- FindStat

Your data matches 30 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000993
(load all 3 compositions to match this statistic)

Mapping

Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: 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]
 => 2
([(0,2),(2,1)],3)
 => [4]
 => [1,1,1,1]
 => 4
([(0,2),(1,2)],3)
 => [3,2]
 => [2,2,1]
 => 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]
 => 2
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [4,4,1]
 => 2
([(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]
 => 2
([(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]
 => 2
([(1,3),(2,3)],4)
 => [6,2,2]
 => [3,3,1,1,1,1]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [4,4,1]
 => 2
([(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]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [3,3,1]
 => 2
([(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]
 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [2,2,1,1,1]
 => 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [3,3,3,1]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [2,2,2,2,1]
 => 4
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [3,3,1,1]
 => 2
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [4,4,1,1]
 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [3,3,1,1]
 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [4,4,1,1]
 => 2
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [2,2,2,2,1]
 => 4
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => [2,2,1,1,1,1,1]
 => 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [3,3,1,1]
 => 2
([(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]
 => 4
([(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),(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]
 => 2
([(1,4),(3,2),(4,3)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => 10
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => 2
([(0,4),(1,2),(2,4),(4,3)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 8
([(0,4),(3,2),(4,1),(4,3)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 8
([(0,4),(1,2),(2,3),(2,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => 10
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => [1,1,1,1,1,1]
 => 6
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,3,3]
 => [3,3,3,1]
 => 3
([(0,4),(1,2),(2,3),(3,4)],5)
 => [5,4]
 => [2,2,2,2,1]
 => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [5,2,2]
 => [3,3,1,1,1]
 => 2

Description

The multiplicity of the largest part of an integer partition.

Matching statistic: St000297

Mapping

Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

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 => 2
([(0,2),(2,1)],3)
 => [4]
 => [1,1,1,1]
 => 11110 => 4
([(0,2),(1,2)],3)
 => [3,2]
 => [2,2,1]
 => 11010 => 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 => 2
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [4,4,1]
 => 1100010 => 2
([(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 => 2
([(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 => 2
([(1,3),(2,3)],4)
 => [6,2,2]
 => [3,3,1,1,1,1]
 => 110011110 => 2
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => 110110 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [4,4,1]
 => 1100010 => 2
([(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 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [3,3,1]
 => 110010 => 2
([(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 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [2,2,1,1,1]
 => 1101110 => 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [3,3,3,1]
 => 1110010 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [2,2,2,2,1]
 => 1111010 => 4
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [3,3,1,1]
 => 1100110 => 2
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [4,4,1,1]
 => 11000110 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [3,3,1,1]
 => 1100110 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [4,4,1,1]
 => 11000110 => 2
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => 1101110 => 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [2,2,2,2,1]
 => 1111010 => 4
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => [2,2,1,1,1,1,1]
 => 110111110 => 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [3,3,1,1]
 => 1100110 => 2
([(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 => 4
([(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),(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 => 2
([(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 => 2
([(0,4),(1,2),(2,4),(4,3)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 111111110 => 8
([(0,4),(3,2),(4,1),(4,3)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 111111110 => 8
([(0,4),(1,2),(2,3),(2,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => 11111111110 => 10
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => [1,1,1,1,1,1]
 => 1111110 => 6
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,3,3]
 => [3,3,3,1]
 => 1110010 => 3
([(0,4),(1,2),(2,3),(3,4)],5)
 => [5,4]
 => [2,2,2,2,1]
 => 1111010 => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => 1101110 => 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [5,2,2]
 => [3,3,1,1,1]
 => 11001110 => 2

Description

The number of leading ones in a binary word.

