Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001785
Mapping
St001785: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1
[2]
 => 0
[1,1]
 => 2
[3]
 => 0
[2,1]
 => 1
[1,1,1]
 => 2
[4]
 => 0
[3,1]
 => 0
[2,2]
 => 0
[2,1,1]
 => 3
[1,1,1,1]
 => 2
[5]
 => 0
[4,1]
 => 0
[3,2]
 => 0
[3,1,1]
 => 0
[2,2,1]
 => 3
[2,1,1,1]
 => 2
[1,1,1,1,1]
 => 2
[6]
 => 0
[5,1]
 => 0
[4,2]
 => 0
[4,1,1]
 => 0
[3,3]
 => 0
[3,2,1]
 => 1
[3,1,1,1]
 => 0
[2,2,2]
 => 0
[2,2,1,1]
 => 6
[2,1,1,1,1]
 => 2
[1,1,1,1,1,1]
 => 2
[7]
 => 0
[6,1]
 => 0
[5,2]
 => 0
[5,1,1]
 => 0
[4,3]
 => 0
[4,2,1]
 => 0
[4,1,1,1]
 => 0
[3,3,1]
 => 0
[3,2,2]
 => 0
[3,2,1,1]
 => 4
[3,1,1,1,1]
 => 0
[2,2,2,1]
 => 3
[2,2,1,1,1]
 => 4
[2,1,1,1,1,1]
 => 2
[1,1,1,1,1,1,1]
 => 2
[8]
 => 0
[7,1]
 => 0
[6,2]
 => 0
[6,1,1]
 => 0
[5,3]
 => 0
[5,2,1]
 => 0
Description
The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. Given a partition $\lambda\vdash n$, let $\alpha(\lambda)$ be the partition given by the lengths of the antidiagonals of the Ferrers diagram of $\lambda$. Then, the value of the statistic on $\mu$ is the number of times $\mu$ appears in the multiset $\{\{\alpha(\lambda)\mid \lambda\vdash n\}\}$.
Matching statistic: St000264
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 6%
Values
[1]
 => [1,0]
 => [1] => ([],1)
 => ? = 1 + 3
[2]
 => [1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => ? = 0 + 3
[1,1]
 => [1,1,0,0]
 => [2] => ([],2)
 => ? = 2 + 3
[3]
 => [1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
[2,1]
 => [1,0,1,1,0,0]
 => [1,2] => ([(1,2)],3)
 => ? = 1 + 3
[1,1,1]
 => [1,1,0,1,0,0]
 => [3] => ([],3)
 => ? = 2 + 3
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[2,2]
 => [1,1,1,0,0,0]
 => [3] => ([],3)
 => ? = 0 + 3
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3] => ([(2,3)],4)
 => ? = 3 + 3
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [4] => ([],4)
 => ? = 2 + 3
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 0 + 3
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 0 + 3
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => ([(2,3)],4)
 => ? = 0 + 3
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 0 + 3
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [4] => ([],4)
 => ? = 3 + 3
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => ? = 2 + 3
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5] => ([],5)
 => ? = 2 + 3
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 0 + 3
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [4] => ([],4)
 => ? = 0 + 3
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => ? = 1 + 3
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [4] => ([],4)
 => ? = 0 + 3
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [5] => ([],5)
 => ? = 6 + 3
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,5] => ([(4,5)],6)
 => ? = 2 + 3
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [6] => ([],6)
 => ? = 2 + 3
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 0 + 3
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 0 + 3
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 0 + 3
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ? = 0 + 3
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 0 + 3
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [5] => ([],5)
 => ? = 0 + 3
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ? = 0 + 3
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,5] => ([(4,5)],6)
 => ? = 4 + 3
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 3 = 0 + 3
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [5] => ([],5)
 => ? = 3 + 3
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [6] => ([],6)
 => ? = 4 + 3
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,6] => ([(5,6)],7)
 => ? = 2 + 3
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [7] => ([],7)
 => ? = 2 + 3
[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] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 3
[7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 3
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 0 + 3
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 3
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 0 + 3
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 3
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [5] => ([],5)
 => ? = 0 + 3
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,5] => ([(4,5)],6)
 => ? = 0 + 3
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 3 = 0 + 3
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 3
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5] => ([],5)
 => ? = 0 + 3
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [6] => ([],6)
 => ? = 0 + 3
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,5] => ([(4,5)],6)
 => ? = 6 + 3
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,6] => ([(5,6)],7)
 => ? = 2 + 3
[3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,6] => ([(5,6),(5,7),(6,7)],8)
 => ? = 0 + 3
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5] => ([],5)
 => ? = 0 + 3
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [6] => ([],6)
 => ? = 6 + 3
[2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [7] => ([],7)
 => ? = 4 + 3
[2,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,7] => ([(6,7)],8)
 => ? = 2 + 3
[1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [8] => ([],8)
 => ? = 2 + 3
[9]
 => [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] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0 + 3
[8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0 + 3
[7,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 3
[7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0 + 3
[6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 0 + 3
[6,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 3
[6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0 + 3
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,5] => ([(4,5)],6)
 => ? = 0 + 3
[5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 3 = 0 + 3
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 0 + 3
[4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 3 = 0 + 3
[6,4]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 3 = 0 + 3
[5,3,2]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 3 = 0 + 3
[4,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 3 = 0 + 3
[5,3,3]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 3 = 0 + 3
Description
The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.