Your data matches 8 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000120
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000120: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1] => [[1,1],[]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[2] => [[2],[]]
 => [2]
 => [1,1,0,0,1,0]
 => 2
[1,1,1] => [[1,1,1],[]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,2] => [[2,1],[]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 1
[2,1] => [[2,2],[1]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[3] => [[3],[]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3
[1,1,1,1] => [[1,1,1,1],[]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,2] => [[2,1,1],[]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,2,1] => [[2,2,1],[1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[1,3] => [[3,1],[]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[2,1,1] => [[2,2,2],[1,1]]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[2,2] => [[3,2],[1]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[3,1] => [[3,3],[2]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[4] => [[4],[]]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,1,2] => [[2,1,1,1],[]]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[1,1,2,1] => [[2,2,1,1],[1]]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[1,1,3] => [[3,1,1],[]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,2,2] => [[3,2,1],[1]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 2
[1,3,1] => [[3,3,1],[2]]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
[1,4] => [[4,1],[]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2
[2,1,2] => [[3,2,2],[1,1]]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[2,2,1] => [[3,3,2],[2,1]]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[2,3] => [[4,2],[1]]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3
[3,1,1] => [[3,3,3],[2,2]]
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 3
[3,2] => [[4,3],[2]]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[4,1] => [[4,4],[3]]
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 4
[5] => [[5],[]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 5
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 2
[1,1,1,3] => [[3,1,1,1],[]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 2
[1,1,2,2] => [[3,2,1,1],[1]]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,1,3,1] => [[3,3,1,1],[2]]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3
[1,1,4] => [[4,1,1],[]]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2
[1,2,3] => [[4,2,1],[1]]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 3
[1,3,2] => [[4,3,1],[2]]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3
[1,4,1] => [[4,4,1],[3]]
 => [4,4,1]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => 4
[1,5] => [[5,1],[]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 4
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => 2
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 2
[2,1,3] => [[4,2,2],[1,1]]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [3,3,3,2]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 3
Description
The number of left tunnels of a Dyck path. A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b
Matching statistic: St000668
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 40%
Values
[1] => [[1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1] => [[1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[2] => [[2],[]]
 => []
 => ?
 => ? = 2 - 1
[1,1,1] => [[1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,2] => [[2,1],[]]
 => []
 => ?
 => ? = 1 - 1
[2,1] => [[2,2],[1]]
 => [1]
 => []
 => ? = 2 - 1
[3] => [[3],[]]
 => []
 => ?
 => ? = 3 - 1
[1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1,2] => [[2,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => []
 => ? = 2 - 1
[1,3] => [[3,1],[]]
 => []
 => ?
 => ? = 2 - 1
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 1
[2,2] => [[3,2],[1]]
 => [1]
 => []
 => ? = 2 - 1
[3,1] => [[3,3],[2]]
 => [2]
 => []
 => ? = 3 - 1
[4] => [[4],[]]
 => []
 => ?
 => ? = 4 - 1
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 1
[1,1,3] => [[3,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 1
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => []
 => ? = 2 - 1
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => []
 => ? = 3 - 1
[1,4] => [[4,1],[]]
 => []
 => ?
 => ? = 3 - 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 1
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => [1]
 => ? = 2 - 1
[2,3] => [[4,2],[1]]
 => [1]
 => []
 => ? = 3 - 1
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[3,2] => [[4,3],[2]]
 => [2]
 => []
 => ? = 3 - 1
[4,1] => [[4,4],[3]]
 => [3]
 => []
 => ? = 4 - 1
[5] => [[5],[]]
 => []
 => ?
 => ? = 5 - 1
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 1
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 1
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 1
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => []
 => ? = 3 - 1
[1,1,4] => [[4,1,1],[]]
 => []
 => ?
 => ? = 2 - 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1]
 => ? = 2 - 1
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => []
 => ? = 3 - 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => []
 => ? = 3 - 1
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => []
 => ? = 4 - 1
[1,5] => [[5,1],[]]
 => []
 => ?