Matching statistic: St000733
(load all 2 compositions to match this statistic)

Mapping

Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

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]
 => [[1,3],[2,5],[4]]
 => 2
([(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]
 => [[1,3],[2,5],[4]]
 => 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]
 => [[1,6,7],[2,9,10],[3],[4],[5],[8]]
 => 2
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [4,4,1]
 => [[1,3,4,5],[2,7,8,9],[6]]
 => 2
([(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]
 => [[1,4],[2,6],[3],[5]]
 => 2
([(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]
 => [[1,4],[2,6],[3],[5]]
 => 2
([(1,3),(2,3)],4)
 => [6,2,2]
 => [3,3,1,1,1,1]
 => [[1,6,7],[2,9,10],[3],[4],[5],[8]]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [4,4,1]
 => [[1,3,4,5],[2,7,8,9],[6]]
 => 2
([(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]
 => [[1,4],[2,6],[3,8],[5],[7]]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => 2
([(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]
 => [[1,4,5,6],[2,8,9,10],[3],[7]]
 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [3,3,3,1]
 => [[1,3,4],[2,6,7],[5,9,10],[8]]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8]]
 => 4
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => 2
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [4,4,1,1]
 => [[1,4,5,6],[2,8,9,10],[3],[7]]
 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [4,4,1,1]
 => [[1,4,5,6],[2,8,9,10],[3],[7]]
 => 2
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8]]
 => 4
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => [2,2,1,1,1,1,1]
 => [[1,7],[2,9],[3],[4],[5],[6],[8]]
 => 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => 2
([(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]
 => [[1,3],[2,5],[4,7],[6,9],[8]]
 => 4
([(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),(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]
 => [[1,7],[2,9],[3],[4],[5],[6],[8]]
 => 2
([(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]
 => [[1,5],[2,7],[3],[4],[6]]
 => 2
([(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
([(0,4),(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
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 6
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,3,3]
 => [3,3,3,1]
 => [[1,3,4],[2,6,7],[5,9,10],[8]]
 => 3
([(0,4),(1,2),(2,3),(3,4)],5)
 => [5,4]
 => [2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8]]
 => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [5,2,2]
 => [3,3,1,1,1]
 => [[1,5,6],[2,8,9],[3],[4],[7]]
 => 2

Description

The row containing the largest entry of a standard tableau.

