Your data matches 26 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001557
Mp00202: Integer partitions first row removalInteger partitions
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
St001557: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[2,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[3,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 2
[4,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[3,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 3
[5,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[4,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[4,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[3,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 2
[2,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 3
[6,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[5,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[5,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[4,4]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[4,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[4,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 2
[3,3,2]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[3,3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[3,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 2
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 3
[7,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[7,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[6,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[6,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[6,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
Description
The number of inversions of the second entry of a permutation. This is, for a permutation $\pi$ of length $n$, $$\# \{2 < k \leq n \mid \pi(2) > \pi(k)\}.$$ The number of inversions of the first entry is [[St000054]] and the number of inversions of the third entry is [[St001556]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000937: Integer partitions ⟶ ℤResult quality: 53% values known / values provided: 53%distinct values known / distinct values provided: 75%
Values
[1,1,1]
=> [1,1]
=> [1]
=> ? = 0
[2,2]
=> [2]
=> []
=> ? = 0
[2,1,1]
=> [1,1]
=> [1]
=> ? = 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,2]
=> [2]
=> []
=> ? = 0
[3,1,1]
=> [1,1]
=> [1]
=> ? = 0
[2,2,1]
=> [2,1]
=> [1]
=> ? = 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[4,2]
=> [2]
=> []
=> ? = 0
[4,1,1]
=> [1,1]
=> [1]
=> ? = 0
[3,3]
=> [3]
=> []
=> ? = 0
[3,2,1]
=> [2,1]
=> [1]
=> ? = 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,2,2]
=> [2,2]
=> [2]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[5,2]
=> [2]
=> []
=> ? = 0
[5,1,1]
=> [1,1]
=> [1]
=> ? = 0
[4,3]
=> [3]
=> []
=> ? = 0
[4,2,1]
=> [2,1]
=> [1]
=> ? = 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,3,1]
=> [3,1]
=> [1]
=> ? = 0
[3,2,2]
=> [2,2]
=> [2]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[6,2]
=> [2]
=> []
=> ? = 0
[6,1,1]
=> [1,1]
=> [1]
=> ? = 0
[5,3]
=> [3]
=> []
=> ? = 0
[5,2,1]
=> [2,1]
=> [1]
=> ? = 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,4]
=> [4]
=> []
=> ? = 0
[4,3,1]
=> [3,1]
=> [1]
=> ? = 0
[4,2,2]
=> [2,2]
=> [2]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[3,3,2]
=> [3,2]
=> [2]
=> 2
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[7,2]
=> [2]
=> []
=> ? = 0
[7,1,1]
=> [1,1]
=> [1]
=> ? = 0
[6,3]
=> [3]
=> []
=> ? = 0
[6,2,1]
=> [2,1]
=> [1]
=> ? = 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[5,4]
=> [4]
=> []
=> ? = 0
[5,3,1]
=> [3,1]
=> [1]
=> ? = 0
[5,2,2]
=> [2,2]
=> [2]
=> 2
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[4,4,1]
=> [4,1]
=> [1]
=> ? = 0
[4,3,2]
=> [3,2]
=> [2]
=> 2
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[8,2]
=> [2]
=> []
=> ? = 0
[8,1,1]
=> [1,1]
=> [1]
=> ? = 0
[7,3]
=> [3]
=> []
=> ? = 0
[7,2,1]
=> [2,1]
=> [1]
=> ? = 0
[7,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[6,4]
=> [4]
=> []
=> ? = 0
[6,3,1]
=> [3,1]
=> [1]
=> ? = 0
[6,2,2]
=> [2,2]
=> [2]
=> 2
[6,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[5,5]
=> [5]
=> []
=> ? = 0
[5,4,1]
=> [4,1]
=> [1]
=> ? = 0
[5,3,2]
=> [3,2]
=> [2]
=> 2
[5,3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[5,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[9,2]
=> [2]
=> []
=> ? = 0
[9,1,1]
=> [1,1]
=> [1]
=> ? = 0
[8,3]
=> [3]
=> []
=> ? = 0
[8,2,1]
=> [2,1]
=> [1]
=> ? = 0
[8,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[7,4]
=> [4]
=> []
=> ? = 0
[7,3,1]
=> [3,1]
=> [1]
=> ? = 0
[7,2,2]
=> [2,2]
=> [2]
=> 2
[7,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[7,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[6,5]
=> [5]
=> []
=> ? = 0
[6,4,1]
=> [4,1]
=> [1]
=> ? = 0
[6,3,2]
=> [3,2]
=> [2]
=> 2
