Your data matches 21 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000749
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[[]],[]]
 => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => [1]
 => 0
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => 0
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1]
 => 0
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => 0
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => 0
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => 0
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 0
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => [1]
 => 0
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => 0
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => 0
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => [2]
 => 0
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1]
 => 0
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => [1]
 => 0
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => [1,1]
 => 0
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [2,1]
 => 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => [1]
 => 0
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => 0
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => [2,2]
 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 0
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => [2]
 => 0
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => 0
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => [3]
 => 0
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => [2,1]
 => 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => 0
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2],[1]]
 => [1]
 => 0
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => 0
[[[],[],[[]]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [[4,4],[2]]
 => [2]
 => 0
[[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [[4,3],[1]]
 => [1]
 => 0
[[[],[[],[]]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => [1,1]
 => 0
[[[],[[[]]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => [1]
 => 0
[[[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => [1]
 => 0
[[[[[]],[]]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [[3,3,3],[1]]
 => [1]
 => 0
[[],[],[],[[]],[]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1,1],[1]]
 => [1]
 => 0
[[],[],[[]],[],[]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 0
[[],[],[[]],[[]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => [1]
 => 0
[[],[],[[],[]],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1,1],[2]]
 => [2]
 => 0
[[],[],[[[]]],[]]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1,1],[1]]
 => [1]
 => 0
[[],[],[[],[[]]]]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1,1],[1]]
 => [1]
 => 0
[[],[[]],[],[],[]]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 0
[[],[[]],[],[[]]]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1],[1,1]]
 => [1,1]
 => 0
[[],[[]],[[]],[]]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2,1],[2,1]]
 => [2,1]
 => 1
[[],[[]],[[],[]]]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [[4,2,1],[1]]
 => [1]
 => 0
[[],[[]],[[[]]]]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2,1],[1,1]]
 => [1,1]
 => 0
[[],[[],[]],[],[]]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3,1],[2,2]]
 => [2,2]
 => 2
[[],[[[]]],[],[]]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2,1],[1,1]]
 => [1,1]
 => 0
[[],[[],[]],[[]]]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [[4,3,1],[2]]
 => [2]
 => 0
[[],[[[]]],[[]]]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2,1],[1]]
 => [1]
 => 0
[[],[[],[],[]],[]]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [[4,4,1],[3]]
 => [3]
 => 0
[[],[[],[[]]],[]]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3,1],[2,1]]
 => [2,1]
 => 1
Description
The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. For example, restricting $S_{(6,3)}$ to $\mathfrak S_8$ yields $$S_{(5,3)}\oplus S_{(6,2)}$$ of degrees (number of standard Young tableaux) 28 and 20, none of which are odd. Restricting to $\mathfrak S_7$ yields $$S_{(4,3)}\oplus 2S_{(5,2)}\oplus S_{(6,1)}$$ of degrees 14, 14 and 6. However, restricting to $\mathfrak S_6$ yields $$S_{(3,3)}\oplus 3S_{(4,2)}\oplus 3S_{(5,1)}\oplus S_6$$ of degrees 5,9,5 and 1. Therefore, the statistic on the partition $(6,3)$ gives 3. This is related to $2$-saturations of Welter's game, see [1, Corollary 1.2].
Matching statistic: St001435
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 17%
Values
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2]
 => [[2],[]]
 => 0
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 0
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [[3,2],[]]
 => 0
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => [[2,2],[]]
 => 0
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => [[3,1],[]]
 => 0
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => [[3],[]]
 => 0
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => 0
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [[4,2,2,1],[]]
 => ? = 0
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [[4,3,1,1],[]]
 => ? = 0
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [[3,3,1,1],[]]
 => ? = 0
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [[4,2,1,1],[]]
 => ? = 0
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [[4,1,1,1],[]]
 => ? = 0
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [[2,2,1,1],[]]
 => ? = 0
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [[4,3,2],[]]
 => ? = 0
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [[3,3,2],[]]
 => ? = 0
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [[4,2,2],[]]
 => ? = 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [[3,2,2],[]]
 => ? = 0
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [[2,2,2],[]]
 => ? = 0
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [[4,3,1],[]]
 => ? = 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [[4,3],[]]
 => ? = 0
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [[3,3,1],[]]
 => ? = 0
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [[3,3],[]]
 => ? = 0
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [[4,2,1],[]]
 => ? = 0
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [[4,1,1],[]]
 => ? = 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [[4,2],[]]
 => ? = 0
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [[4,1],[]]
 => 0
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [[4],[]]
 => 0
[[[],[],[[]]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => 0
[[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 0
[[[],[[],[]]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => 0
[[[],[[[]]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0
[[[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [[2,2],[]]
 => 0
[[[[[]],[]]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [[2],[]]
 => 0
[[],[],[],[[]],[]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [[5,3,3,2,1],[]]
 => ? = 0
[[],[],[[]],[],[]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [[5,4,2,2,1],[]]
 => ? = 0
[[],[],[[]],[[]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [[4,4,2,2,1],[]]
 => ? = 0
[[],[],[[],[]],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [[5,3,2,2,1],[]]
 => ? = 0
[[],[],[[[]]],[]]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [[5,2,2,2,1],[]]
 => ? = 0
[[],[],[[],[[]]]]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => [[3,3,2,2,1],[]]
 => ? = 0
[[],[[]],[],[],[]]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [[5,4,3,1,1],[]]
 => ? = 0
[[],[[]],[],[[]]]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [[4,4,3,1,1],[]]
 => ? = 0
[[],[[]],[[]],[]]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [[5,3,3,1,1],[]]
 => ? = 1
[[],[[]],[[],[]]]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1,1]
 => [[4,3,3,1,1],[]]
 => ? = 0
[[],[[]],[[[]]]]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [[3,3,3,1,1],[]]
 => ? = 0
[[],[[],[]],[],[]]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [[5,4,2,1,1],[]]
 => ? = 2
[[],[[[]]],[],[]]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => [[5,4,1,1,1],[]]
 => ? = 0
[[],[[],[]],[[]]]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1,1]
 => [[4,4,2,1,1],[]]
 => ? = 0
[[],[[[]]],[[]]]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [4,4,1,1,1]
 => [[4,4,1,1,1],[]]
 => ? = 0
[[],[[],[],[]],[]]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [[5,3,2,1,1],[]]
 => ? = 0