Matching statistic: St000745

Mapping

Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 2
([],2)
 => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
([(0,1)],2)
 => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 3
([],3)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [[1,3,5,7],[2,4,6,8]]
 => 2
([(1,2)],3)
 => [6]
 => [[1,2,3,4,5,6]]
 => [[1],[2],[3],[4],[5],[6]]
 => 6
([(0,1),(0,2)],3)
 => [3,2]
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 2
([(0,2),(2,1)],3)
 => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4
([(0,2),(1,2)],3)
 => [3,2]
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 2
([(2,3)],4)
 => [6,6]
 => [[1,2,3,4,5,6],[7,8,9,10,11,12]]
 => [[1,7],[2,8],[3,9],[4,10],[5,11],[6,12]]
 => 6
([(1,2),(1,3)],4)
 => [6,2,2]
 => [[1,2,7,8,9,10],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7],[8],[9],[10]]
 => 2
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [[1,2,9],[3,4],[5,6],[7,8]]
 => [[1,3,5,7],[2,4,6,8],[9]]
 => 2
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [[1,2,3,4,5,6,7]]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 7
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => [[1,3],[2,4],[5],[6]]
 => 2
([(1,2),(2,3)],4)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [[1,5],[2,6],[3,7],[4,8]]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => [[1,3],[2,4],[5],[6]]
 => 2
([(1,3),(2,3)],4)
 => [6,2,2]
 => [[1,2,7,8,9,10],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7],[8],[9],[10]]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => [[1,3],[2,4],[5],[6]]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [[1,2,9],[3,4],[5,6],[7,8]]
 => [[1,3,5,7],[2,4,6,8],[9]]
 => 2
([(0,3),(1,2)],4)
 => [3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => [[1,4,7],[2,5,8],[3,6,9]]
 => 3
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => [[1,4],[2,5],[3,6],[7],[8]]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7]]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 5
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [[1,2,3,4,5,6,7]]
 => [[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]
 => [[1,2,9,10],[3,4],[5,6],[7,8]]
 => [[1,3,5,7],[2,4,6,8],[9],[10]]
 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [[1,3],[2,4],[5],[6],[7]]
 => 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [[1,2,3,10],[4,5,6],[7,8,9]]
 => [[1,4,7],[2,5,8],[3,6,9],[10]]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => [[1,5],[2,6],[3,7],[4,8],[9]]
 => 4
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7],[8]]
 => 2
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [[1,2,9,10],[3,4],[5,6],[7,8]]
 => [[1,3,5,7],[2,4,6,8],[9],[10]]
 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7],[8]]
 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [[1,2,9,10],[3,4],[5,6],[7,8]]
 => [[1,3,5,7],[2,4,6,8],[9],[10]]
 => 2
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [[1,3],[2,4],[5],[6],[7]]
 => 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => [[1,5],[2,6],[3,7],[4,8],[9]]
 => 4
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => [[1,2,5,6,7,8,9],[3,4]]
 => [[1,3],[2,4],[5],[6],[7],[8],[9]]
 => 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7],[8]]
 => 2
([(0,4),(1,2),(1,4),(4,3)],5)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => 10
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => [[1,5],[2,6],[3,7],[4,8],[9]]
 => 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => [[1,2,3,4,5,6,7,8]]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => 8
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => 10
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => [7,2]
 => [[1,2,5,6,7,8,9],[3,4]]
 => [[1,3],[2,4],[5],[6],[7],[8],[9]]
 => 2
([(1,4),(3,2),(4,3)],5)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => 10
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [[1,3],[2,4],[5],[6],[7]]
 => 2
([(0,4),(1,2),(2,4),(4,3)],5)
 => [8]
 => [[1,2,3,4,5,6,7,8]]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => 8
([(0,4),(3,2),(4,1),(4,3)],5)
 => [8]
 => [[1,2,3,4,5,6,7,8]]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => 8
([(0,4),(1,2),(2,3),(2,4)],5)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => 10
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => [[1,2,3,4,5,6]]
 => [[1],[2],[3],[4],[5],[6]]
 => 6
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,3,3]
 => [[1,2,3,10],[4,5,6],[7,8,9]]
 => [[1,4,7],[2,5,8],[3,6,9],[10]]
 => 3
([(0,4),(1,2),(2,3),(3,4)],5)
 => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => [[1,5],[2,6],[3,7],[4,8],[9]]
 => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [[1,3],[2,4],[5],[6],[7]]
 => 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [5,2,2]
 => [[1,2,7,8,9],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7],[8],[9]]
 => 2

Description

The index of the last row whose first entry is the row number in a standard Young tableau.