[6,3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[6,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[6,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[10,2]
=> [2]
=> []
=> ? = 0
[10,1,1]
=> [1,1]
=> [1]
=> ? = 0
[9,3]
=> [3]
=> []
=> ? = 0
[9,2,1]
=> [2,1]
=> [1]
=> ? = 0
[8,4]
=> [4]
=> []
=> ? = 0
[8,3,1]
=> [3,1]
=> [1]
=> ? = 0
Description
The number of positive values of the symmetric group character corresponding to the partition. For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
Matching statistic: St000678
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St000678: Dyck paths ⟶ ℤResult quality: 53% values known / values provided: 53%distinct values known / distinct values provided: 75%
Values
[1,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> ? = 0
[2,2]
=> [2]
=> []
=> []
=> ? = 0
[2,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> ? = 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,2]
=> [2]
=> []
=> []
=> ? = 0
[3,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> ? = 0
[2,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> ? = 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[4,2]
=> [2]
=> []
=> []
=> ? = 0
[4,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> ? = 0
[3,3]
=> [3]
=> []
=> []
=> ? = 0
[3,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> ? = 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
[5,2]
=> [2]
=> []
=> []
=> ? = 0
[5,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> ? = 0
[4,3]
=> [3]
=> []
=> []
=> ? = 0
[4,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> ? = 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,3,1]
=> [3,1]
=> [1]
=> [1,0]
=> ? = 0
[3,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
[6,2]
=> [2]
=> []
=> []
=> ? = 0
[6,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> ? = 0
[5,3]
=> [3]
=> []
=> []
=> ? = 0
[5,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> ? = 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[4,4]
=> [4]
=> []
=> []
=> ? = 0
[4,3,1]
=> [3,1]
=> [1]
=> [1,0]
=> ? = 0
[4,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[3,3,2]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
[7,2]
=> [2]
=> []
=> []
=> ? = 0
[7,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> ? = 0
[6,3]
=> [3]
=> []
=> []
=> ? = 0
[6,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> ? = 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[5,4]
=> [4]
=> []
=> []
=> ? = 0
[5,3,1]
=> [3,1]
=> [1]
=> [1,0]
=> ? = 0
[5,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[4,4,1]
=> [4,1]
=> [1]
=> [1,0]
=> ? = 0
[4,3,2]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
[8,2]
=> [2]
=> []
=> []
=> ? = 0
[8,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> ? = 0
[7,3]
=> [3]
=> []
=> []
=> ? = 0
[7,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> ? = 0
[7,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[6,4]
=> [4]
=> []
=> []
=> ? = 0
[6,3,1]
=> [3,1]
=> [1]
=> [1,0]
=> ? = 0
[6,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[6,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[5,5]
=> [5]
=> []
=> []
=> ? = 0
[5,4,1]
=> [4,1]
=> [1]
=> [1,0]
=> ? = 0
[5,3,2]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[5,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[5,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
[9,2]
=> [2]
=> []
=> []
=> ? = 0
[9,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> ? = 0
[8,3]
=> [3]
=> []
=> []
=> ? = 0
[8,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> ? = 0
[8,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[7,4]
=> [4]
=> []
=> []
=> ? = 0
[7,3,1]
=> [3,1]
=> [1]
=> [1,0]
=> ? = 0
[7,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[7,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[7,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[6,5]
=> [5]
=> []
=> []
=> ? = 0
[6,4,1]
=> [4,1]
=> [1]
=> [1,0]
=> ? = 0
[6,3,2]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[6,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[6,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[6,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[10,2]
=> [2]
=> []
=> []
=> ? = 0
[10,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> ? = 0
[9,3]
=> [3]
=> []
=> []
=> ? = 0
[9,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> ? = 0
[8,4]
=> [4]
=> []
=> []
=> ? = 0
[8,3,1]
=> [3,1]
=> [1]
=> [1,0]
=> ? = 0
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000460
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000460: Integer partitions ⟶ ℤResult quality: 42% values known / values provided: 42%distinct values known / distinct values provided: 75%
Values
[1,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[2,2]
=> [2]
=> []
=> ?