[[],[[],[[]]],[]]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1,1]
 => [[5,2,2,1,1],[]]
 => ? = 1
[[],[[[]],[]],[]]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1,1]
 => [[5,3,1,1,1],[]]
 => ? = 0
[[],[[[],[]]],[]]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,1,1]
 => [[5,2,1,1,1],[]]
 => ? = 0
[[],[[[[]]]],[]]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [5,1,1,1,1]
 => [[5,1,1,1,1],[]]
 => ? = 0
[[],[[],[],[[]]]]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1,1]
 => [[3,3,2,1,1],[]]
 => ? = 0
[[],[[],[[]],[]]]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1,1]
 => [[4,2,2,1,1],[]]
 => ? = 0
[[],[[],[[],[]]]]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1,1]
 => [[3,2,2,1,1],[]]
 => ? = 0
[[],[[],[[[]]]]]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => ? = 0
[[],[[[]],[[]]]]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1,1]
 => [[3,3,1,1,1],[]]
 => ? = 0
[[],[[[[]],[]]]]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1,1]
 => [[3,1,1,1,1],[]]
 => ? = 0
[[[]],[],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => [[5,4,3,2],[]]
 => ? = 0
[[[]],[],[],[[]]]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => [[4,4,3,2],[]]
 => ? = 0
[[[]],[],[[]],[]]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2]
 => [[5,3,3,2],[]]
 => ? = 0
[[[]],[],[[],[]]]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2]
 => [[4,3,3,2],[]]
 => ? = 0
[[[]],[],[[[]]]]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2]
 => [[3,3,3,2],[]]
 => ? = 0
[[[]],[[]],[],[]]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2]
 => [[5,4,2,2],[]]
 => ? = 0
[[[[[[]]]]],[]]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => [[5],[]]
 => 0
[[[],[[[],[]]]]]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1]
 => [[2,1,1,1],[]]
 => 0
[[[],[[[[]]]]]]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0
[[[[],[[]],[]]]]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 0
[[[[[]],[],[]]]]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => [[3,2],[]]
 => 0
[[[[[]],[[]]]]]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => [[2,2],[]]
 => 0
[[[[[],[]],[]]]]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [3,1]
 => [[3,1],[]]
 => 0
[[[[[[]]],[]]]]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => [[3],[]]
 => 0
[[[[[],[[]]]]]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => 0
[[[],[[[[[]]]]]]]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => 0
[[[[[[],[]]],[]]]]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [4,1]
 => [[4,1],[]]
 => 0
[[[[[[[]]]],[]]]]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [4]
 => [[4],[]]
 => 0
[[[[[],[],[[]]]]]]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => 0
[[[[[],[[]],[]]]]]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 0
[[[[[],[[],[]]]]]]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => 0
[[[[[],[[[]]]]]]]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0
[[[[[[]],[[]]]]]]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [2,2]
 => [[2,2],[]]
 => 0
[[[[[[[]],[]]]]]]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [2]
 => [[2],[]]
 => 0
Description
The number of missing boxes in the first row.
Matching statistic: St001438
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 17%
Values
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2]
 => [[2],[]]
 => 0
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 0
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [[3,2],[]]
 => 0
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => [[2,2],[]]
 => 0
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => [[3,1],[]]
 => 0
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => [[3],[]]
 => 0
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => 0
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [[4,2,2,1],[]]
 => ? = 0
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [[4,3,1,1],[]]
 => ? = 0
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [[3,3,1,1],[]]
 => ? = 0
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [[4,2,1,1],[]]
 => ? = 0
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [[4,1,1,1],[]]
 => ? = 0
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [[2,2,1,1],[]]
 => ? = 0
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [[4,3,2],[]]
 => ? = 0
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [[3,3,2],[]]
 => ? = 0
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [[4,2,2],[]]
 => ? = 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [[3,2,2],[]]
 => ? = 0
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [[2,2,2],[]]
 => ? = 0
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [[4,3,1],[]]
 => ? = 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [[4,3],[]]
 => ? = 0
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [[3,3,1],[]]
 => ? = 0
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [[3,3],[]]
 => ? = 0
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [[4,2,1],[]]
 => ? = 0
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [[4,1,1],[]]
 => ? = 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [[4,2],[]]
 => ? = 0
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [[4,1],[]]
 => 0
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [[4],[]]
 => 0
[[[],[],[[]]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => 0
[[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 0
[[[],[[],[]]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => 0
[[[],[[[]]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0
[[[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [[2,2],[]]
 => 0
[[[[[]],[]]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [[2],[]]
 => 0
[[],[],[],[[]],[]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [[5,3,3,2,1],[]]
 => ? = 0
[[],[],[[]],[],[]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [[5,4,2,2,1],[]]
 => ? = 0
[[],[],[[]],[[]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [[4,4,2,2,1],[]]
 => ? = 0
[[],[],[[],[]],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [[5,3,2,2,1],[]]
 => ? = 0
[[],[],[[[]]],[]]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [[5,2,2,2,1],[]]
 => ? = 0
[[],[],[[],[[]]]]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => [[3,3,2,2,1],[]]
 => ? = 0
[[],[[]],[],[],[]]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [[5,4,3,1,1],[]]
 => ? = 0
[[],[[]],[],[[]]]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [[4,4,3,1,1],[]]
 => ? = 0
[[],[[]],[[]],[]]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [[5,3,3,1,1],[]]
 => ? = 1
[[],[[]],[[],[]]]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1,1]
 => [[4,3,3,1,1],[]]
 => ? = 0
[[],[[]],[[[]]]]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [[3,3,3,1,1],[]]
 => ? = 0
[[],[[],[]],[],[]]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [[5,4,2,1,1],[]]
 => ? = 2
[[],[[[]]],[],[]]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => [[5,4,1,1,1],[]]
 => ? = 0
[[],[[],[]],[[]]]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1,1]
 => [[4,4,2,1,1],[]]
 => ? = 0
[[],[[[]]],[[]]]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [4,4,1,1,1]
 => [[4,4,1,1,1],[]]
 => ? = 0
[[],[[],[],[]],[]]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [[5,3,2,1,1],[]]
 => ? = 0
[[],[[],[[]]],[]]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1,1]
 => [[5,2,2,1,1],[]]
 => ? = 1
[[],[[[]],[]],[]]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1,1]
 => [[5,3,1,1,1],[]]
 => ? = 0
[[],[[[],[]]],[]]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,1,1]
 => [[5,2,1,1,1],[]]
 => ? = 0
[[],[[[[]]]],[]]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [5,1,1,1,1]
 => [[5,1,1,1,1],[]]
 => ? = 0
[[],[[],[],[[]]]]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1,1]
 => [[3,3,2,1,1],[]]
 => ? = 0
[[],[[],[[]],[]]]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1,1]
 => [[4,2,2,1,1],[]]
 => ? = 0
[[],[[],[[],[]]]]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1,1]
 => [[3,2,2,1,1],[]]
 => ? = 0
[[],[[],[[[]]]]]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => ? = 0
[[],[[[]],[[]]]]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1,1]
 => [[3,3,1,1,1],[]]
 => ? = 0
[[],[[[[]],[]]]]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1,1]
 => [[3,1,1,1,1],[]]
 => ? = 0
[[[]],[],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => [[5,4,3,2],[]]
 => ? = 0
[[[]],[],[],[[]]]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => [[4,4,3,2],[]]
 => ? = 0