Matching statistic: St000326

Mapping

Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [2]
 => 100 => 001 => 3 = 2 + 1
([],2)
 => [2,2]
 => 1100 => 0011 => 3 = 2 + 1
([(0,1)],2)
 => [3]
 => 1000 => 0001 => 4 = 3 + 1
([],3)
 => [2,2,2,2]
 => 111100 => 001111 => 3 = 2 + 1
([(1,2)],3)
 => [6]
 => 1000000 => 0000001 => 7 = 6 + 1
([(0,1),(0,2)],3)
 => [3,2]
 => 10100 => 00101 => 3 = 2 + 1
([(0,2),(2,1)],3)
 => [4]
 => 10000 => 00001 => 5 = 4 + 1
([(0,2),(1,2)],3)
 => [3,2]
 => 10100 => 00101 => 3 = 2 + 1
([(2,3)],4)
 => [6,6]
 => 11000000 => 00000011 => 7 = 6 + 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => 100001100 => 001100001 => 3 = 2 + 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => 1011100 => 0011101 => 3 = 2 + 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => 10000000 => 00000001 => 8 = 7 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => 100100 => 001001 => 3 = 2 + 1
([(1,2),(2,3)],4)
 => [4,4]
 => 110000 => 000011 => 5 = 4 + 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => 100100 => 001001 => 3 = 2 + 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => 100001100 => 001100001 => 3 = 2 + 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => 100100 => 001001 => 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => 1011100 => 0011101 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => 111000 => 000111 => 4 = 3 + 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => 1001000 => 0001001 => 4 = 3 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => 101100 => 001101 => 3 = 2 + 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => 100000 => 000001 => 6 = 5 + 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => 10000000 => 00000001 => 8 = 7 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => 10011100 => 00111001 => 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => 1000100 => 0010001 => 3 = 2 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => 1011000 => 0001101 => 4 = 3 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => 1010000 => 0000101 => 5 = 4 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => 1001100 => 0011001 => 3 = 2 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => 10011100 => 00111001 => 3 = 2 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => 1001100 => 0011001 => 3 = 2 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => 10011100 => 00111001 => 3 = 2 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => 1000100 => 0010001 => 3 = 2 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => 1010000 => 0000101 => 5 = 4 + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => 100000100 => 001000001 => 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => 1001100 => 0011001 => 3 = 2 + 1
([(0,4),(1,2),(1,4),(4,3)],5)
 => [10]
 => 10000000000 => 00000000001 => 11 = 10 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => 1010000 => 0000101 => 5 = 4 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => 100000000 => 000000001 => 9 = 8 + 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [10]
 => 10000000000 => 00000000001 => 11 = 10 + 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => [7,2]
 => 100000100 => 001000001 => 3 = 2 + 1
([(1,4),(3,2),(4,3)],5)
 => [10]
 => 10000000000 => 00000000001 => 11 = 10 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => 1000100 => 0010001 => 3 = 2 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => [8]
 => 100000000 => 000000001 => 9 = 8 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => [8]
 => 100000000 => 000000001 => 9 = 8 + 1
([(0,4),(1,2),(2,3),(2,4)],5)
 => [10]
 => 10000000000 => 00000000001 => 11 = 10 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1000000 => 0000001 => 7 = 6 + 1
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,3,3]
 => 1011000 => 0001101 => 4 = 3 + 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => [5,4]
 => 1010000 => 0000101 => 5 = 4 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => 1000100 => 0010001 => 3 = 2 + 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [5,2,2]
 => 10001100 => 00110001 => 3 = 2 + 1

Description

The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of

$\{1,\dots,n,n+1\}$ that contains

$n+1$ , this is the minimal element of the set.

Matching statistic: St000617

Mapping

Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000617: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 2
([],2)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2
([(0,1)],2)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3
([],3)
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 2
([(1,2)],3)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 6
([(0,1),(0,2)],3)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 2
([(0,2),(2,1)],3)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4
([(0,2),(1,2)],3)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 2
([(2,3)],4)
 => [6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6
([(1,2),(1,3)],4)
 => [6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => 2
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 2
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 7
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
([(1,2),(2,3)],4)
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
([(1,3),(2,3)],4)
 => [6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 2
([(0,3),(1,2)],4)
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 3
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 5
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 4
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => 2
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 4
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
([(0,4),(1,2),(1,4),(4,3)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 10
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 8
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 10
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => [7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => 2
([(1,4),(3,2),(4,3)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 10
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 2
([(0,4),(1,2),(2,4),(4,3)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 8
([(0,4),(3,2),(4,1),(4,3)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 8
([(0,4),(1,2),(2,3),(2,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 10
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 6
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 3
([(0,4),(1,2),(2,3),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 2
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => [8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => 2
([(0,5),(1,4),(4,2),(4,5),(5,3)],6)
 => [6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6
([(1,5),(3,4),(4,2),(5,3)],6)
 => [6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6

Description

The number of global maxima of a Dyck path.