 => ? = 4 - 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 2 = 3 - 1
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => [1]
 => ? = 2 - 1
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => [1]
 => ? = 3 - 1
[2,4] => [[5,2],[1]]
 => [1]
 => []
 => ? = 4 - 1
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => [2]
 => 2 = 3 - 1
[3,3] => [[5,3],[2]]
 => [2]
 => []
 => ? = 4 - 1
[4,2] => [[5,4],[3]]
 => [3]
 => []
 => ? = 4 - 1
[1,1,1,1,3] => [[3,1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1 = 2 - 1
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 2 = 3 - 1
[1,3,1,2] => [[4,3,3,1],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[1,3,2,1] => [[4,4,3,1],[3,2]]
 => [3,2]
 => [2]
 => 2 = 3 - 1
[2,1,1,3] => [[4,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
 => [3,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 2 = 3 - 1
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
 => [3,2,1]
 => [2,1]
 => 2 = 3 - 1
[3,1,3] => [[5,3,3],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[3,2,2] => [[5,4,3],[3,2]]
 => [3,2]
 => [2]
 => 2 = 3 - 1
[1,1,3,1,2] => [[4,3,3,1,1],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[1,1,3,2,1] => [[4,4,3,1,1],[3,2]]
 => [3,2]
 => [2]
 => 2 = 3 - 1
[1,2,1,2,2] => [[4,3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1 = 2 - 1
[1,2,1,3,1] => [[4,4,2,2,1],[3,1,1]]
 => [3,1,1]
 => [1,1]
 => 1 = 2 - 1
[1,2,2,1,2] => [[4,3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 2 = 3 - 1
[1,2,2,2,1] => [[4,4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [2,1]
 => 2 = 3 - 1
[1,3,1,3] => [[5,3,3,1],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[1,3,2,2] => [[5,4,3,1],[3,2]]
 => [3,2]
 => [2]
 => 2 = 3 - 1
[2,1,2,3] => [[5,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,1,3,2] => [[5,4,2,2],[3,1,1]]
 => [3,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,2,1,3] => [[5,3,3,2],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 2 = 3 - 1
[2,2,2,2] => [[5,4,3,2],[3,2,1]]
 => [3,2,1]
 => [2,1]
 => 2 = 3 - 1
Description
The least common multiple of the parts of the partition.
Matching statistic: St000707
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 40%
Values
[1] => [[1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1] => [[1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[2] => [[2],[]]
 => []
 => ?
 => ? = 2 - 1
[1,1,1] => [[1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,2] => [[2,1],[]]
 => []
 => ?
 => ? = 1 - 1
[2,1] => [[2,2],[1]]
 => [1]
 => []
 => ? = 2 - 1
[3] => [[3],[]]
 => []
 => ?
 => ? = 3 - 1
[1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1,2] => [[2,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => []
 => ? = 2 - 1
[1,3] => [[3,1],[]]
 => []
 => ?
 => ? = 2 - 1
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 1
[2,2] => [[3,2],[1]]
 => [1]
 => []
 => ? = 2 - 1
[3,1] => [[3,3],[2]]
 => [2]
 => []
 => ? = 3 - 1
[4] => [[4],[]]
 => []
 => ?
 => ? = 4 - 1
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 1
[1,1,3] => [[3,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 1
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => []
 => ? = 2 - 1
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => []
 => ? = 3 - 1
[1,4] => [[4,1],[]]
 => []
 => ?
 => ? = 3 - 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 1
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => [1]
 => ? = 2 - 1
[2,3] => [[4,2],[1]]
 => [1]
 => []
 => ? = 3 - 1
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[3,2] => [[4,3],[2]]
 => [2]
 => []
 => ? = 3 - 1
[4,1] => [[4,4],[3]]
 => [3]
 => []
 => ? = 4 - 1
[5] => [[5],[]]
 => []
 => ?
 => ? = 5 - 1
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 1
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 1
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 1
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => []
 => ? = 3 - 1
[1,1,4] => [[4,1,1],[]]
 => []
 => ?
 => ? = 2 - 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1]
 => ? = 2 - 1
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => []
 => ? = 3 - 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => []
 => ? = 3 - 1
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => []
 => ? = 4 - 1
[1,5] => [[5,1],[]]
 => []
 => ?