[[[]],[],[[]],[]]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2]
 => [[5,3,3,2],[]]
 => ? = 0
[[[]],[],[[],[]]]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2]
 => [[4,3,3,2],[]]
 => ? = 0
[[[]],[],[[[]]]]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2]
 => [[3,3,3,2],[]]
 => ? = 0
[[[]],[[]],[],[]]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2]
 => [[5,4,2,2],[]]
 => ? = 0
[[[[[[]]]]],[]]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => [[5],[]]
 => 0
[[[],[[[],[]]]]]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1]
 => [[2,1,1,1],[]]
 => 0
[[[],[[[[]]]]]]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0
[[[[],[[]],[]]]]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 0
[[[[[]],[],[]]]]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => [[3,2],[]]
 => 0
[[[[[]],[[]]]]]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => [[2,2],[]]
 => 0
[[[[[],[]],[]]]]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [3,1]
 => [[3,1],[]]
 => 0
[[[[[[]]],[]]]]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => [[3],[]]
 => 0
[[[[[],[[]]]]]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => 0
[[[],[[[[[]]]]]]]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => 0
[[[[[[],[]]],[]]]]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [4,1]
 => [[4,1],[]]
 => 0
[[[[[[[]]]],[]]]]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [4]
 => [[4],[]]
 => 0
[[[[[],[],[[]]]]]]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => 0
[[[[[],[[]],[]]]]]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 0
[[[[[],[[],[]]]]]]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => 0
[[[[[],[[[]]]]]]]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0
[[[[[[]],[[]]]]]]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [2,2]
 => [[2,2],[]]
 => 0
[[[[[[[]],[]]]]]]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [2]
 => [[2],[]]
 => 0
Description
The number of missing boxes of a skew partition.
Matching statistic: St001487
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 17%
Values
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2]
 => [[2],[]]
 => 1 = 0 + 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 1 = 0 + 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [[3,2],[]]
 => 1 = 0 + 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => [[2,2],[]]
 => 1 = 0 + 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => [[3,1],[]]
 => 1 = 0 + 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => [[3],[]]
 => 1 = 0 + 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => 1 = 0 + 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [[4,2,2,1],[]]
 => ? = 0 + 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [[4,3,1,1],[]]
 => ? = 0 + 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [[3,3,1,1],[]]
 => ? = 0 + 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [[4,2,1,1],[]]
 => ? = 0 + 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [[4,1,1,1],[]]
 => ? = 0 + 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [[2,2,1,1],[]]
 => ? = 0 + 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [[4,3,2],[]]
 => ? = 0 + 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [[3,3,2],[]]
 => ? = 0 + 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [[4,2,2],[]]
 => ? = 1 + 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [[3,2,2],[]]
 => ? = 0 + 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [[2,2,2],[]]
 => ? = 0 + 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [[4,3,1],[]]
 => ? = 2 + 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [[4,3],[]]
 => ? = 0 + 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [[3,3,1],[]]
 => ? = 0 + 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [[3,3],[]]
 => ? = 0 + 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [[4,2,1],[]]
 => ? = 0 + 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [[4,1,1],[]]
 => ? = 1 + 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [[4,2],[]]
 => ? = 0 + 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [[4,1],[]]
 => 1 = 0 + 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [[4],[]]
 => 1 = 0 + 1
[[[],[],[[]]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => 1 = 0 + 1
[[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 1 = 0 + 1
[[[],[[],[]]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1 = 0 + 1
[[[],[[[]]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1 = 0 + 1
[[[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [[2,2],[]]
 => 1 = 0 + 1
[[[[[]],[]]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [[2],[]]
 => 1 = 0 + 1
[[],[],[],[[]],[]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [[5,3,3,2,1],[]]
 => ? = 0 + 1
[[],[],[[]],[],[]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [[5,4,2,2,1],[]]
 => ? = 0 + 1
[[],[],[[]],[[]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [[4,4,2,2,1],[]]
 => ? = 0 + 1
[[],[],[[],[]],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [[5,3,2,2,1],[]]
 => ? = 0 + 1
[[],[],[[[]]],[]]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [[5,2,2,2,1],[]]
 => ? = 0 + 1
[[],[],[[],[[]]]]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => [[3,3,2,2,1],[]]
 => ? = 0 + 1
[[],[[]],[],[],[]]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [[5,4,3,1,1],[]]
 => ? = 0 + 1
[[],[[]],[],[[]]]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [[4,4,3,1,1],[]]
 => ? = 0 + 1
[[],[[]],[[]],[]]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [[5,3,3,1,1],[]]
 => ? = 1 + 1
[[],[[]],[[],[]]]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1,1]
 => [[4,3,3,1,1],[]]
 => ? = 0 + 1
[[],[[]],[[[]]]]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [[3,3,3,1,1],[]]
 => ? = 0 + 1
[[],[[],[]],[],[]]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [[5,4,2,1,1],[]]
 => ? = 2 + 1
[[],[[[]]],[],[]]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => [[5,4,1,1,1],[]]
 => ? = 0 + 1
[[],[[],[]],[[]]]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1,1]
 => [[4,4,2,1,1],[]]
 => ? = 0 + 1
[[],[[[]]],[[]]]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [4,4,1,1,1]
 => [[4,4,1,1,1],[]]
 => ? = 0 + 1
[[],[[],[],[]],[]]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [[5,3,2,1,1],[]]
 => ? = 0 + 1
[[],[[],[[]]],[]]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1,1]
 => [[5,2,2,1,1],[]]
 => ? = 1 + 1
[[],[[[]],[]],[]]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1,1]
 => [[5,3,1,1,1],[]]
 => ? = 0 + 1
[[],[[[],[]]],[]]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,1,1]
 => [[5,2,1,1,1],[]]
 => ? = 0 + 1
[[],[[[[]]]],[]]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [5,1,1,1,1]
 => [[5,1,1,1,1],[]]
 => ? = 0 + 1
[[],[[],[],[[]]]]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1,1]
 => [[3,3,2,1,1],[]]
 => ? = 0 + 1
[[],[[],[[]],[]]]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1,1]
 => [[4,2,2,1,1],[]]
 => ? = 0 + 1
[[],[[],[[],[]]]]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1,1]
 => [[3,2,2,1,1],[]]
 => ? = 0 + 1
[[],[[],[[[]]]]]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => ? = 0 + 1
[[],[[[]],[[]]]]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1,1]
 => [[3,3,1,1,1],[]]
 => ? = 0 + 1
[[],[[[[]],[]]]]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1,1]
 => [[3,1,1,1,1],[]]
 => ? = 0 + 1
[[[]],[],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => [[5,4,3,2],[]]
 => ? = 0 + 1
[[[]],[],[],[[]]]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => [[4,4,3,2],[]]
 => ? = 0 + 1
[[[]],[],[[]],[]]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2]
 => [[5,3,3,2],[]]
 => ? = 0 + 1
[[[]],[],[[],[]]]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2]
 => [[4,3,3,2],[]]
 => ? = 0 + 1
[[[]],[],[[[]]]]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2]
 => [[3,3,3,2],[]]
 => ? = 0 + 1
[[[]],[[]],[],[]]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2]
 => [[5,4,2,2],[]]
 => ? = 0 + 1
[[[[[[]]]]],[]]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => [[5],[]]
 => 1 = 0 + 1
[[[],[[[],[]]]]]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1]
 => [[2,1,1,1],[]]
 => 1 = 0 + 1
[[[],[[[[]]]]]]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 1 = 0 + 1
[[[[],[[]],[]]]]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 1 = 0 + 1
[[[[[]],[],[]]]]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => [[3,2],[]]
 => 1 = 0 + 1
[[[[[]],[[]]]]]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => [[2,2],[]]
 => 1 = 0 + 1
[[[[[],[]],[]]]]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [3,1]