Matching statistic: St000642
(load all 2 compositions to match this statistic)

Mapping

St000642: Posets ⟶ ℤResult quality: 85% ●values known / values provided: 85%●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)
 => 2
([(0,2),(2,1)],3)
 => 4
([(0,2),(1,2)],3)
 => 2
([(2,3)],4)
 => 6
([(1,2),(1,3)],4)
 => 2
([(0,1),(0,2),(0,3)],4)
 => 2
([(0,2),(0,3),(3,1)],4)
 => 7
([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(1,2),(2,3)],4)
 => 4
([(0,3),(3,1),(3,2)],4)
 => 2
([(1,3),(2,3)],4)
 => 2
([(0,3),(1,3),(3,2)],4)
 => 2
([(0,3),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2)],4)
 => 3
([(0,3),(1,2),(1,3)],4)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(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)
 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 4
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
([(0,4),(4,1),(4,2),(4,3)],5)
 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => 2
([(0,4),(1,4),(2,3),(4,2)],5)
 => 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => 2
([(0,4),(1,2),(1,4),(4,3)],5)
 => 10
([(0,2),(0,4),(3,1),(4,3)],5)
 => 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 8
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => 10
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => 2
([(1,4),(3,2),(4,3)],5)
 => 10
([(0,3),(3,4),(4,1),(4,2)],5)
 => 2
([(0,4),(1,2),(2,4),(4,3)],5)
 => 8
([(0,4),(3,2),(4,1),(4,3)],5)
 => 8
([(0,4),(1,2),(2,3),(2,4)],5)
 => 10
([(0,4),(2,3),(3,1),(4,2)],5)
 => 6
([(0,3),(1,2),(2,4),(3,4)],5)
 => 3
([(0,4),(1,2),(2,3),(3,4)],5)
 => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => 2
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => 2
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => ? = 2
([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
 => ? = 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
 => ? = 2
([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
 => ? = 2
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => ? = 10
([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
 => ? = 10
([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
 => ? = 2
([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
 => ? = 2
([(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)
 => ? = 2
([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ? = 2

Description

The size of the smallest orbit of antichains under Panyushev complementation.

Matching statistic: St001038
(load all 2 compositions to match this statistic)

Mapping

Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 81%●distinct values known / distinct values provided: 78%

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]
 => 2
([(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]
 => 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]
 => 2
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 2
([(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]
 => 2
([(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]
 => 2
([(1,3),(2,3)],4)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 2
([(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]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2
([(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]
 => 2
([(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]
 => 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 3
([(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]
 => 4
([(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]
 => 2
([(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]
 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2
([(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]
 => 2
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2
([(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]
 => 4
([(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]
 => 2
([(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]
 => 2
([(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]
 => 4
([(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),(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]
 => 2
([(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]
 => 2
([(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
([(0,4),(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
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 3
([(0,4),(1,2),(2,3),(3,4)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => 2
([(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]
 => ? = 2
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => 2
([(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]
 => ? = 2
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => 2
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => 2
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => [5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => 2
([(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]
 => ? = 2
([(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]
 => ? = 2
([(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]
 => ? = 2
([(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]
 => ? = 2
([(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
([(0,5),(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
([(0,3),(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
([(0,5),(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 minimal height of a column in the parallelogram polyomino associated with the Dyck path.

Matching statistic: St000474

Mapping

Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000474: Integer partitions ⟶ ℤResult quality: 56% ●values known / values provided: 75%●distinct values known / distinct values provided: 56%