=> ? = 0
[2,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[3,2]
=> [2]
=> []
=> ?
=> ? = 0
[3,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[2,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[4,2]
=> [2]
=> []
=> ?
=> ? = 0
[4,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[3,3]
=> [3]
=> []
=> ?
=> ? = 0
[3,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[2,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[5,2]
=> [2]
=> []
=> ?
=> ? = 0
[5,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[4,3]
=> [3]
=> []
=> ?
=> ? = 0
[4,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[3,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[3,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[6,2]
=> [2]
=> []
=> ?
=> ? = 0
[6,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[5,3]
=> [3]
=> []
=> ?
=> ? = 0
[5,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[4,4]
=> [4]
=> []
=> ?
=> ? = 0
[4,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[4,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[3,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 2
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[7,2]
=> [2]
=> []
=> ?
=> ? = 0
[7,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[6,3]
=> [3]
=> []
=> ?
=> ? = 0
[6,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[5,4]
=> [4]
=> []
=> ?
=> ? = 0
[5,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[5,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[4,4,1]
=> [4,1]
=> [1]
=> []
=> ? = 0
[4,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 2
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[8,2]
=> [2]
=> []
=> ?
=> ? = 0
[8,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[7,3]
=> [3]
=> []
=> ?
=> ? = 0
[7,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[7,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[6,4]
=> [4]
=> []
=> ?
=> ? = 0
[6,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[6,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[6,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[5,5]
=> [5]
=> []
=> ?
=> ? = 0
[5,4,1]
=> [4,1]
=> [1]
=> []
=> ? = 0
[5,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 2
[5,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[5,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[9,2]
=> [2]
=> []
=> ?
=> ? = 0
[9,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[8,3]
=> [3]
=> []
=> ?
=> ? = 0
[8,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[8,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[7,4]
=> [4]
=> []
=> ?
=> ? = 0
[7,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[7,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[7,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[6,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[6,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[6,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[6,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[9,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[8,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[8,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[7,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[7,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[7,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[7,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[10,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[9,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000870
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000870: Integer partitions ⟶ ℤResult quality: 42% values known / values provided: 42%distinct values known / distinct values provided: 75%
Values
[1,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[2,2]
=> [2]
=> []
=> ?
=> ? = 0
[2,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[3,2]
=> [2]
=> []
=> ?
=> ? = 0
[3,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[2,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[4,2]
=> [2]
=> []
=> ?
=> ? = 0
[4,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[3,3]
=> [3]
=> []
=> ?
=> ? = 0
[3,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[2,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[5,2]
=> [2]
=> []
=> ?
=> ? = 0
[5,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[4,3]
=> [3]
=> []
=> ?
=> ? = 0
[4,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[3,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[3,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[6,2]
=> [2]
=> []
=> ?
=> ? = 0
[6,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[5,3]
=> [3]
=> []
=> ?
=> ? = 0
[5,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[4,4]
=> [4]
=> []
=> ?
=> ? = 0
[4,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[4,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[3,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 2
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[7,2]
=> [2]
=> []
=> ?
=> ? = 0
[7,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[6,3]
=> [3]
=> []
=> ?
=> ? = 0
[6,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[5,4]
=> [4]
=> []
=> ?
=> ? = 0
[5,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[5,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[4,4,1]
=> [4,1]
=> [1]
=> []
=> ? = 0
[4,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 2
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[8,2]
=> [2]
=> []
=> ?
=> ? = 0
[8,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[7,3]
=> [3]
=> []
=> ?
=> ? = 0
[7,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[7,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[6,4]
=> [4]
=> []
=> ?
=> ? = 0
[6,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[6,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[6,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[5,5]
=> [5]
=> []
=> ?
=> ? = 0
[5,4,1]
=> [4,1]
=> [1]
=> []
=> ? = 0
[5,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 2
[5,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[5,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[9,2]
=> [2]
=> []
=> ?