 => [[3,1],[]]
 => 1 = 0 + 1
[[[[[[]]],[]]]]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => [[3],[]]
 => 1 = 0 + 1
[[[[[],[[]]]]]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => 1 = 0 + 1
[[[],[[[[[]]]]]]]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => 1 = 0 + 1
[[[[[[],[]]],[]]]]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [4,1]
 => [[4,1],[]]
 => 1 = 0 + 1
[[[[[[[]]]],[]]]]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [4]
 => [[4],[]]
 => 1 = 0 + 1
[[[[[],[],[[]]]]]]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => 1 = 0 + 1
[[[[[],[[]],[]]]]]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 1 = 0 + 1
[[[[[],[[],[]]]]]]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1 = 0 + 1
[[[[[],[[[]]]]]]]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1 = 0 + 1
[[[[[[]],[[]]]]]]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [2,2]
 => [[2,2],[]]
 => 1 = 0 + 1
[[[[[[[]],[]]]]]]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [2]
 => [[2],[]]
 => 1 = 0 + 1
Description
The number of inner corners of a skew partition.
Matching statistic: St001490
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 17%
Values
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2]
 => [[2],[]]
 => 1 = 0 + 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 1 = 0 + 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [[3,2],[]]
 => 1 = 0 + 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => [[2,2],[]]
 => 1 = 0 + 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => [[3,1],[]]
 => 1 = 0 + 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => [[3],[]]
 => 1 = 0 + 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => 1 = 0 + 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [[4,2,2,1],[]]
 => ? = 0 + 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [[4,3,1,1],[]]
 => ? = 0 + 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [[3,3,1,1],[]]
 => ? = 0 + 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [[4,2,1,1],[]]
 => ? = 0 + 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [[4,1,1,1],[]]
 => ? = 0 + 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [[2,2,1,1],[]]
 => ? = 0 + 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [[4,3,2],[]]
 => ? = 0 + 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [[3,3,2],[]]
 => ? = 0 + 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [[4,2,2],[]]
 => ? = 1 + 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [[3,2,2],[]]
 => ? = 0 + 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [[2,2,2],[]]
 => ? = 0 + 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [[4,3,1],[]]
 => ? = 2 + 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [[4,3],[]]
 => ? = 0 + 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [[3,3,1],[]]
 => ? = 0 + 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [[3,3],[]]
 => ? = 0 + 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [[4,2,1],[]]
 => ? = 0 + 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [[4,1,1],[]]
 => ? = 1 + 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [[4,2],[]]
 => ? = 0 + 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [[4,1],[]]
 => 1 = 0 + 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [[4],[]]
 => 1 = 0 + 1
[[[],[],[[]]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => 1 = 0 + 1
[[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 1 = 0 + 1
[[[],[[],[]]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1 = 0 + 1
[[[],[[[]]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1 = 0 + 1
[[[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [[2,2],[]]
 => 1 = 0 + 1
[[[[[]],[]]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [[2],[]]
 => 1 = 0 + 1
[[],[],[],[[]],[]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [[5,3,3,2,1],[]]
 => ? = 0 + 1
[[],[],[[]],[],[]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [[5,4,2,2,1],[]]
 => ? = 0 + 1
[[],[],[[]],[[]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [[4,4,2,2,1],[]]
 => ? = 0 + 1
[[],[],[[],[]],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [[5,3,2,2,1],[]]
 => ? = 0 + 1
[[],[],[[[]]],[]]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [[5,2,2,2,1],[]]
 => ? = 0 + 1
[[],[],[[],[[]]]]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => [[3,3,2,2,1],[]]
 => ? = 0 + 1
[[],[[]],[],[],[]]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [[5,4,3,1,1],[]]
 => ? = 0 + 1
[[],[[]],[],[[]]]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [[4,4,3,1,1],[]]
 => ? = 0 + 1
[[],[[]],[[]],[]]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [[5,3,3,1,1],[]]
 => ? = 1 + 1
[[],[[]],[[],[]]]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1,1]
 => [[4,3,3,1,1],[]]
 => ? = 0 + 1
[[],[[]],[[[]]]]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [[3,3,3,1,1],[]]
 => ? = 0 + 1
[[],[[],[]],[],[]]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [[5,4,2,1,1],[]]
 => ? = 2 + 1
[[],[[[]]],[],[]]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => [[5,4,1,1,1],[]]
 => ? = 0 + 1
[[],[[],[]],[[]]]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1,1]
 => [[4,4,2,1,1],[]]
 => ? = 0 + 1
[[],[[[]]],[[]]]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [4,4,1,1,1]
 => [[4,4,1,1,1],[]]
 => ? = 0 + 1
[[],[[],[],[]],[]]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [[5,3,2,1,1],[]]
 => ? = 0 + 1
[[],[[],[[]]],[]]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1,1]
 => [[5,2,2,1,1],[]]
 => ? = 1 + 1
[[],[[[]],[]],[]]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1,1]
 => [[5,3,1,1,1],[]]
 => ? = 0 + 1
[[],[[[],[]]],[]]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,1,1]
 => [[5,2,1,1,1],[]]
 => ? = 0 + 1
[[],[[[[]]]],[]]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [5,1,1,1,1]
 => [[5,1,1,1,1],[]]
 => ? = 0 + 1
[[],[[],[],[[]]]]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1,1]
 => [[3,3,2,1,1],[]]
 => ? = 0 + 1
[[],[[],[[]],[]]]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1,1]
 => [[4,2,2,1,1],[]]
 => ? = 0 + 1
[[],[[],[[],[]]]]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1,1]
 => [[3,2,2,1,1],[]]
 => ? = 0 + 1
[[],[[],[[[]]]]]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => ? = 0 + 1
[[],[[[]],[[]]]]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1,1]
 => [[3,3,1,1,1],[]]
 => ? = 0 + 1
[[],[[[[]],[]]]]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1,1]
 => [[3,1,1,1,1],[]]
 => ? = 0 + 1
[[[]],[],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => [[5,4,3,2],[]]
 => ? = 0 + 1
[[[]],[],[],[[]]]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => [[4,4,3,2],[]]
 => ? = 0 + 1
[[[]],[],[[]],[]]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2]
 => [[5,3,3,2],[]]
 => ? = 0 + 1
[[[]],[],[[],[]]]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2]
 => [[4,3,3,2],[]]
 => ? = 0 + 1
[[[]],[],[[[]]]]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2]
 => [[3,3,3,2],[]]
 => ? = 0 + 1
[[[]],[[]],[],[]]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2]
 => [[5,4,2,2],[]]
 => ? = 0 + 1
[[[[[[]]]]],[]]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => [[5],[]]
 => 1 = 0 + 1
[[[],[[[],[]]]]]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1]
 => [[2,1,1,1],[]]
 => 1 = 0 + 1
[[[],[[[[]]]]]]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 1 = 0 + 1
[[[[],[[]],[]]]]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 1 = 0 + 1
[[[[[]],[],[]]]]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => [[3,2],[]]
 => 1 = 0 + 1
[[[[[]],[[]]]]]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => [[2,2],[]]
 => 1 = 0 + 1
[[[[[],[]],[]]]]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [3,1]
 => [[3,1],[]]
 => 1 = 0 + 1
[[[[[[]]],[]]]]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => [[3],[]]
 => 1 = 0 + 1
[[[[[],[[]]]]]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => 1 = 0 + 1
[[[],[[[[[]]]]]]]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => 1 = 0 + 1
[[[[[[],[]]],[]]]]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [4,1]
 => [[4,1],[]]
 => 1 = 0 + 1
[[[[[[[]]]],[]]]]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [4]
 => [[4],[]]
 => 1 = 0 + 1
[[[[[],[],[[]]]]]]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => 1 = 0 + 1
[[[[[],[[]],[]]]]]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 1 = 0 + 1
[[[[[],[[],[]]]]]]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1 = 0 + 1
[[[[[],[[[]]]]]]]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1 = 0 + 1