 => ? = 4 - 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 2 = 3 - 1
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => [1]
 => ? = 2 - 1
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => [1]
 => ? = 3 - 1
[2,4] => [[5,2],[1]]
 => [1]
 => []
 => ? = 4 - 1
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => [2]
 => 2 = 3 - 1
[3,3] => [[5,3],[2]]
 => [2]
 => []
 => ? = 4 - 1
[4,2] => [[5,4],[3]]
 => [3]
 => []
 => ? = 4 - 1
[1,1,1,1,3] => [[3,1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1 = 2 - 1
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 2 = 3 - 1
[1,3,1,2] => [[4,3,3,1],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[1,3,2,1] => [[4,4,3,1],[3,2]]
 => [3,2]
 => [2]
 => 2 = 3 - 1
[2,1,1,3] => [[4,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
 => [3,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 2 = 3 - 1
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
 => [3,2,1]
 => [2,1]
 => 2 = 3 - 1
[3,1,3] => [[5,3,3],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[3,2,2] => [[5,4,3],[3,2]]
 => [3,2]
 => [2]
 => 2 = 3 - 1
[1,1,3,1,2] => [[4,3,3,1,1],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[1,1,3,2,1] => [[4,4,3,1,1],[3,2]]
 => [3,2]
 => [2]
 => 2 = 3 - 1
[1,2,1,2,2] => [[4,3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1 = 2 - 1
[1,2,1,3,1] => [[4,4,2,2,1],[3,1,1]]
 => [3,1,1]
 => [1,1]
 => 1 = 2 - 1
[1,2,2,1,2] => [[4,3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 2 = 3 - 1
[1,2,2,2,1] => [[4,4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [2,1]
 => 2 = 3 - 1
[1,3,1,3] => [[5,3,3,1],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[1,3,2,2] => [[5,4,3,1],[3,2]]
 => [3,2]
 => [2]
 => 2 = 3 - 1
[2,1,2,3] => [[5,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,1,3,2] => [[5,4,2,2],[3,1,1]]
 => [3,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,2,1,3] => [[5,3,3,2],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 2 = 3 - 1
[2,2,2,2] => [[5,4,3,2],[3,2,1]]
 => [3,2,1]
 => [2,1]
 => 2 = 3 - 1
Description
The product of the factorials of the parts.
Matching statistic: St000708
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 40%
Values
[1] => [[1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1] => [[1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[2] => [[2],[]]
 => []
 => ?
 => ? = 2 - 1
[1,1,1] => [[1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,2] => [[2,1],[]]
 => []
 => ?
 => ? = 1 - 1
[2,1] => [[2,2],[1]]
 => [1]
 => []
 => ? = 2 - 1
[3] => [[3],[]]
 => []
 => ?
 => ? = 3 - 1
[1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1,2] => [[2,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => []
 => ? = 2 - 1
[1,3] => [[3,1],[]]
 => []
 => ?
 => ? = 2 - 1
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 1
[2,2] => [[3,2],[1]]
 => [1]
 => []
 => ? = 2 - 1
[3,1] => [[3,3],[2]]
 => [2]
 => []
 => ? = 3 - 1
[4] => [[4],[]]
 => []
 => ?
 => ? = 4 - 1
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 1
[1,1,3] => [[3,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 1
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => []
 => ? = 2 - 1
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => []
 => ? = 3 - 1
[1,4] => [[4,1],[]]
 => []
 => ?
 => ? = 3 - 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 1
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => [1]
 => ? = 2 - 1
[2,3] => [[4,2],[1]]
 => [1]
 => []
 => ? = 3 - 1
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[3,2] => [[4,3],[2]]
 => [2]
 => []
 => ? = 3 - 1
[4,1] => [[4,4],[3]]
 => [3]
 => []
 => ? = 4 - 1
[5] => [[5],[]]
 => []
 => ?
 => ? = 5 - 1
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 1
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 1
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 1
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => []
 => ? = 3 - 1
[1,1,4] => [[4,1,1],[]]
 => []
 => ?
 => ? = 2 - 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1]
 => ? = 2 - 1
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => []
 => ? = 3 - 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => []
 => ? = 3 - 1
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => []
 => ? = 4 - 1
[1,5] => [[5,1],[]]
 => []
 => ?
 => ? = 4 - 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 2 = 3 - 1
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => [1]
 => ? = 2 - 1
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => [1]
 => ? = 3 - 1
[2,4] => [[5,2],[1]]
 => [1]
 => []
 => ? = 4 - 1
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => [2]
 => 2 = 3 - 1
[3,3] => [[5,3],[2]]
 => [2]
 => []
 => ? = 4 - 1
[4,2] => [[5,4],[3]]
 => [3]
 => []
 => ? = 4 - 1
[1,1,1,1,3] => [[3,1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1 = 2 - 1
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 2 = 3 - 1
[1,3,1,2] => [[4,3,3,1],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[1,3,2,1] => [[4,4,3,1],[3,2]]
 => [3,2]
 => [2]
 => 2 = 3 - 1
[2,1,1,3] => [[4,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
 => [3,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 2 = 3 - 1
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
 => [3,2,1]
 => [2,1]
 => 2 = 3 - 1
[3,1,3] => [[5,3,3],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[3,2,2] => [[5,4,3],[3,2]]
 => [3,2]
 => [2]
 => 2 = 3 - 1
[1,1,3,1,2] => [[4,3,3,1,1],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[1,1,3,2,1] => [[4,4,3,1,1],[3,2]]
 => [3,2]
 => [2]
 => 2 = 3 - 1
[1,2,1,2,2] => [[4,3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1 = 2 - 1
[1,2,1,3,1] => [[4,4,2,2,1],[3,1,1]]
 => [3,1,1]
 => [1,1]
 => 1 = 2 - 1
[1,2,2,1,2] => [[4,3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 2 = 3 - 1
[1,2,2,2,1] => [[4,4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [2,1]
 => 2 = 3 - 1
[1,3,1,3] => [[5,3,3,1],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[1,3,2,2] => [[5,4,3,1],[3,2]]
 => [3,2]
 => [2]
 => 2 = 3 - 1
[2,1,2,3] => [[5,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,1,3,2] => [[5,4,2,2],[3,1,1]]
 => [3,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,2,1,3] => [[5,3,3,2],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 2 = 3 - 1
[2,2,2,2] => [[5,4,3,2],[3,2,1]]
 => [3,2,1]
 => [2,1]
 => 2 = 3 - 1
Description
The product of the parts of an integer partition.
