Your data matches 43 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 = 1 + 1
([],2)
 => [2,2]
 => [2,2]
 => 2 = 1 + 1
([(0,1)],2)
 => [3]
 => [1,1,1]
 => 3 = 2 + 1
([],3)
 => [2,2,2,2]
 => [4,4]
 => 2 = 1 + 1
([(1,2)],3)
 => [6]
 => [1,1,1,1,1,1]
 => 6 = 5 + 1
([(0,1),(0,2)],3)
 => [3,2]
 => [2,2,1]
 => 2 = 1 + 1
([(0,2),(2,1)],3)
 => [4]
 => [1,1,1,1]
 => 4 = 3 + 1
([(0,2),(1,2)],3)
 => [3,2]
 => [2,2,1]
 => 2 = 1 + 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,1,1,1,1,1,1]
 => 7 = 6 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [2,2,1,1]
 => 2 = 1 + 1
([(1,2),(2,3)],4)
 => [4,4]
 => [2,2,2,2]
 => 4 = 3 + 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => 2 = 1 + 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => 2 = 1 + 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [2,2,2,1,1]
 => 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [3,3,1]
 => 2 = 1 + 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,1,1,1,1]
 => 5 = 4 + 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,1,1,1,1,1,1]
 => 7 = 6 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [2,2,1,1,1]
 => 2 = 1 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [3,3,1,1]
 => 2 = 1 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [3,3,1,1]
 => 2 = 1 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [3,3,1,1]
 => 2 = 1 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 8 = 7 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => 2 = 1 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 8 = 7 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 8 = 7 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => [1,1,1,1,1,1]
 => 6 = 5 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => 2 = 1 + 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [6,2]
 => [2,2,1,1,1,1]
 => 2 = 1 + 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [6,2]
 => [2,2,1,1,1,1]
 => 2 = 1 + 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => [6,2]
 => [2,2,1,1,1,1]
 => 2 = 1 + 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [6,2]
 => [2,2,1,1,1,1]
 => 2 = 1 + 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => [1,1,1,1,1,1,1]
 => 7 = 6 + 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [6,2]
 => [2,2,1,1,1,1]
 => 2 = 1 + 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 8 = 7 + 1
Description
The multiplicity of the largest part of an integer partition.
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => [1,0,1,0]
 => 2 = 1 + 1
([],2)
 => [2,2]
 => [1,1,1,0,0,0]
 => 2 = 1 + 1
([(0,1)],2)
 => [3]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
([],3)
 => [2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
([(1,2)],3)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
([(0,1),(0,2)],3)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
([(0,2),(2,1)],3)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
([(0,2),(1,2)],3)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
([(1,2),(2,3)],4)
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4 = 3 + 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
([(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 = 1 + 1
([(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 = 1 + 1
([(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 = 1 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
([(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 = 1 + 1
([(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 = 7 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
([(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 = 7 + 1
([(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 = 7 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
([(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 = 1 + 1
([(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 = 1 + 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 8 = 7 + 1
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St001803
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => [1,1]
 => [[1],[2]]
 => 1
([],2)
 => [2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 1
([(0,1)],2)
 => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 2
([],3)
 => [2,2,2,2]
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => 1
([(1,2)],3)
 => [6]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 5
([(0,1),(0,2)],3)
 => [3,2]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 1
([(0,2),(2,1)],3)
 => [4]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 3
([(0,2),(1,2)],3)
 => [3,2]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 6
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 1
([(1,2),(2,3)],4)
 => [4,4]
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => 3
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5],[7]]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 4
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 6
([(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]]
 => 1
([(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]]
 => 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => 1
([(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]]
 => 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
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => 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
([(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
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 5
([(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]]
 => 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [6,2]
 => [2,2,1,1,1,1]
 => [[1,6],[2,8],[3],[4],[5],[7]]
 => 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [6,2]
 => [2,2,1,1,1,1]
 => [[1,6],[2,8],[3],[4],[5],[7]]
 => 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => [6,2]
 => [2,2,1,1,1,1]
 => [[1,6],[2,8],[3],[4],[5],[7]]