[[[[[[]],[[]]]]]]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [2,2]
 => [[2,2],[]]
 => 1 = 0 + 1
[[[[[[[]],[]]]]]]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [2]
 => [[2],[]]
 => 1 = 0 + 1
Description
The number of connected components of a skew partition.
Matching statistic: St000475
Mapping
Mp00047: Ordered trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 17%
Values
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 0
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => 0
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => 0
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,2]
 => 0
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [8]
 => 0
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => 0
[[[],[[]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => 0
[[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 0
[[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 0
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [5,5,5,5,5,5]
 => ? = 0
[[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [10,10,10,10]
 => ? = 0
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [5,5,5,5]
 => ? = 0
[[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 0
[[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 0
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [5,5,5,5,5,5]
 => ? = 0
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [5,5,5,5,5,5]
 => ? = 1
[[[]],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => [10,10]
 => ? = 0
[[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => [5,5]
 => 0
[[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [10,10,10,10]
 => ? = 2
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [5,5,5,5]
 => ? = 0
[[[],[]],[[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => [10,10]
 => ? = 0
[[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => [5,5]
 => 0
[[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => [15,15]
 => ? = 0
[[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 1
[[[[]],[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 0
[[[[],[]]],[]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => [5,5]
 => 0
[[[[[]]]],[]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => [5]
 => 0
[[[],[],[[]]]]
 => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
 => [4,4,4]
 => 0
[[[],[[]],[]]]
 => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
 => [4,4,4]
 => 0
[[[],[[],[]]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [8]
 => 0
[[[],[[[]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [4]
 => 0
[[[[]],[[]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => 0
[[[[[]],[]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [3]
 => 0
[[],[],[],[[]],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[],[[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[],[[]],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[],[[],[]],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12]
 => ? = 0
[[],[],[[[]]],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[],[[],[[]]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
 => [18,18,18,18,18]
 => ? = 0
[[],[[]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[[]],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[[]],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 1
[[],[[]],[[],[]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 0
[[],[[]],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[[],[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12]
 => ? = 2
[[],[[[]]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[[],[]],[[]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 0
[[],[[[]]],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[[],[],[]],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => [18,18,18,18,18,18,18,18,18,18]
 => ? = 0
[[],[[],[[]]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
 => [18,18,18,18,18]
 => ? = 1
[[],[[[]],[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
 => [18,18,18,18,18]
 => ? = 0
[[],[[[],[]]],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,6),(5,4)],7)
 => [6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[[[[]]]],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => [6,6,6,6,6]
 => ? = 0
[[],[[],[],[[]]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7)
 => [24,24,24]
 => ? = 0
[[],[[],[[]],[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7)
 => [24,24,24]
 => ? = 0
[[],[[],[[],[]]]]
 => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
 => [48]
 => ? = 0
[[],[[],[[[]]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [24]
 => ? = 0
[[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => [24,12]
 => ? = 0
[[],[[[[]],[]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [18]
 => ? = 0
[[[]],[],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[[]],[],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[[]],[],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[[]],[],[[],[]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 0
[[[]],[],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[[]],[[]],[],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[[]],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,3,3]
 => ? = 1
[[[]],[[],[]],[]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 0
[[[[[],[]]]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
 => [12]
 => 0
[[[[[[]]]]],[]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => [6]
 => 0
[[[],[[[],[]]]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => [5,5]
 => 0
[[[],[[[[]]]]]]
 => ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
 => [5]
 => 0
[[[[]],[[[]]]]]
 => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
 => [5,5]
 => 0
[[[[[]]],[[]]]]
 => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
 => [5,5]
 => 0
[[[[],[[]],[]]]]
 => ([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7)
 => [4,4,4]
 => 0
[[[[[]],[],[]]]]
 => ([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7)
 => [4,4,4]
 => 0
[[[[[]],[[]]]]]
 => ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
 => [4,2]
 => 0
[[[[[],[]],[]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
 => [8]
 => 0
[[[[[[]]],[]]]]
 => ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
 => [4]
 => 0
[[[[[],[[]]]]]]
 => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
 => [3]
 => 0
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000929
Mapping
Mp00047: Ordered trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 17%
Values
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 0
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => 0
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => 0
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,2]
 => 0
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [8]
 => 0
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => 0
[[[],[[]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => 0
[[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 0
[[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 0
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [5,5,5,5,5,5]
 => ? = 0
[[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [10,10,10,10]
 => ? = 0
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [5,5,5,5]
 => ? = 0
[[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 0
[[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 0
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [5,5,5,5,5,5]
 => ? = 0
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [5,5,5,5,5,5]
 => ? = 1
[[[]],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => [10,10]
 => ? = 0
[[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => [5,5]
 => 0
[[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [10,10,10,10]
 => ? = 2
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [5,5,5,5]
 => ? = 0
[[[],[]],[[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => [10,10]
 => ? = 0
[[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => [5,5]
 => 0
[[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => [15,15]
 => ? = 0
[[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 1
[[[[]],[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 0
[[[[],[]]],[]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => [5,5]
 => 0
[[[[[]]]],[]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => [5]
 => 0
[[[],[],[[]]]]
 => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
 => [4,4,4]
 => 0
[[[],[[]],[]]]
 => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