Matching statistic: St000933
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000933: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 40%
Values
[1] => [[1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1] => [[1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[2] => [[2],[]]
 => []
 => ?
 => ? = 2 - 1
[1,1,1] => [[1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,2] => [[2,1],[]]
 => []
 => ?
 => ? = 1 - 1
[2,1] => [[2,2],[1]]
 => [1]
 => []
 => ? = 2 - 1
[3] => [[3],[]]
 => []
 => ?
 => ? = 3 - 1
[1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1,2] => [[2,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => []
 => ? = 2 - 1
[1,3] => [[3,1],[]]
 => []
 => ?
 => ? = 2 - 1
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 1
[2,2] => [[3,2],[1]]
 => [1]
 => []
 => ? = 2 - 1
[3,1] => [[3,3],[2]]
 => [2]
 => []
 => ? = 3 - 1
[4] => [[4],[]]
 => []
 => ?
 => ? = 4 - 1
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 1
[1,1,3] => [[3,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 1
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => []
 => ? = 2 - 1
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => []
 => ? = 3 - 1
[1,4] => [[4,1],[]]
 => []
 => ?
 => ? = 3 - 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 1
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => [1]
 => ? = 2 - 1
[2,3] => [[4,2],[1]]
 => [1]
 => []
 => ? = 3 - 1
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[3,2] => [[4,3],[2]]
 => [2]
 => []
 => ? = 3 - 1
[4,1] => [[4,4],[3]]
 => [3]
 => []
 => ? = 4 - 1
[5] => [[5],[]]
 => []
 => ?
 => ? = 5 - 1
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 1
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 1
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 1
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => []
 => ? = 3 - 1
[1,1,4] => [[4,1,1],[]]
 => []
 => ?
 => ? = 2 - 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1]
 => ? = 2 - 1
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => []
 => ? = 3 - 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => []
 => ? = 3 - 1
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => []
 => ? = 4 - 1
[1,5] => [[5,1],[]]
 => []
 => ?
 => ? = 4 - 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 2 = 3 - 1
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => [1]
 => ? = 2 - 1
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => [1]
 => ? = 3 - 1
[2,4] => [[5,2],[1]]
 => [1]
 => []
 => ? = 4 - 1
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => [2]
 => 2 = 3 - 1
[3,3] => [[5,3],[2]]
 => [2]
 => []
 => ? = 4 - 1
[4,2] => [[5,4],[3]]
 => [3]
 => []
 => ? = 4 - 1
[1,1,1,1,3] => [[3,1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1 = 2 - 1
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 2 = 3 - 1
[1,3,1,2] => [[4,3,3,1],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[1,3,2,1] => [[4,4,3,1],[3,2]]
 => [3,2]
 => [2]
 => 2 = 3 - 1
[2,1,1,3] => [[4,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
 => [3,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 2 = 3 - 1
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
 => [3,2,1]
 => [2,1]
 => 2 = 3 - 1
[3,1,3] => [[5,3,3],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[3,2,2] => [[5,4,3],[3,2]]
 => [3,2]
 => [2]
 => 2 = 3 - 1
[1,1,3,1,2] => [[4,3,3,1,1],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[1,1,3,2,1] => [[4,4,3,1,1],[3,2]]
 => [3,2]
 => [2]
 => 2 = 3 - 1
[1,2,1,2,2] => [[4,3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1 = 2 - 1
[1,2,1,3,1] => [[4,4,2,2,1],[3,1,1]]
 => [3,1,1]
 => [1,1]
 => 1 = 2 - 1
[1,2,2,1,2] => [[4,3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 2 = 3 - 1
[1,2,2,2,1] => [[4,4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [2,1]
 => 2 = 3 - 1
[1,3,1,3] => [[5,3,3,1],[2,2]]
 => [2,2]
 => [2]
 => 2 = 3 - 1
[1,3,2,2] => [[5,4,3,1],[3,2]]
 => [3,2]
 => [2]
 => 2 = 3 - 1
[2,1,2,3] => [[5,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,1,3,2] => [[5,4,2,2],[3,1,1]]
 => [3,1,1]
 => [1,1]
 => 1 = 2 - 1
[2,2,1,3] => [[5,3,3,2],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 2 = 3 - 1
[2,2,2,2] => [[5,4,3,2],[3,2,1]]
 => [3,2,1]
 => [2,1]
 => 2 = 3 - 1
Description
The number of multipartitions of sizes given by an integer partition. This is, for $\lambda = (\lambda_1,\ldots,\lambda_n)$, this is the number of $n$-tuples $(\lambda^{(1)},\ldots,\lambda^{(n)})$ of partitions $\lambda^{(i)}$ such that $\lambda^{(i)} \vdash \lambda_i$.
