Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000378
Mapping
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
 => [1]
 => []
 => 0
[1,2,3] => [1,1,1]
 => [1,1]
 => [1]
 => 1
[1,3,2] => [2,1]
 => [1]
 => []
 => 0
[2,1,3] => [2,1]
 => [1]
 => []
 => 0
[3,2,1] => [2,1]
 => [1]
 => []
 => 0
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,4,3] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[1,3,2,4] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[1,3,4,2] => [3,1]
 => [1]
 => []
 => 0
[1,4,2,3] => [3,1]
 => [1]
 => []
 => 0
[1,4,3,2] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[2,1,3,4] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[2,1,4,3] => [2,2]
 => [2]
 => []
 => 0
[2,3,1,4] => [3,1]
 => [1]
 => []
 => 0
[2,4,3,1] => [3,1]
 => [1]
 => []
 => 0
[3,1,2,4] => [3,1]
 => [1]
 => []
 => 0
[3,2,1,4] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[3,2,4,1] => [3,1]
 => [1]
 => []
 => 0
[3,4,1,2] => [2,2]
 => [2]
 => []
 => 0
[4,1,3,2] => [3,1]
 => [1]
 => []
 => 0
[4,2,1,3] => [3,1]
 => [1]
 => []
 => 0
[4,2,3,1] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[4,3,2,1] => [2,2]
 => [2]
 => []
 => 0
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,4,5,3] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,2,5,3,4] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,2,5,4,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,3,2,5,4] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[1,3,4,2,5] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,3,4,5,2] => [4,1]
 => [1]
 => []
 => 0
[1,3,5,2,4] => [4,1]
 => [1]
 => []
 => 0
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,4,2,3,5] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,4,2,5,3] => [4,1]
 => [1]
 => []
 => 0
[1,4,3,2,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,4,5,2,3] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[1,4,5,3,2] => [4,1]
 => [1]
 => []
 => 0
[1,5,2,3,4] => [4,1]
 => [1]
 => []
 => 0
[1,5,2,4,3] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,5,3,2,4] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,5,3,4,2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,5,4,2,3] => [4,1]
 => [1]
 => []
 => 0
[1,5,4,3,2] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[2,1,3,5,4] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[2,1,4,3,5] => [2,2,1]
 => [2,1]
 => [1]
 => 1
Description
The diagonal inversion number of an integer partition. The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$. See also exercise 3.19 of [2]. This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St001200
(load all 2 compositions to match this statistic)
Mapping
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 29%
Values
[1,2] => [1,2] => [1,2] => [1,0,1,0]
 => 2 = 0 + 2
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 3 = 1 + 2
[1,3,2] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2 = 0 + 2
[2,1,3] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 0 + 2
[3,2,1] => [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
 => 2 = 0 + 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 3 = 1 + 2
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 3 = 1 + 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 3 = 1 + 2
[1,3,4,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[1,4,3,2] => [1,3,4,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 3 = 1 + 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 3 = 1 + 2
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2 = 0 + 2
[2,3,1,4] => [3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
[2,4,3,1] => [3,4,1,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2 = 0 + 2
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
[3,2,1,4] => [2,3,1,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 3 = 1 + 2
[3,2,4,1] => [2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2 = 0 + 2
[3,4,1,2] => [3,1,4,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2 = 0 + 2
[4,1,3,2] => [3,4,2,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[4,2,1,3] => [2,4,3,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[4,2,3,1] => [2,3,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 3 = 1 + 2
[4,3,2,1] => [3,2,4,1] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2 = 0 + 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 3 = 1 + 2
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 3 = 1 + 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => 3 = 1 + 2
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 1 + 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 1 + 2
[1,2,5,4,3] => [1,2,4,5,3] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 3 = 1 + 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => 3 = 1 + 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 1 + 2
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => 3 = 1 + 2
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,3,5,2,4] => [1,5,4,2,3] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,3,5,4,2] => [1,4,5,2,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => 3 = 1 + 2
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 3 = 1 + 2
[1,4,2,5,3] => [1,5,3,2,4] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,4,3,2,5] => [1,3,4,2,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => 3 = 1 + 2