 => [4,4,4]
 => 0
[[[],[[],[]]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [8]
 => 0
[[[],[[[]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [4]
 => 0
[[[[]],[[]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => 0
[[[[[]],[]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [3]
 => 0
[[],[],[],[[]],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[],[[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[],[[]],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[],[[],[]],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12]
 => ? = 0
[[],[],[[[]]],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[],[[],[[]]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
 => [18,18,18,18,18]
 => ? = 0
[[],[[]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[[]],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[[]],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 1
[[],[[]],[[],[]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 0
[[],[[]],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[[],[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12]
 => ? = 2
[[],[[[]]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[[],[]],[[]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 0
[[],[[[]]],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[[],[],[]],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => [18,18,18,18,18,18,18,18,18,18]
 => ? = 0
[[],[[],[[]]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
 => [18,18,18,18,18]
 => ? = 1
[[],[[[]],[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
 => [18,18,18,18,18]
 => ? = 0
[[],[[[],[]]],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,6),(5,4)],7)
 => [6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[[[[]]]],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => [6,6,6,6,6]
 => ? = 0
[[],[[],[],[[]]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7)
 => [24,24,24]
 => ? = 0
[[],[[],[[]],[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7)
 => [24,24,24]
 => ? = 0
[[],[[],[[],[]]]]
 => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
 => [48]
 => ? = 0
[[],[[],[[[]]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [24]
 => ? = 0
[[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => [24,12]
 => ? = 0
[[],[[[[]],[]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [18]
 => ? = 0
[[[]],[],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[[]],[],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[[]],[],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[[]],[],[[],[]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 0
[[[]],[],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[[]],[[]],[],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[[]],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,3,3]
 => ? = 1
[[[]],[[],[]],[]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 0
[[[[[],[]]]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
 => [12]
 => 0
[[[[[[]]]]],[]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => [6]
 => 0
[[[],[[[],[]]]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => [5,5]
 => 0
[[[],[[[[]]]]]]
 => ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
 => [5]
 => 0
[[[[]],[[[]]]]]
 => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
 => [5,5]
 => 0
[[[[[]]],[[]]]]
 => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
 => [5,5]
 => 0
[[[[],[[]],[]]]]
 => ([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7)
 => [4,4,4]
 => 0
[[[[[]],[],[]]]]
 => ([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7)
 => [4,4,4]
 => 0
[[[[[]],[[]]]]]
 => ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
 => [4,2]
 => 0
[[[[[],[]],[]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
 => [8]
 => 0
[[[[[[]]],[]]]]
 => ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
 => [4]
 => 0
[[[[[],[[]]]]]]
 => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
 => [3]
 => 0
Description
The constant term of the character polynomial of an integer partition. The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Matching statistic: St001122
(load all 2 compositions to match this statistic)
Mapping
Mp00047: Ordered trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St001122: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 17%
Values
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 0
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => 0
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => 0
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,2]
 => 0
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [8]
 => 0
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => 0
[[[],[[]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => 0
[[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 0
[[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 0
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [5,5,5,5,5,5]
 => ? = 0
[[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [10,10,10,10]
 => ? = 0
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [5,5,5,5]
 => ? = 0
[[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 0
[[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 0
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [5,5,5,5,5,5]
 => ? = 0
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [5,5,5,5,5,5]
 => ? = 1
[[[]],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => [10,10]
 => ? = 0
[[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => [5,5]
 => 0
[[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [10,10,10,10]
 => ? = 2
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [5,5,5,5]
 => ? = 0
[[[],[]],[[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => [10,10]
 => ? = 0
[[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => [5,5]
 => 0
[[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => [15,15]
 => ? = 0
[[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 1
[[[[]],[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 0
[[[[],[]]],[]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => [5,5]
 => 0
[[[[[]]]],[]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => [5]
 => 0
[[[],[],[[]]]]
 => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
 => [4,4,4]
 => 0
[[[],[[]],[]]]
 => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
 => [4,4,4]
 => 0
[[[],[[],[]]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [8]
 => 0
[[[],[[[]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [4]
 => 0
[[[[]],[[]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => 0
[[[[[]],[]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [3]
 => 0
[[],[],[],[[]],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[],[[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[],[[]],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[],[[],[]],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12]
 => ? = 0
[[],[],[[[]]],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[],[[],[[]]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
 => [18,18,18,18,18]
 => ? = 0
[[],[[]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[[]],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[[]],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 1
[[],[[]],[[],[]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 0
[[],[[]],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[[],[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12]
 => ? = 2
[[],[[[]]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[[],[]],[[]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 0
[[],[[[]]],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[[],[],[]],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => [18,18,18,18,18,18,18,18,18,18]
 => ? = 0
[[],[[],[[]]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
 => [18,18,18,18,18]
 => ? = 1
[[],[[[]],[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
 => [18,18,18,18,18]
 => ? = 0
[[],[[[],[]]],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,6),(5,4)],7)
 => [6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[[[[]]]],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => [6,6,6,6,6]
 => ? = 0
[[],[[],[],[[]]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7)
 => [24,24,24]
 => ? = 0
[[],[[],[[]],[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7)
 => [24,24,24]
 => ? = 0
[[],[[],[[],[]]]]
 => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
 => [48]
 => ? = 0
[[],[[],[[[]]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [24]