Matching statistic: St000566
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 40%
Values
[1] => [[1],[]]
 => []
 => ?
 => ? = 1 - 2
[1,1] => [[1,1],[]]
 => []
 => ?
 => ? = 1 - 2
[2] => [[2],[]]
 => []
 => ?
 => ? = 2 - 2
[1,1,1] => [[1,1,1],[]]
 => []
 => ?
 => ? = 1 - 2
[1,2] => [[2,1],[]]
 => []
 => ?
 => ? = 1 - 2
[2,1] => [[2,2],[1]]
 => [1]
 => []
 => ? = 2 - 2
[3] => [[3],[]]
 => []
 => ?
 => ? = 3 - 2
[1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 2
[1,1,2] => [[2,1,1],[]]
 => []
 => ?
 => ? = 1 - 2
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => []
 => ? = 2 - 2
[1,3] => [[3,1],[]]
 => []
 => ?
 => ? = 2 - 2
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 2
[2,2] => [[3,2],[1]]
 => [1]
 => []
 => ? = 2 - 2
[3,1] => [[3,3],[2]]
 => [2]
 => []
 => ? = 3 - 2
[4] => [[4],[]]
 => []
 => ?
 => ? = 4 - 2
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 2
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 2
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 2
[1,1,3] => [[3,1,1],[]]
 => []
 => ?
 => ? = 1 - 2
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 2
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => []
 => ? = 2 - 2
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => []
 => ? = 3 - 2
[1,4] => [[4,1],[]]
 => []
 => ?
 => ? = 3 - 2
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 0 = 2 - 2
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 2
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => [1]
 => ? = 2 - 2
[2,3] => [[4,2],[1]]
 => [1]
 => []
 => ? = 3 - 2
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => [2]
 => 1 = 3 - 2
[3,2] => [[4,3],[2]]
 => [2]
 => []
 => ? = 3 - 2
[4,1] => [[4,4],[3]]
 => [3]
 => []
 => ? = 4 - 2
[5] => [[5],[]]
 => []
 => ?
 => ? = 5 - 2
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 2
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 2
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 2
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 2
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 2
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => []
 => ? = 3 - 2
[1,1,4] => [[4,1,1],[]]
 => []
 => ?
 => ? = 2 - 2
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 0 = 2 - 2
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 2
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1]
 => ? = 2 - 2
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => []
 => ? = 3 - 2
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => [2]
 => 1 = 3 - 2
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => []
 => ? = 3 - 2
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => []
 => ? = 4 - 2
[1,5] => [[5,1],[]]
 => []
 => ?