[1,4,3,5,2] => [1,3,5,2,4] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => 3 = 1 + 2
[1,4,5,2,3] => [1,4,2,5,3] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 1 + 2
[1,4,5,3,2] => [1,5,2,4,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,5,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,5,2,4,3] => [1,4,5,3,2] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 1 + 2
[1,5,3,2,4] => [1,3,5,4,2] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 1 + 2
[1,5,3,4,2] => [1,3,4,5,2] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 3 = 1 + 2
[1,5,4,2,3] => [1,5,3,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,5,4,3,2] => [1,4,3,5,2] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 1 + 2
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 1 + 2
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 1 + 2
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 1 + 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 + 2
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 2
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 2
[1,2,3,5,6,4] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 2
[1,2,3,6,4,5] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 2
[1,2,3,6,5,4] => [1,2,3,5,6,4] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 2
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1 + 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 2
[1,2,4,5,3,6] => [1,2,5,3,4,6] => [1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1 + 2
[1,2,4,5,6,3] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 2
[1,2,4,6,3,5] => [1,2,6,5,3,4] => [1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 2
[1,2,4,6,5,3] => [1,2,5,6,3,4] => [1,2,4,6,3,5] => [1,0,1,0,1,1,0,1,1,0,0,0]
 => ? = 1 + 2
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1 + 2
[1,2,5,3,6,4] => [1,2,6,4,3,5] => [1,2,6,4,3,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 2
[1,2,5,4,3,6] => [1,2,4,5,3,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 2
[1,2,5,4,6,3] => [1,2,4,6,3,5] => [1,2,4,6,3,5] => [1,0,1,0,1,1,0,1,1,0,0,0]
 => ? = 1 + 2
[1,2,5,6,3,4] => [1,2,5,3,6,4] => [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 2
[1,2,5,6,4,3] => [1,2,6,3,5,4] => [1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 2
[1,2,6,3,4,5] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 2
[1,2,6,3,5,4] => [1,2,5,6,4,3] => [1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 2
[1,2,6,4,3,5] => [1,2,4,6,5,3] => [1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 2
[1,2,6,4,5,3] => [1,2,4,5,6,3] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 2
[1,2,6,5,3,4] => [1,2,6,4,5,3] => [1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 2
[1,2,6,5,4,3] => [1,2,5,4,6,3] => [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 2
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 1 + 2
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1 + 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1 + 2
[1,3,2,5,6,4] => [1,3,2,6,4,5] => [1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 1 + 2
[1,3,2,6,4,5] => [1,3,2,6,5,4] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 1 + 2
[1,3,2,6,5,4] => [1,3,2,5,6,4] => [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1 + 2
[1,3,4,2,5,6] => [1,4,2,3,5,6] => [1,4,2,3,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 1 + 2
[1,3,4,2,6,5] => [1,4,2,3,6,5] => [1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1 + 2
[1,3,4,5,2,6] => [1,5,2,3,4,6] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 1 + 2
[1,3,4,5,6,2] => [1,6,2,3,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 2
[1,3,4,6,2,5] => [1,6,5,2,3,4] => [1,6,5,2,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 2
[1,3,4,6,5,2] => [1,5,6,2,3,4] => [1,4,6,2,3,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 1 + 2
[1,3,5,2,4,6] => [1,5,4,2,3,6] => [1,5,4,2,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 1 + 2
[1,3,5,2,6,4] => [1,6,4,2,3,5] => [1,6,4,2,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 2
[1,3,5,4,2,6] => [1,4,5,2,3,6] => [1,3,5,2,4,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
 => ? = 1 + 2
[1,3,5,4,6,2] => [1,4,6,2,3,5] => [1,4,6,2,3,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 1 + 2
[1,3,5,6,2,4] => [1,5,2,3,6,4] => [1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1 + 2
[1,3,5,6,4,2] => [1,6,2,3,5,4] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 2
[1,3,6,2,4,5] => [1,6,5,4,2,3] => [1,6,5,4,2,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 2
[1,3,6,2,5,4] => [1,5,6,4,2,3] => [1,3,6,5,2,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
 => ? = 1 + 2
[1,3,6,4,2,5] => [1,4,6,5,2,3] => [1,3,6,5,2,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
 => ? = 1 + 2
[1,3,6,4,5,2] => [1,4,5,6,2,3] => [1,2,4,6,3,5] => [1,0,1,0,1,1,0,1,1,0,0,0]
 => ? = 1 + 2
[1,3,6,5,2,4] => [1,6,4,5,2,3] => [1,6,3,5,2,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 2
[1,3,6,5,4,2] => [1,5,4,6,2,3] => [1,4,3,6,2,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => ? = 1 + 2
[1,4,2,3,5,6] => [1,4,3,2,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 1 + 2
[1,4,2,3,6,5] => [1,4,3,2,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1 + 2
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.