 => ? = 0
[[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => [24,12]
 => ? = 0
[[],[[[[]],[]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [18]
 => ? = 0
[[[]],[],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[[]],[],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[[]],[],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[[]],[],[[],[]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 0
[[[]],[],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[[]],[[]],[],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[[]],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,3,3]
 => ? = 1
[[[]],[[],[]],[]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 0
[[[[[],[]]]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
 => [12]
 => 0
[[[[[[]]]]],[]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => [6]
 => 0
[[[],[[[],[]]]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => [5,5]
 => 0
[[[],[[[[]]]]]]
 => ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
 => [5]
 => 0
[[[[]],[[[]]]]]
 => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
 => [5,5]
 => 0
[[[[[]]],[[]]]]
 => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
 => [5,5]
 => 0
[[[[],[[]],[]]]]
 => ([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7)
 => [4,4,4]
 => 0
[[[[[]],[],[]]]]
 => ([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7)
 => [4,4,4]
 => 0
[[[[[]],[[]]]]]
 => ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
 => [4,2]
 => 0
[[[[[],[]],[]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
 => [8]
 => 0
[[[[[[]]],[]]]]
 => ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
 => [4]
 => 0
[[[[[],[[]]]]]]
 => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
 => [3]
 => 0
Description
The multiplicity of the sign representation in the Kronecker square corresponding to a partition. The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$: $$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$ This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{1^n}$, for $\lambda\vdash n$. It equals $1$ if and only if $\lambda$ is self-conjugate.
Matching statistic: St001123
(load all 2 compositions to match this statistic)
Mapping
Mp00047: Ordered trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St001123: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 17%
Values
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 0
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => 0
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => 0
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,2]
 => 0
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [8]
 => 0
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => 0
[[[],[[]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => 0
[[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 0
[[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 0
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [5,5,5,5,5,5]
 => ? = 0
[[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [10,10,10,10]
 => ? = 0
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [5,5,5,5]
 => ? = 0
[[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 0
[[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 0
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [5,5,5,5,5,5]
 => ? = 0
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [5,5,5,5,5,5]
 => ? = 1
[[[]],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => [10,10]
 => ? = 0
[[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => [5,5]
 => 0
[[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [10,10,10,10]
 => ? = 2
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [5,5,5,5]
 => ? = 0
[[[],[]],[[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => [10,10]
 => ? = 0
[[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => [5,5]
 => 0
[[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => [15,15]
 => ? = 0
[[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 1
[[[[]],[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 0
[[[[],[]]],[]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => [5,5]
 => 0
[[[[[]]]],[]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => [5]
 => 0
[[[],[],[[]]]]
 => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
 => [4,4,4]
 => 0
[[[],[[]],[]]]
 => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
 => [4,4,4]
 => 0
[[[],[[],[]]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [8]
 => 0
[[[],[[[]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [4]
 => 0
[[[[]],[[]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => 0
[[[[[]],[]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [3]
 => 0
[[],[],[],[[]],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[],[[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[],[[]],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[],[[],[]],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12]
 => ? = 0
[[],[],[[[]]],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[],[[],[[]]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
 => [18,18,18,18,18]
 => ? = 0
[[],[[]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[[]],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[[]],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 1
[[],[[]],[[],[]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 0
[[],[[]],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[[],[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12]
 => ? = 2
[[],[[[]]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[[],[]],[[]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 0
[[],[[[]]],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[[],[],[]],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => [18,18,18,18,18,18,18,18,18,18]
 => ? = 0
[[],[[],[[]]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
 => [18,18,18,18,18]
 => ? = 1
[[],[[[]],[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
 => [18,18,18,18,18]
 => ? = 0
[[],[[[],[]]],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,6),(5,4)],7)
 => [6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[],[[[[]]]],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => [6,6,6,6,6]
 => ? = 0
[[],[[],[],[[]]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7)
 => [24,24,24]
 => ? = 0
[[],[[],[[]],[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7)
 => [24,24,24]
 => ? = 0
[[],[[],[[],[]]]]
 => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
 => [48]
 => ? = 0
[[],[[],[[[]]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [24]
 => ? = 0
[[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => [24,12]
 => ? = 0
[[],[[[[]],[]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [18]
 => ? = 0
[[[]],[],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[[]],[],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[[]],[],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[[]],[],[[],[]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 0
[[[]],[],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[[]],[[]],[],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0
[[[]],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,3,3]
 => ? = 1
[[[]],[[],[]],[]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 0
[[[[[],[]]]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
 => [12]
 => 0
[[[[[[]]]]],[]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => [6]
 => 0
[[[],[[[],[]]]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => [5,5]
 => 0
[[[],[[[[]]]]]]
 => ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
 => [5]
 => 0
[[[[]],[[[]]]]]
 => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
 => [5,5]
 => 0
[[[[[]]],[[]]]]
 => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
 => [5,5]
 => 0
[[[[],[[]],[]]]]
 => ([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7)
 => [4,4,4]
 => 0
[[[[[]],[],[]]]]
 => ([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7)
 => [4,4,4]
 => 0
[[[[[]],[[]]]]]
 => ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
 => [4,2]
 => 0
[[[[[],[]],[]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
 => [8]
 => 0
[[[[[[]]],[]]]]
 => ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
 => [4]
 => 0
[[[[[],[[]]]]]]
 => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
 => [3]
 => 0
Description
The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$: $$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$ This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{21^{n-2}}$, for $\lambda\vdash n$.