 => ? = 4 - 2
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 0 = 2 - 2
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 0 = 2 - 2
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 2
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 1 = 3 - 2
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => [1]
 => ? = 2 - 2
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => [1]
 => ? = 3 - 2
[2,4] => [[5,2],[1]]
 => [1]
 => []
 => ? = 4 - 2
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [2]
 => 1 = 3 - 2
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => [2]
 => 1 = 3 - 2
[3,3] => [[5,3],[2]]
 => [2]
 => []
 => ? = 4 - 2
[4,2] => [[5,4],[3]]
 => [3]
 => []
 => ? = 4 - 2
[1,1,1,1,3] => [[3,1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 2
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 2
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
 => [2,2]
 => [2]
 => 1 = 3 - 2
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 0 = 2 - 2
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 0 = 2 - 2
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 1 = 3 - 2
[1,3,1,2] => [[4,3,3,1],[2,2]]
 => [2,2]
 => [2]
 => 1 = 3 - 2
[1,3,2,1] => [[4,4,3,1],[3,2]]
 => [3,2]
 => [2]
 => 1 = 3 - 2
[2,1,1,3] => [[4,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 0 = 2 - 2
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 0 = 2 - 2
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
 => [3,1,1]
 => [1,1]
 => 0 = 2 - 2
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 1 = 3 - 2
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
 => [3,2,1]
 => [2,1]
 => 1 = 3 - 2
[3,1,3] => [[5,3,3],[2,2]]
 => [2,2]
 => [2]
 => 1 = 3 - 2
[3,2,2] => [[5,4,3],[3,2]]
 => [3,2]
 => [2]
 => 1 = 3 - 2
[1,1,3,1,2] => [[4,3,3,1,1],[2,2]]
 => [2,2]
 => [2]
 => 1 = 3 - 2
[1,1,3,2,1] => [[4,4,3,1,1],[3,2]]
 => [3,2]
 => [2]
 => 1 = 3 - 2
[1,2,1,2,2] => [[4,3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 0 = 2 - 2
[1,2,1,3,1] => [[4,4,2,2,1],[3,1,1]]
 => [3,1,1]
 => [1,1]
 => 0 = 2 - 2
[1,2,2,1,2] => [[4,3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 1 = 3 - 2
[1,2,2,2,1] => [[4,4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [2,1]
 => 1 = 3 - 2
[1,3,1,3] => [[5,3,3,1],[2,2]]
 => [2,2]
 => [2]
 => 1 = 3 - 2
[1,3,2,2] => [[5,4,3,1],[3,2]]
 => [3,2]
 => [2]
 => 1 = 3 - 2
[2,1,2,3] => [[5,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 0 = 2 - 2
[2,1,3,2] => [[5,4,2,2],[3,1,1]]
 => [3,1,1]
 => [1,1]
 => 0 = 2 - 2
[2,2,1,3] => [[5,3,3,2],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 1 = 3 - 2
[2,2,2,2] => [[5,4,3,2],[3,2,1]]
 => [3,2,1]
 => [2,1]
 => 1 = 3 - 2
Description
The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is $$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$
Matching statistic: St000621
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 40%
Values
[1] => [[1],[]]
 => []
 => ?
 => ? = 1 - 2
[1,1] => [[1,1],[]]
 => []
 => ?
 => ? = 1 - 2
[2] => [[2],[]]
 => []
 => ?
 => ? = 2 - 2
[1,1,1] => [[1,1,1],[]]
 => []
 => ?
 => ? = 1 - 2
[1,2] => [[2,1],[]]
 => []
 => ?
 => ? = 1 - 2
[2,1] => [[2,2],[1]]
 => [1]
 => []
 => ? = 2 - 2
[3] => [[3],[]]
 => []
 => ?
 => ? = 3 - 2
[1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 2
[1,1,2] => [[2,1,1],[]]
 => []
 => ?
 => ? = 1 - 2
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => []
 => ? = 2 - 2
[1,3] => [[3,1],[]]
 => []
 => ?
 => ? = 2 - 2
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 2
[2,2] => [[3,2],[1]]
 => [1]
 => []
 => ? = 2 - 2
[3,1] => [[3,3],[2]]
 => [2]
 => []
 => ? = 3 - 2
[4] => [[4],[]]
 => []
 => ?
 => ? = 4 - 2
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 2
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 2
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 2
[1,1,3] => [[3,1,1],[]]
 => []
 => ?
 => ? = 1 - 2
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 2
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => []
 => ? = 2 - 2
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => []
 => ? = 3 - 2
[1,4] => [[4,1],[]]
 => []
 => ?
 => ? = 3 - 2
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 0 = 2 - 2
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 2
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => [1]
 => ? = 2 - 2
[2,3] => [[4,2],[1]]
 => [1]
 => []
 => ? = 3 - 2
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => [2]
 => 1 = 3 - 2
[3,2] => [[4,3],[2]]
 => [2]
 => []
 => ? = 3 - 2
[4,1] => [[4,4],[3]]
 => [3]
 => []
 => ? = 4 - 2
[5] => [[5],[]]
 => []
 => ?
 => ? = 5 - 2
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 2
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 2
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 2
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 2
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 2
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => []
 => ? = 3 - 2
[1,1,4] => [[4,1,1],[]]
 => []
 => ?
 => ? = 2 - 2
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 0 = 2 - 2
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 2
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1]
 => ? = 2 - 2
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => []
 => ? = 3 - 2
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => [2]
 => 1 = 3 - 2
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => []
 => ? = 3 - 2
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => []
 => ? = 4 - 2
[1,5] => [[5,1],[]]
 => []
 => ?