=> ? = 0
[9,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[8,3]
=> [3]
=> []
=> ?
=> ? = 0
[8,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[8,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[7,4]
=> [4]
=> []
=> ?
=> ? = 0
[7,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[7,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[7,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[6,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[6,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[6,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[6,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[9,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[8,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[8,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[7,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[7,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[7,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[7,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[10,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[9,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
Description
The product of the hook lengths of the diagonal cells in an integer partition. For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells $(i,i)$ of a partition.
Matching statistic: St001247
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St001247: Integer partitions ⟶ ℤResult quality: 42% values known / values provided: 42%distinct values known / distinct values provided: 75%
Values
[1,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[2,2]
=> [2]
=> []
=> ?
=> ? = 0
[2,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[3,2]
=> [2]
=> []
=> ?
=> ? = 0
[3,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[2,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[4,2]
=> [2]
=> []
=> ?
=> ? = 0
[4,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[3,3]
=> [3]
=> []
=> ?
=> ? = 0
[3,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[2,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[5,2]
=> [2]
=> []
=> ?
=> ? = 0
[5,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[4,3]
=> [3]
=> []
=> ?
=> ? = 0
[4,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[3,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[3,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[6,2]
=> [2]
=> []
=> ?
=> ? = 0
[6,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[5,3]
=> [3]
=> []
=> ?
=> ? = 0
[5,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[4,4]
=> [4]
=> []
=> ?
=> ? = 0
[4,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[4,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[3,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 2
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[7,2]
=> [2]
=> []
=> ?
=> ? = 0
[7,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[6,3]
=> [3]
=> []
=> ?
=> ? = 0
[6,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[5,4]
=> [4]
=> []
=> ?
=> ? = 0
[5,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[5,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[4,4,1]
=> [4,1]
=> [1]
=> []
=> ? = 0
[4,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 2
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[8,2]
=> [2]
=> []
=> ?
=> ? = 0
[8,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[7,3]
=> [3]
=> []
=> ?
=> ? = 0
[7,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[7,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[6,4]
=> [4]
=> []
=> ?
=> ? = 0
[6,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[6,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[6,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[5,5]
=> [5]
=> []
=> ?
=> ? = 0
[5,4,1]
=> [4,1]
=> [1]
=> []
=> ? = 0
[5,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 2
[5,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[5,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[9,2]
=> [2]
=> []
=> ?
=> ? = 0
[9,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[8,3]
=> [3]
=> []
=> ?
=> ? = 0
[8,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[8,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[7,4]
=> [4]
=> []
=> ?
=> ? = 0
[7,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[7,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[7,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[6,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[6,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[6,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[6,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[9,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[8,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[8,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[7,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[7,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[7,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[7,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[10,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[9,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
Description
The number of parts of a partition that are not congruent 2 modulo 3.
Matching statistic: St001249
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St001249: Integer partitions ⟶ ℤResult quality: 42% values known / values provided: 42%distinct values known / distinct values provided: 75%
Values
[1,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[2,2]
=> [2]
=> []
=> ?
=> ? = 0
[2,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[3,2]
=> [2]
=> []
=> ?
=> ? = 0
[3,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[2,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[4,2]
=> [2]
=> []
=> ?
=> ? = 0
[4,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[3,3]
=> [3]
=> []
=> ?
=> ? = 0
[3,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[2,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[5,2]
=> [2]
=> []
=> ?
=> ? = 0
[5,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[4,3]
=> [3]
=> []
=> ?
=> ? = 0
[4,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[3,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[3,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[6,2]
=> [2]
=> []
=> ?
=> ? = 0
[6,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[5,3]
=> [3]
=> []
=> ?
=> ? = 0
[5,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[4,4]
=> [4]
=> []
=> ?
=> ? = 0
[4,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[4,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[3,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 2
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[7,2]
=> [2]
=> []
=> ?
=> ? = 0
[7,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[6,3]
=> [3]
=> []
=> ?
=> ? = 0
[6,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[5,4]
=> [4]
=> []
=> ?