Matching statistic: St000759
Mapping
Mp00047: Ordered trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000759: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 17%
Values
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 1 = 0 + 1
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => 1 = 0 + 1
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => 1 = 0 + 1
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,2]
 => 1 = 0 + 1
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [8]
 => 1 = 0 + 1
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => 1 = 0 + 1
[[[],[[]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => 1 = 0 + 1
[[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 0 + 1
[[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 0 + 1
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [5,5,5,5,5,5]
 => ? = 0 + 1
[[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [10,10,10,10]
 => ? = 0 + 1
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [5,5,5,5]
 => ? = 0 + 1
[[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 0 + 1
[[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 0 + 1
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [5,5,5,5,5,5]
 => ? = 0 + 1
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [5,5,5,5,5,5]
 => ? = 1 + 1
[[[]],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => [10,10]
 => ? = 0 + 1
[[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => [5,5]
 => 1 = 0 + 1
[[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [10,10,10,10]
 => ? = 2 + 1
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [5,5,5,5]
 => ? = 0 + 1
[[[],[]],[[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => [10,10]
 => ? = 0 + 1
[[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => [5,5]
 => 1 = 0 + 1
[[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => [15,15]
 => ? = 0 + 1
[[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 1 + 1
[[[[]],[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => [15]
 => ? = 0 + 1
[[[[],[]]],[]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => [5,5]
 => 1 = 0 + 1
[[[[[]]]],[]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => [5]
 => 1 = 0 + 1
[[[],[],[[]]]]
 => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
 => [4,4,4]
 => 1 = 0 + 1
[[[],[[]],[]]]
 => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
 => [4,4,4]
 => 1 = 0 + 1
[[[],[[],[]]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => [8]
 => 1 = 0 + 1
[[[],[[[]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [4]
 => 1 = 0 + 1
[[[[]],[[]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => [4,2]
 => 1 = 0 + 1
[[[[[]],[]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => [3]
 => 1 = 0 + 1
[[],[],[],[[]],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0 + 1
[[],[],[[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0 + 1
[[],[],[[]],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0 + 1
[[],[],[[],[]],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12]
 => ? = 0 + 1
[[],[],[[[]]],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0 + 1
[[],[],[[],[[]]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
 => [18,18,18,18,18]
 => ? = 0 + 1
[[],[[]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0 + 1
[[],[[]],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0 + 1
[[],[[]],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 1 + 1
[[],[[]],[[],[]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 0 + 1
[[],[[]],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6]
 => ? = 0 + 1
[[],[[],[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12]
 => ? = 2 + 1
[[],[[[]]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0 + 1
[[],[[],[]],[[]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 0 + 1
[[],[[[]]],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6]
 => ? = 0 + 1
[[],[[],[],[]],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
 => [18,18,18,18,18,18,18,18,18,18]
 => ? = 0 + 1
[[],[[],[[]]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
 => [18,18,18,18,18]
 => ? = 1 + 1
[[],[[[]],[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
 => [18,18,18,18,18]
 => ? = 0 + 1
[[],[[[],[]]],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,6),(5,4)],7)
 => [6,6,6,6,6,6,6,6,6,6]
 => ? = 0 + 1
[[],[[[[]]]],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => [6,6,6,6,6]
 => ? = 0 + 1
[[],[[],[],[[]]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7)
 => [24,24,24]
 => ? = 0 + 1
[[],[[],[[]],[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7)
 => [24,24,24]
 => ? = 0 + 1
[[],[[],[[],[]]]]
 => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
 => [48]
 => ? = 0 + 1
[[],[[],[[[]]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => [24]
 => ? = 0 + 1
[[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => [24,12]
 => ? = 0 + 1
[[],[[[[]],[]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => [18]
 => ? = 0 + 1
[[[]],[],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0 + 1
[[[]],[],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0 + 1
[[[]],[],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0 + 1
[[[]],[],[[],[]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 0 + 1
[[[]],[],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6]
 => ? = 0 + 1
[[[]],[[]],[],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 0 + 1
[[[]],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,3,3]
 => ? = 1 + 1
[[[]],[[],[]],[]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 0 + 1
[[[[[],[]]]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
 => [12]
 => 1 = 0 + 1
[[[[[[]]]]],[]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => [6]
 => 1 = 0 + 1
[[[],[[[],[]]]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => [5,5]
 => 1 = 0 + 1
[[[],[[[[]]]]]]
 => ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
 => [5]
 => 1 = 0 + 1
[[[[]],[[[]]]]]
 => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
 => [5,5]
 => 1 = 0 + 1
[[[[[]]],[[]]]]
 => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
 => [5,5]
 => 1 = 0 + 1
[[[[],[[]],[]]]]
 => ([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7)
 => [4,4,4]
 => 1 = 0 + 1
[[[[[]],[],[]]]]
 => ([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7)
 => [4,4,4]
 => 1 = 0 + 1
[[[[[]],[[]]]]]
 => ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
 => [4,2]
 => 1 = 0 + 1
[[[[[],[]],[]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
 => [8]
 => 1 = 0 + 1
[[[[[[]]],[]]]]
 => ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
 => [4]
 => 1 = 0 + 1
[[[[[],[[]]]]]]
 => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
 => [3]
 => 1 = 0 + 1
Description
The smallest missing part in an integer partition. In [3], this is referred to as the mex, the minimal excluded part of the partition. For compositions, this is studied in [sec.3.2., 1].
The following 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000897The number of different multiplicities of parts of an integer partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St001330The hat guessing number of a graph. St000781The number of proper colouring schemes of a Ferrers diagram. St001568The smallest positive integer that does not appear twice in the partition. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001964The interval resolution global dimension of a poset. St000454The largest eigenvalue of a graph if it is integral. St000422The energy of a graph, if it is integral.