Values

([],1)
 => [2]
 => []
 => ? = 2
([],2)
 => [2,2]
 => [2]
 => 2
([(0,1)],2)
 => [3]
 => []
 => ? = 3
([],3)
 => [2,2,2,2]
 => [2,2,2]
 => 2
([(1,2)],3)
 => [6]
 => []
 => ? = 6
([(0,1),(0,2)],3)
 => [3,2]
 => [2]
 => 2
([(0,2),(2,1)],3)
 => [4]
 => []
 => ? = 4
([(0,2),(1,2)],3)
 => [3,2]
 => [2]
 => 2
([(2,3)],4)
 => [6,6]
 => [6]
 => 6
([(1,2),(1,3)],4)
 => [6,2,2]
 => [2,2]
 => 2
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [2,2,2]
 => 2
([(0,2),(0,3),(3,1)],4)
 => [7]
 => []
 => ? = 7
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [2]
 => 2
([(1,2),(2,3)],4)
 => [4,4]
 => [4]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [2]
 => 2
([(1,3),(2,3)],4)
 => [6,2,2]
 => [2,2]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [2]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [2,2,2]
 => 2
([(0,3),(1,2)],4)
 => [3,3,3]
 => [3,3]
 => 3
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [3]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [2,2]
 => 2
([(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]
 => [2,2,2]
 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [2]
 => 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [3,3]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [4]
 => 4
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [2,2]
 => 2
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [2,2,2]
 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [2,2]
 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [2,2,2]
 => 2
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [2]
 => 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [4]
 => 4
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => [2]
 => 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [2,2]
 => 2
([(0,4),(1,2),(1,4),(4,3)],5)
 => [10]
 => []
 => ? = 10
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [4]
 => 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => []
 => ? = 8
([(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]
 => [2]
 => 2
([(1,4),(3,2),(4,3)],5)
 => [10]
 => []
 => ? = 10
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [2]
 => 2
([(0,4),(1,2),(2,4),(4,3)],5)
 => [8]
 => []
 => ? = 8
([(0,4),(3,2),(4,1),(4,3)],5)
 => [8]
 => []
 => ? = 8
([(0,4),(1,2),(2,3),(2,4)],5)
 => [10]
 => []
 => ? = 10
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => []
 => ? = 6
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,3,3]
 => [3,3]
 => 3
([(0,4),(1,2),(2,3),(3,4)],5)
 => [5,4]
 => [4]
 => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [2]
 => 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [5,2,2]
 => [2,2]
 => 2
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => [8,2]
 => [2]
 => 2
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [5,2,2]
 => [2,2]
 => 2
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
 => [8,2]
 => [2]
 => 2
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [5,2,2]
 => [2,2]
 => 2
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [6,2]
 => [2]
 => 2
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => [5,2,2]
 => [2,2]
 => 2
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => [5,5]
 => [5]
 => 5
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => [4,2,2,2]
 => [2,2,2]
 => 2
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => [5,2,2]
 => [2,2]
 => 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => [8,2]
 => [2]
 => 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [5,5]
 => [5]
 => 5
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => [9]
 => []
 => ? = 9
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => [6,4]
 => [4]
 => 4
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [6,2]
 => [2]
 => 2
([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
 => [8,2]
 => [2]
 => 2
([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [8,2]
 => [2]
 => 2
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
 => [9]
 => []
 => ? = 9
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => []
 => ? = 7
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [9]
 => []
 => ? = 9
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => [9]
 => []
 => ? = 9
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => [10]
 => []
 => ? = 10
([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
 => [10]
 => []
 => ? = 10
([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
 => [10]
 => []
 => ? = 10
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [8]
 => []
 => ? = 8
([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
 => [10]
 => []
 => ? = 10
([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
 => [10]
 => []
 => ? = 10

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

$crank(\lambda) = \begin{cases} \text{ largest part of }\lambda & o(\lambda) = 0\\ \mu(\lambda) - o(\lambda) & o(\lambda) > 0. \end{cases}$

Matching statistic: St000667

Mapping

Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 56% ●values known / values provided: 75%●distinct values known / distinct values provided: 56%