 => 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [6,2]
 => [2,2,1,1,1,1]
 => [[1,6],[2,8],[3],[4],[5],[7]]
 => 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 6
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [6,2]
 => [2,2,1,1,1,1]
 => [[1,6],[2,8],[3],[4],[5],[7]]
 => 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => 7
Description
The maximal overlap of the cylindrical tableau associated with a tableau. A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle. The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux. In particular, the statistic equals $0$, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
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 = 1 + 1
([],2)
 => [2,2]
 => [2,2]
 => 1100 => 2 = 1 + 1
([(0,1)],2)
 => [3]
 => [1,1,1]
 => 1110 => 3 = 2 + 1
([],3)
 => [2,2,2,2]
 => [4,4]
 => 110000 => 2 = 1 + 1
([(1,2)],3)
 => [6]
 => [1,1,1,1,1,1]
 => 1111110 => 6 = 5 + 1
([(0,1),(0,2)],3)
 => [3,2]
 => [2,2,1]
 => 11010 => 2 = 1 + 1
([(0,2),(2,1)],3)
 => [4]
 => [1,1,1,1]
 => 11110 => 4 = 3 + 1
([(0,2),(1,2)],3)
 => [3,2]
 => [2,2,1]
 => 11010 => 2 = 1 + 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,1,1,1,1,1,1]
 => 11111110 => 7 = 6 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [2,2,1,1]
 => 110110 => 2 = 1 + 1
([(1,2),(2,3)],4)
 => [4,4]
 => [2,2,2,2]
 => 111100 => 4 = 3 + 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => 110110 => 2 = 1 + 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => 110110 => 2 = 1 + 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [2,2,2,1,1]
 => 1110110 => 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [3,3,1]
 => 110010 => 2 = 1 + 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,1,1,1,1]
 => 111110 => 5 = 4 + 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,1,1,1,1,1,1]
 => 11111110 => 7 = 6 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [2,2,1,1,1]
 => 1101110 => 2 = 1 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [3,3,1,1]
 => 1100110 => 2 = 1 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [3,3,1,1]
 => 1100110 => 2 = 1 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => 1101110 => 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [3,3,1,1]
 => 1100110 => 2 = 1 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 111111110 => 8 = 7 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => 1101110 => 2 = 1 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 111111110 => 8 = 7 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 111111110 => 8 = 7 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => [1,1,1,1,1,1]
 => 1111110 => 6 = 5 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => 1101110 => 2 = 1 + 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [6,2]
 => [2,2,1,1,1,1]
 => 11011110 => 2 = 1 + 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [6,2]
 => [2,2,1,1,1,1]
 => 11011110 => 2 = 1 + 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => [6,2]
 => [2,2,1,1,1,1]
 => 11011110 => 2 = 1 + 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [6,2]
 => [2,2,1,1,1,1]
 => 11011110 => 2 = 1 + 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => [1,1,1,1,1,1,1]
 => 11111110 => 7 = 6 + 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [6,2]
 => [2,2,1,1,1,1]
 => 11011110 => 2 = 1 + 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 111111110 => 8 = 7 + 1
Description
The number of leading ones in a binary word.
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
([],2)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
([(0,1)],2)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
([],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 + 1
([(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 = 5 + 1
([(0,1),(0,2)],3)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
([(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 = 3 + 1
([(0,2),(1,2)],3)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
([(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 = 6 + 1
([(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 + 1
([(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 = 3 + 1
([(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 + 1
([(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 = 1 + 1
([(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 = 2 + 1
([(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 = 1 + 1
([(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 = 4 + 1
([(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 = 6 + 1
([(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 = 1 + 1
([(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 = 1 + 1
([(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 = 1 + 1
([(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 = 1 + 1
([(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 = 1 + 1
([(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 = 7 + 1
([(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 = 1 + 1
([(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 = 7 + 1
([(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 = 7 + 1
([(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 = 5 + 1
([(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 = 1 + 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => 2 = 1 + 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => 2 = 1 + 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => 2 = 1 + 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => 2 = 1 + 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [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 = 6 + 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => 2 = 1 + 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [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 = 7 + 1
Description
The number of global maxima of a Dyck path.
