1,5),(1,6),(3,5),(4,2),(5,4),(6,4)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,3),(1,4),(1,6),(2,5),(3,4),(3,6),(4,2),(6,5)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,6),(3,5),(4,3),(5,1),(6,2),(6,4)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,5),(2,6),(3,4),(4,1),(4,6),(5,2),(5,3)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,6),(1,3),(3,6),(5,2),(6,4),(6,5)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
Description
The height of a Dyck path. The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Mp00306: Posets rowmotion cycle typeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St000676: Dyck paths ⟶ ℤResult quality: 70% values known / values provided: 78%distinct values known / distinct values provided: 70%
Values
([],1)
=> [2]
=> [1,0,1,0]
=> 2
([],2)
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
([(0,1)],2)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([],3)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(1,2)],3)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
([(0,2),(2,1)],3)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
([],4)
=> [2,2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2
([(2,3)],4)
=> [6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 6
([(1,2),(1,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 6
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
([(1,2),(2,3)],4)
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
([(1,3),(2,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 6
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3
([(0,3),(1,2)],4)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 4
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 4
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 4
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> 4
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 4
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 4
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 5
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 8
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 7
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 5
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 8
([(0,3),(1,4),(4,2)],5)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 12
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 8
([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 8
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 8
([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 8
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 8
([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 8
([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 8
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 12
([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 12
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5),(4,6),(5,6)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 12
([(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(6,1)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 12
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 12
([(0,5),(2,6),(4,1),(4,6),(5,2),(5,4),(6,3)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 12
([(0,5),(1,4),(3,6),(4,3),(4,5),(5,6),(6,2)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 12
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
([(0,6),(1,3),(1,6),(2,5),(3,5),(5,4),(6,2)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 12
([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 12
([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
([(0,4),(1,5),(2,5),(2,6),(3,1),(3,6),(4,2),(4,3)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 12
([(0,3),(0,6),(1,5),(1,6),(3,5),(4,2),(5,4),(6,4)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 12
([(0,3),(1,4),(1,6),(2,5),(3,4),(3,6),(4,2),(6,5)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 12
([(0,6),(3,5),(4,3),(5,1),(6,2),(6,4)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 12
([(0,5),(2,6),(3,4),(4,1),(4,6),(5,2),(5,3)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 12
([(0,6),(1,3),(3,6),(5,2),(6,4),(6,5)],7)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 12
([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
Description
The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
The following 25 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000439The position of the first down step of a Dyck path. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St000025The number of initial rises of a Dyck path. St000444The length of the maximal rise of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000874The position of the last double rise in a Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000738The first entry in the last row of a standard tableau. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one.