=> ? = 0
[5,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[5,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[4,4,1]
=> [4,1]
=> [1]
=> []
=> ? = 0
[4,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 2
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[8,2]
=> [2]
=> []
=> ?
=> ? = 0
[8,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[7,3]
=> [3]
=> []
=> ?
=> ? = 0
[7,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[7,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[6,4]
=> [4]
=> []
=> ?
=> ? = 0
[6,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[6,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[6,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[5,5]
=> [5]
=> []
=> ?
=> ? = 0
[5,4,1]
=> [4,1]
=> [1]
=> []
=> ? = 0
[5,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 2
[5,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[5,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[9,2]
=> [2]
=> []
=> ?
=> ? = 0
[9,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[8,3]
=> [3]
=> []
=> ?
=> ? = 0
[8,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[8,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[7,4]
=> [4]
=> []
=> ?
=> ? = 0
[7,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[7,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[7,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[6,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[6,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[6,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[6,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[9,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[8,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[8,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[7,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[7,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[7,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[7,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[10,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[9,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
Description
Sum of the odd parts of a partition.
Matching statistic: St001250
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St001250: Integer partitions ⟶ ℤResult quality: 42% values known / values provided: 42%distinct values known / distinct values provided: 75%
Values
[1,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[2,2]
=> [2]
=> []
=> ?
=> ? = 0
[2,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[3,2]
=> [2]
=> []
=> ?
=> ? = 0
[3,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[2,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[4,2]
=> [2]
=> []
=> ?
=> ? = 0
[4,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[3,3]
=> [3]
=> []
=> ?
=> ? = 0
[3,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[2,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[5,2]
=> [2]
=> []
=> ?
=> ? = 0
[5,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[4,3]
=> [3]
=> []
=> ?
=> ? = 0
[4,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[3,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[3,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[6,2]
=> [2]
=> []
=> ?
=> ? = 0
[6,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[5,3]
=> [3]
=> []
=> ?
=> ? = 0
[5,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[4,4]
=> [4]
=> []
=> ?
=> ? = 0
[4,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[4,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[3,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 2
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[7,2]
=> [2]
=> []
=> ?
=> ? = 0
[7,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[6,3]
=> [3]
=> []
=> ?
=> ? = 0
[6,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[5,4]
=> [4]
=> []
=> ?
=> ? = 0
[5,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[5,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[4,4,1]
=> [4,1]
=> [1]
=> []
=> ? = 0
[4,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 2
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[8,2]
=> [2]
=> []
=> ?
=> ? = 0
[8,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[7,3]
=> [3]
=> []
=> ?
=> ? = 0
[7,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[7,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[6,4]
=> [4]
=> []
=> ?
=> ? = 0
[6,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[6,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[6,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[5,5]
=> [5]
=> []
=> ?
=> ? = 0
[5,4,1]
=> [4,1]
=> [1]
=> []
=> ? = 0
[5,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 2
[5,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[5,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[9,2]
=> [2]
=> []
=> ?
=> ? = 0
[9,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[8,3]
=> [3]
=> []
=> ?
=> ? = 0
[8,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[8,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[7,4]
=> [4]
=> []
=> ?
=> ? = 0
[7,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[7,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[7,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[6,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[6,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[6,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[6,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[9,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[8,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[8,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[7,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[7,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[7,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[7,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[10,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[9,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
Description
The number of parts of a partition that are not congruent 0 modulo 3.
Matching statistic: St001360
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St001360: Integer partitions ⟶ ℤResult quality: 42% values known / values provided: 42%distinct values known / distinct values provided: 75%
Values
[1,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[2,2]
=> [2]
=> []
=> ?
=> ? = 0
[2,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[3,2]
=> [2]
=> []
=> ?
=> ? = 0
[3,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[2,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[4,2]
=> [2]
=> []
=> ?
=> ? = 0
[4,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[3,3]
=> [3]
=> []
=> ?
=> ? = 0
[3,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[2,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[5,2]
=> [2]
=> []
=> ?
=> ? = 0
[5,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[4,3]
=> [3]
=> []
=> ?
=> ? = 0
[4,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[3,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[3,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[6,2]
=> [2]
=> []
=> ?
=> ? = 0
[6,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[5,3]
=> [3]
=> []
=> ?