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 = 1 + 1
([],2)
 => [2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 2 = 1 + 1
([(0,1)],2)
 => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 3 = 2 + 1
([],3)
 => [2,2,2,2]
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => 2 = 1 + 1
([(1,2)],3)
 => [6]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 6 = 5 + 1
([(0,1),(0,2)],3)
 => [3,2]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2 = 1 + 1
([(0,2),(2,1)],3)
 => [4]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 4 = 3 + 1
([(0,2),(1,2)],3)
 => [3,2]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2 = 1 + 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 7 = 6 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 2 = 1 + 1
([(1,2),(2,3)],4)
 => [4,4]
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => 4 = 3 + 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 2 = 1 + 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 2 = 1 + 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5],[7]]
 => 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => 2 = 1 + 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5 = 4 + 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 7 = 6 + 1
([(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 = 1 + 1
([(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 = 1 + 1
([(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 = 1 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => 2 = 1 + 1
([(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 = 1 + 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]]
 => 8 = 7 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => 2 = 1 + 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]]
 => 8 = 7 + 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]]
 => 8 = 7 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 6 = 5 + 1
([(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 = 1 + 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [6,2]
 => [2,2,1,1,1,1]
 => [[1,6],[2,8],[3],[4],[5],[7]]
 => 2 = 1 + 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [6,2]
 => [2,2,1,1,1,1]
 => [[1,6],[2,8],[3],[4],[5],[7]]
 => 2 = 1 + 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => [6,2]
 => [2,2,1,1,1,1]
 => [[1,6],[2,8],[3],[4],[5],[7]]
 => 2 = 1 + 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [6,2]
 => [2,2,1,1,1,1]
 => [[1,6],[2,8],[3],[4],[5],[7]]
 => 2 = 1 + 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 7 = 6 + 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [6,2]
 => [2,2,1,1,1,1]
 => [[1,6],[2,8],[3],[4],[5],[7]]
 => 2 = 1 + 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => 8 = 7 + 1
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 = 1 + 1
([],2)
 => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2 = 1 + 1
([(0,1)],2)
 => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 3 = 2 + 1
([],3)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [[1,3,5,7],[2,4,6,8]]
 => 2 = 1 + 1
([(1,2)],3)
 => [6]
 => [[1,2,3,4,5,6]]
 => [[1],[2],[3],[4],[5],[6]]
 => 6 = 5 + 1
([(0,1),(0,2)],3)
 => [3,2]
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 2 = 1 + 1
([(0,2),(2,1)],3)
 => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4 = 3 + 1
([(0,2),(1,2)],3)
 => [3,2]
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 2 = 1 + 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [[1,2,3,4,5,6,7]]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 7 = 6 + 1
([(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 + 1
([(1,2),(2,3)],4)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [[1,5],[2,6],[3,7],[4,8]]
 => 4 = 3 + 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => [[1,3],[2,4],[5],[6]]
 => 2 = 1 + 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => [[1,3],[2,4],[5],[6]]
 => 2 = 1 + 1
([(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 = 2 + 1
([(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 = 1 + 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 5 = 4 + 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [[1,2,3,4,5,6,7]]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 7 = 6 + 1
([(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 = 1 + 1
([(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 = 1 + 1
([(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 = 1 + 1
([(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 = 1 + 1
([(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 = 1 + 1
([(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 = 7 + 1
([(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 = 1 + 1
([(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 = 7 + 1
([(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 = 7 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => [[1,2,3,4,5,6]]
 => [[1],[2],[3],[4],[5],[6]]
 => 6 = 5 + 1
([(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 = 1 + 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [[1,3],[2,4],[5],[6],[7],[8]]
 => 2 = 1 + 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [[1,3],[2,4],[5],[6],[7],[8]]
 => 2 = 1 + 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [[1,3],[2,4],[5],[6],[7],[8]]
 => 2 = 1 + 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [[1,3],[2,4],[5],[6],[7],[8]]
 => 2 = 1 + 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => [[1,2,3,4,5,6,7]]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 7 = 6 + 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [[1,3],[2,4],[5],[6],[7],[8]]
 => 2 = 1 + 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [8]
 => [[1,2,3,4,5,6,7,8]]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => 8 = 7 + 1
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 = 1 + 2
([],2)
 => [2,2]
 => 1100 => 0011 => 3 = 1 + 2
([(0,1)],2)
 => [3]
 => 1000 => 0001 => 4 = 2 + 2
([],3)
 => [2,2,2,2]
 => 111100 => 001111 => 3 = 1 + 2
([(1,2)],3)
 => [6]
 => 1000000 => 0000001 => 7 = 5 + 2
([(0,1),(0,2)],3)
 => [3,2]
 => 10100 => 00101 => 3 = 1 + 2
([(0,2),(2,1)],3)
 => [4]
 => 10000 => 00001 => 5 = 3 + 2
([(0,2),(1,2)],3)
 => [3,2]
 => 10100 => 00101 => 3 = 1 + 2
([(0,2),(0,3),(3,1)],4)
 => [7]
 => 10000000 => 00000001 => 8 = 6 + 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => 100100 => 001001 => 3 = 1 + 2
([(1,2),(2,3)],4)
 => [4,4]
 => 110000 => 000011 => 5 = 3 + 2
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => 100100 => 001001 => 3 = 1 + 2
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => 100100 => 001001 => 3 = 1 + 2
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => 1001000 => 0001001 => 4 = 2 + 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => 101100 => 001101 => 3 = 1 + 2
([(0,3),(2,1),(3,2)],4)
 => [5]
 => 100000 => 000001 => 6 = 4 + 2
([(0,3),(1,2),(2,3)],4)
 => [7]