Values

([],1)
 => [2]
 => []
 => ? = 2
([],2)
 => [2,2]
 => [2]
 => 2
([(0,1)],2)
 => [3]
 => []
 => ? = 3
([],3)
 => [2,2,2,2]
 => [2,2,2]
 => 2
([(1,2)],3)
 => [6]
 => []
 => ? = 6
([(0,1),(0,2)],3)
 => [3,2]
 => [2]
 => 2
([(0,2),(2,1)],3)
 => [4]
 => []
 => ? = 4
([(0,2),(1,2)],3)
 => [3,2]
 => [2]
 => 2
([(2,3)],4)
 => [6,6]
 => [6]
 => 6
([(1,2),(1,3)],4)
 => [6,2,2]
 => [2,2]
 => 2
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [2,2,2]
 => 2
([(0,2),(0,3),(3,1)],4)
 => [7]
 => []
 => ? = 7
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [2]
 => 2
([(1,2),(2,3)],4)
 => [4,4]
 => [4]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [2]
 => 2
([(1,3),(2,3)],4)
 => [6,2,2]
 => [2,2]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [2]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [2,2,2]
 => 2
([(0,3),(1,2)],4)
 => [3,3,3]
 => [3,3]
 => 3
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [3]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [2,2]
 => 2
([(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]
 => [2,2,2]
 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [2]
 => 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [3,3]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [4]
 => 4
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [2,2]
 => 2
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [2,2,2]
 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [2,2]
 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [2,2,2]
 => 2
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [2]
 => 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [4]
 => 4
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => [2]
 => 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [2,2]
 => 2
([(0,4),(1,2),(1,4),(4,3)],5)
 => [10]
 => []
 => ? = 10
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [4]
 => 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => []
 => ? = 8
([(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]
 => [2]
 => 2
([(1,4),(3,2),(4,3)],5)
 => [10]
 => []
 => ? = 10
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [2]
 => 2
([(0,4),(1,2),(2,4),(4,3)],5)
 => [8]
 => []
 => ? = 8
([(0,4),(3,2),(4,1),(4,3)],5)
 => [8]
 => []
 => ? = 8
([(0,4),(1,2),(2,3),(2,4)],5)
 => [10]
 => []
 => ? = 10
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => []
 => ? = 6
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,3,3]
 => [3,3]
 => 3
([(0,4),(1,2),(2,3),(3,4)],5)
 => [5,4]
 => [4]
 => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [2]
 => 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [5,2,2]
 => [2,2]
 => 2
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => [8,2]
 => [2]
 => 2
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [5,2,2]
 => [2,2]
 => 2
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
 => [8,2]
 => [2]
 => 2
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [5,2,2]
 => [2,2]
 => 2
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [6,2]
 => [2]
 => 2
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => [5,2,2]
 => [2,2]
 => 2
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => [5,5]
 => [5]
 => 5
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => [4,2,2,2]
 => [2,2,2]
 => 2
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => [5,2,2]
 => [2,2]
 => 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => [8,2]
 => [2]
 => 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [5,5]
 => [5]
 => 5
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => [9]
 => []
 => ? = 9
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => [6,4]
 => [4]
 => 4
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [6,2]
 => [2]
 => 2
([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
 => [8,2]
 => [2]
 => 2
([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [8,2]
 => [2]
 => 2
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
 => [9]
 => []
 => ? = 9
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => []
 => ? = 7
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [9]
 => []
 => ? = 9
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => [9]
 => []
 => ? = 9
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => [10]
 => []
 => ? = 10
([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
 => [10]
 => []
 => ? = 10
([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
 => [10]
 => []
 => ? = 10
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [8]
 => []
 => ? = 8
([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
 => [10]
 => []
 => ? = 10
([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
 => [10]
 => []
 => ? = 10

Description

The greatest common divisor of the parts of the partition.

The following 20 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000668The least common multiple of the parts of the partition. St000770The major index of an integer partition when read from bottom to top. St001571The Cartan determinant of the integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001933The largest multiplicity of a part in an integer partition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000455The second largest eigenvalue of a graph if it is integral.