 => ? = 4 - 2
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 0 = 2 - 2
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 0 = 2 - 2
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 2
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 1 = 3 - 2
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => [1]
 => ? = 2 - 2
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => [1]
 => ? = 3 - 2
[2,4] => [[5,2],[1]]
 => [1]
 => []
 => ? = 4 - 2
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [2]
 => 1 = 3 - 2
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => [2]
 => 1 = 3 - 2
[3,3] => [[5,3],[2]]
 => [2]
 => []
 => ? = 4 - 2
[4,2] => [[5,4],[3]]
 => [3]
 => []
 => ? = 4 - 2
[1,1,1,1,3] => [[3,1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 2
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 2
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
 => [2,2]
 => [2]
 => 1 = 3 - 2
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 0 = 2 - 2
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 0 = 2 - 2
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 1 = 3 - 2
[1,3,1,2] => [[4,3,3,1],[2,2]]
 => [2,2]
 => [2]
 => 1 = 3 - 2
[1,3,2,1] => [[4,4,3,1],[3,2]]
 => [3,2]
 => [2]
 => 1 = 3 - 2
[2,1,1,3] => [[4,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 0 = 2 - 2
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 0 = 2 - 2
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
 => [3,1,1]
 => [1,1]
 => 0 = 2 - 2
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 1 = 3 - 2
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
 => [3,2,1]
 => [2,1]
 => 1 = 3 - 2
[3,1,3] => [[5,3,3],[2,2]]
 => [2,2]
 => [2]
 => 1 = 3 - 2
[3,2,2] => [[5,4,3],[3,2]]
 => [3,2]
 => [2]
 => 1 = 3 - 2
[1,1,3,1,2] => [[4,3,3,1,1],[2,2]]
 => [2,2]
 => [2]
 => 1 = 3 - 2
[1,1,3,2,1] => [[4,4,3,1,1],[3,2]]
 => [3,2]
 => [2]
 => 1 = 3 - 2
[1,2,1,2,2] => [[4,3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 0 = 2 - 2
[1,2,1,3,1] => [[4,4,2,2,1],[3,1,1]]
 => [3,1,1]
 => [1,1]
 => 0 = 2 - 2
[1,2,2,1,2] => [[4,3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 1 = 3 - 2
[1,2,2,2,1] => [[4,4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [2,1]
 => 1 = 3 - 2
[1,3,1,3] => [[5,3,3,1],[2,2]]
 => [2,2]
 => [2]
 => 1 = 3 - 2
[1,3,2,2] => [[5,4,3,1],[3,2]]
 => [3,2]
 => [2]
 => 1 = 3 - 2
[2,1,2,3] => [[5,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 0 = 2 - 2
[2,1,3,2] => [[5,4,2,2],[3,1,1]]
 => [3,1,1]
 => [1,1]
 => 0 = 2 - 2
[2,2,1,3] => [[5,3,3,2],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 1 = 3 - 2
[2,2,2,2] => [[5,4,3,2],[3,2,1]]
 => [3,2,1]
 => [2,1]
 => 1 = 3 - 2
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is even. This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1]. The case of an odd minimum is [[St000620]].
Matching statistic: St000928
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000928: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 40%
Values
[1] => [[1],[]]
 => []
 => ?
 => ? = 1 - 3
[1,1] => [[1,1],[]]
 => []
 => ?
 => ? = 1 - 3
[2] => [[2],[]]
 => []
 => ?
 => ? = 2 - 3
[1,1,1] => [[1,1,1],[]]
 => []
 => ?
 => ? = 1 - 3
[1,2] => [[2,1],[]]
 => []
 => ?
 => ? = 1 - 3
[2,1] => [[2,2],[1]]
 => [1]
 => []
 => ? = 2 - 3
[3] => [[3],[]]
 => []
 => ?
 => ? = 3 - 3
[1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 3
[1,1,2] => [[2,1,1],[]]
 => []
 => ?
 => ? = 1 - 3
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => []
 => ? = 2 - 3
[1,3] => [[3,1],[]]
 => []
 => ?
 => ? = 2 - 3
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 3
[2,2] => [[3,2],[1]]
 => [1]
 => []
 => ? = 2 - 3
[3,1] => [[3,3],[2]]
 => [2]
 => []
 => ? = 3 - 3
[4] => [[4],[]]
 => []
 => ?
 => ? = 4 - 3
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 3
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 3
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 3
[1,1,3] => [[3,1,1],[]]
 => []
 => ?