 => 10000000 => 00000001 => 8 = 6 + 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => 1000100 => 0010001 => 3 = 1 + 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => 1001100 => 0011001 => 3 = 1 + 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => 1001100 => 0011001 => 3 = 1 + 2
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => 1000100 => 0010001 => 3 = 1 + 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => 1001100 => 0011001 => 3 = 1 + 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => 100000000 => 000000001 => 9 = 7 + 2
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => 1000100 => 0010001 => 3 = 1 + 2
([(0,4),(1,2),(2,4),(4,3)],5)
 => [8]
 => 100000000 => 000000001 => 9 = 7 + 2
([(0,4),(3,2),(4,1),(4,3)],5)
 => [8]
 => 100000000 => 000000001 => 9 = 7 + 2
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 1000000 => 0000001 => 7 = 5 + 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => 1000100 => 0010001 => 3 = 1 + 2
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [6,2]
 => 10000100 => 00100001 => 3 = 1 + 2
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [6,2]
 => 10000100 => 00100001 => 3 = 1 + 2
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => [6,2]
 => 10000100 => 00100001 => 3 = 1 + 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [6,2]
 => 10000100 => 00100001 => 3 = 1 + 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 10000000 => 00000001 => 8 = 6 + 2
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [6,2]
 => 10000100 => 00100001 => 3 = 1 + 2
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [8]
 => 100000000 => 000000001 => 9 = 7 + 2
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: St000642
(load all 2 compositions to match this statistic)
Mapping
St000642: Posets ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ? = 1 + 1
([],2)
 => 2 = 1 + 1
([(0,1)],2)
 => 3 = 2 + 1
([],3)
 => 2 = 1 + 1
([(1,2)],3)
 => 6 = 5 + 1
([(0,1),(0,2)],3)
 => 2 = 1 + 1
([(0,2),(2,1)],3)
 => 4 = 3 + 1
([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,2),(0,3),(3,1)],4)
 => 7 = 6 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(1,2),(2,3)],4)
 => 4 = 3 + 1
([(0,3),(3,1),(3,2)],4)
 => 2 = 1 + 1
([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
([(0,3),(1,2),(1,3)],4)
 => 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
([(0,3),(2,1),(3,2)],4)
 => 5 = 4 + 1
([(0,3),(1,2),(2,3)],4)
 => 7 = 6 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2 = 1 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => 2 = 1 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => 2 = 1 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 8 = 7 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => 2 = 1 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => 8 = 7 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => 8 = 7 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => 6 = 5 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2 = 1 + 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => 2 = 1 + 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2 = 1 + 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => 2 = 1 + 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2 = 1 + 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 7 = 6 + 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2 = 1 + 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ? = 7 + 1
Description
The size of the smallest orbit of antichains under Panyushev complementation.
Matching statistic: St001184
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 71%
Values
([],1)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
([],2)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
([(0,1)],2)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
([],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 + 1
([(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 = 5 + 1
([(0,1),(0,2)],3)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
([(0,2),(2,1)],3)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
([(0,2),(1,2)],3)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
([(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]
 => ? = 6 + 1
([(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]
 => 2 = 1 + 1
([(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 = 3 + 1
([(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]
 => 2 = 1 + 1
([(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]
 => 2 = 1 + 1
([(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]
 => 3 = 2 + 1
([(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]
 => 2 = 1 + 1
([(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 = 4 + 1
([(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]
 => ? = 6 + 1
([(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]
 => 2 = 1 + 1
([(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]
 => 2 = 1 + 1
([(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]
 => 2 = 1 + 1
([(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]
 => 2 = 1 + 1
([(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]
 => 2 = 1 + 1
([(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]
 => ? = 7 + 1
([(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]
 => 2 = 1 + 1
([(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]
 => ? = 7 + 1
([(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]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => [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 = 5 + 1
([(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]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2 = 1 + 1
([(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]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 1 + 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 1 + 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 1 + 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 1 + 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [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]
 => ? = 6 + 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 1 + 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [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]
 => ? = 7 + 1
Description
Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.
The following 33 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001481The minimal height of a peak of a Dyck path. St000319The spin of an integer partition. St000320The dinv adjustment of an 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. St000474Dyson's crank of a partition. St000667The greatest common divisor of the parts of the partition. 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. 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. 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. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000699The toughness times the least common multiple of 1,. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000456The monochromatic index of a connected graph. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.