=> ? = 0
[5,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[4,4]
=> [4]
=> []
=> ?
=> ? = 0
[4,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[4,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[3,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 2
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[7,2]
=> [2]
=> []
=> ?
=> ? = 0
[7,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[6,3]
=> [3]
=> []
=> ?
=> ? = 0
[6,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[5,4]
=> [4]
=> []
=> ?
=> ? = 0
[5,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[5,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[4,4,1]
=> [4,1]
=> [1]
=> []
=> ? = 0
[4,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 2
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[8,2]
=> [2]
=> []
=> ?
=> ? = 0
[8,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[7,3]
=> [3]
=> []
=> ?
=> ? = 0
[7,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[7,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[6,4]
=> [4]
=> []
=> ?
=> ? = 0
[6,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[6,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[6,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[5,5]
=> [5]
=> []
=> ?
=> ? = 0
[5,4,1]
=> [4,1]
=> [1]
=> []
=> ? = 0
[5,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 2
[5,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[5,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[9,2]
=> [2]
=> []
=> ?
=> ? = 0
[9,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[8,3]
=> [3]
=> []
=> ?
=> ? = 0
[8,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[8,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[7,4]
=> [4]
=> []
=> ?
=> ? = 0
[7,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[7,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[7,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[6,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[6,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[6,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[6,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[9,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[8,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[8,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[7,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[7,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[7,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[7,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[10,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[9,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
Description
The number of covering relations in Young's lattice below a partition.
Matching statistic: St001380
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St001380: Integer partitions ⟶ ℤResult quality: 42% values known / values provided: 42%distinct values known / distinct values provided: 75%
Values
[1,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[2,2]
=> [2]
=> []
=> ?
=> ? = 0
[2,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[3,2]
=> [2]
=> []
=> ?
=> ? = 0
[3,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[2,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[4,2]
=> [2]
=> []
=> ?
=> ? = 0
[4,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[3,3]
=> [3]
=> []
=> ?
=> ? = 0
[3,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[2,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[5,2]
=> [2]
=> []
=> ?
=> ? = 0
[5,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[4,3]
=> [3]
=> []
=> ?
=> ? = 0
[4,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[3,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[3,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[6,2]
=> [2]
=> []
=> ?
=> ? = 0
[6,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[5,3]
=> [3]
=> []
=> ?
=> ? = 0
[5,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[4,4]
=> [4]
=> []
=> ?
=> ? = 0
[4,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[4,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[3,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 2
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[7,2]
=> [2]
=> []
=> ?
=> ? = 0
[7,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[6,3]
=> [3]
=> []
=> ?
=> ? = 0
[6,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[5,4]
=> [4]
=> []
=> ?
=> ? = 0
[5,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[5,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[4,4,1]
=> [4,1]
=> [1]
=> []
=> ? = 0
[4,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 2
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[8,2]
=> [2]
=> []
=> ?
=> ? = 0
[8,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[7,3]
=> [3]
=> []
=> ?
=> ? = 0
[7,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[7,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[6,4]
=> [4]
=> []
=> ?
=> ? = 0
[6,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[6,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 2
[6,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[5,5]
=> [5]
=> []
=> ?
=> ? = 0
[5,4,1]
=> [4,1]
=> [1]
=> []
=> ? = 0
[5,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 2
[5,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[5,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[9,2]
=> [2]
=> []
=> ?
=> ? = 0
[9,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[8,3]
=> [3]
=> []
=> ?
=> ? = 0
[8,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 0
[8,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[7,4]
=> [4]
=> []
=> ?
=> ? = 0
[7,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 0
[7,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[7,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[6,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[6,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[6,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[6,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[9,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[8,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[8,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[7,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[7,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[7,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[7,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[10,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[9,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
Description
The number of monomer-dimer tilings of a Ferrers diagram. For a hook of length $n$, this is the $n$-th Fibonacci number.
The following 16 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001176The size of a partition minus its first part. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000782The indicator function of whether a given perfect matching is an L & P matching. St001722The number of minimal chains with small intervals between a binary word and the top element. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even. St000371The number of mid points of decreasing subsequences of length 3 in a permutation.