 => ? = 1 - 3
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 3
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => []
 => ? = 2 - 3
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => []
 => ? = 3 - 3
[1,4] => [[4,1],[]]
 => []
 => ?
 => ? = 3 - 3
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => -1 = 2 - 3
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 3
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => [1]
 => ? = 2 - 3
[2,3] => [[4,2],[1]]
 => [1]
 => []
 => ? = 3 - 3
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => [2]
 => 0 = 3 - 3
[3,2] => [[4,3],[2]]
 => [2]
 => []
 => ? = 3 - 3
[4,1] => [[4,4],[3]]
 => [3]
 => []
 => ? = 4 - 3
[5] => [[5],[]]
 => []
 => ?
 => ? = 5 - 3
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 3
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 3
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 3
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 3
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 3
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => []
 => ? = 3 - 3
[1,1,4] => [[4,1,1],[]]
 => []
 => ?
 => ? = 2 - 3
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => -1 = 2 - 3
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 3
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1]
 => ? = 2 - 3
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => []
 => ? = 3 - 3
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => [2]
 => 0 = 3 - 3
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => []
 => ? = 3 - 3
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => []
 => ? = 4 - 3
[1,5] => [[5,1],[]]
 => []
 => ?
 => ? = 4 - 3
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => -1 = 2 - 3
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => -1 = 2 - 3
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [1]
 => ? = 2 - 3
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 0 = 3 - 3
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => [1]
 => ? = 2 - 3
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => [1]
 => ? = 3 - 3
[2,4] => [[5,2],[1]]
 => [1]
 => []
 => ? = 4 - 3
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [2]
 => 0 = 3 - 3
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => [2]
 => 0 = 3 - 3
[3,3] => [[5,3],[2]]
 => [2]
 => []
 => ? = 4 - 3
[4,2] => [[5,4],[3]]
 => [3]
 => []
 => ? = 4 - 3
[1,1,1,1,3] => [[3,1,1,1,1],[]]
 => []
 => ?
 => ? = 1 - 3
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => []
 => ? = 2 - 3
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
 => [2,2]
 => [2]
 => 0 = 3 - 3
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => -1 = 2 - 3
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => -1 = 2 - 3
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 0 = 3 - 3
[1,3,1,2] => [[4,3,3,1],[2,2]]
 => [2,2]
 => [2]
 => 0 = 3 - 3
[1,3,2,1] => [[4,4,3,1],[3,2]]
 => [3,2]
 => [2]
 => 0 = 3 - 3
[2,1,1,3] => [[4,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => -1 = 2 - 3
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => -1 = 2 - 3
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
 => [3,1,1]
 => [1,1]
 => -1 = 2 - 3
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 0 = 3 - 3
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
 => [3,2,1]
 => [2,1]
 => 0 = 3 - 3
[3,1,3] => [[5,3,3],[2,2]]
 => [2,2]
 => [2]
 => 0 = 3 - 3
[3,2,2] => [[5,4,3],[3,2]]
 => [3,2]
 => [2]
 => 0 = 3 - 3
[1,1,3,1,2] => [[4,3,3,1,1],[2,2]]
 => [2,2]
 => [2]
 => 0 = 3 - 3
[1,1,3,2,1] => [[4,4,3,1,1],[3,2]]
 => [3,2]
 => [2]
 => 0 = 3 - 3
[1,2,1,2,2] => [[4,3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => -1 = 2 - 3
[1,2,1,3,1] => [[4,4,2,2,1],[3,1,1]]
 => [3,1,1]
 => [1,1]
 => -1 = 2 - 3
[1,2,2,1,2] => [[4,3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 0 = 3 - 3
[1,2,2,2,1] => [[4,4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [2,1]
 => 0 = 3 - 3
[1,3,1,3] => [[5,3,3,1],[2,2]]
 => [2,2]
 => [2]
 => 0 = 3 - 3
[1,3,2,2] => [[5,4,3,1],[3,2]]
 => [3,2]
 => [2]
 => 0 = 3 - 3
[2,1,2,3] => [[5,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => -1 = 2 - 3
[2,1,3,2] => [[5,4,2,2],[3,1,1]]
 => [3,1,1]
 => [1,1]
 => -1 = 2 - 3
[2,2,1,3] => [[5,3,3,2],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 0 = 3 - 3
[2,2,2,2] => [[5,4,3,2],[3,2,1]]
 => [3,2,1]
 => [2,1]
 => 0 = 3 - 3
Description
The sum of the coefficients of the character polynomial of an integer partition. The definition of the character polynomial can be found in [1].