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Matching statistic: St001839
St001839: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> 0
{{1,2}}
=> 0
{{1},{2}}
=> 0
{{1,2,3}}
=> 0
{{1,2},{3}}
=> 0
{{1,3},{2}}
=> 1
{{1},{2,3}}
=> 0
{{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> 0
{{1,2,4},{3}}
=> 1
{{1,2},{3,4}}
=> 0
{{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> 1
{{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> 1
{{1,4},{2,3}}
=> 1
{{1},{2,3,4}}
=> 0
{{1},{2,3},{4}}
=> 0
{{1,4},{2},{3}}
=> 1
{{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> 0
{{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> 0
{{1,2,3,5},{4}}
=> 1
{{1,2,3},{4,5}}
=> 0
{{1,2,3},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> 1
{{1,2,4},{3,5}}
=> 1
{{1,2,4},{3},{5}}
=> 1
{{1,2,5},{3,4}}
=> 2
{{1,2},{3,4,5}}
=> 0
{{1,2},{3,4},{5}}
=> 0
{{1,2,5},{3},{4}}
=> 2
{{1,2},{3,5},{4}}
=> 1
{{1,2},{3},{4,5}}
=> 0
{{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> 1
{{1,3,4},{2,5}}
=> 1
{{1,3,4},{2},{5}}
=> 1
{{1,3,5},{2,4}}
=> 1
{{1,3},{2,4,5}}
=> 1
{{1,3},{2,4},{5}}
=> 1
{{1,3,5},{2},{4}}
=> 1
{{1,3},{2,5},{4}}
=> 1
{{1,3},{2},{4,5}}
=> 1
{{1,3},{2},{4},{5}}
=> 1
{{1,4,5},{2,3}}
=> 1
{{1,4},{2,3,5}}
=> 1
Description
The number of excedances of a set partition.
The Mahonian representation of a set partition $\{B_1,\dots,B_k\}$ of $\{1,\dots,n\}$ is the restricted growth word $w_1 \dots w_n$ obtained by sorting the blocks of the set partition according to their maximal element, and setting $w_i$ to the index of the block containing $i$.
Let $\bar w$ be the nondecreasing rearrangement of $w$.
The word $w$ has an excedance at position $i$ if $w_i > \bar w_i$.
Matching statistic: St001823
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001823: Signed permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001823: Signed permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Values
{{1}}
=> [1] => [1] => [1] => 0
{{1,2}}
=> [2,1] => [1,2] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [1,3,2] => 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [1,3,4,2] => 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [1,4,2,3] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [1,4,2,3] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => [1,3,4,5,2] => ? = 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => [1,3,4,2,5] => ? = 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => [1,3,4,2,5] => ? = 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => [1,3,5,2,4] => ? = 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => [1,3,5,2,4] => ? = 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => [1,3,2,5,4] => ? = 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => [1,4,5,2,3] => ? = 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 1
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [1,4,5,2,3] => [1,4,5,2,3] => ? = 1
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 2
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 2
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 1
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [1,5,2,4,3] => [1,5,2,4,3] => ? = 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1
Description
The Stasinski-Voll length of a signed permutation.
The Stasinski-Voll length of a signed permutation $\sigma$ is
$$
L(\sigma) = \frac{1}{2} \#\{(i,j) ~\mid -n \leq i < j \leq n,~ i \not\equiv j \operatorname{mod} 2,~ \sigma(i) > \sigma(j)\},
$$
where $n$ is the size of $\sigma$.
Matching statistic: St001946
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001946: Parking functions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001946: Parking functions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Values
{{1}}
=> [1] => [1] => [1] => 0
{{1,2}}
=> [2,1] => [1,2] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [1,3,2] => 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [1,3,4,2] => 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [1,4,2,3] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [1,4,2,3] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => [1,3,4,5,2] => ? = 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => [1,3,4,2,5] => ? = 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => [1,3,4,2,5] => ? = 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => [1,3,5,2,4] => ? = 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => [1,3,5,2,4] => ? = 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => [1,3,2,5,4] => ? = 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => [1,4,5,2,3] => ? = 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 1
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [1,4,5,2,3] => [1,4,5,2,3] => ? = 1
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 2
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 2
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 1
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [1,5,2,4,3] => [1,5,2,4,3] => ? = 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1
Description
The number of descents in a parking function.
This is the number of indices $i$ such that $p_i > p_{i+1}$.
Matching statistic: St001582
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Values
{{1}}
=> [1] => [1] => [1] => ? = 0
{{1,2}}
=> [2,1] => [1,2] => [2,1] => 0
{{1},{2}}
=> [1,2] => [1,2] => [2,1] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [3,2,1] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => [3,2,1] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [2,3,1] => 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => [3,2,1] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [3,2,1] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => [4,3,2,1] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [3,4,2,1] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [4,3,2,1] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [2,4,3,1] => 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [4,2,3,1] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [3,2,4,1] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => [4,3,2,1] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [3,2,4,1] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [3,4,2,1] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [4,5,3,2,1] => ? = 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => [3,5,4,2,1] => ? = 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => [5,3,4,2,1] => ? = 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => [5,3,4,2,1] => ? = 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => [4,3,5,2,1] => ? = 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => [4,3,5,2,1] => ? = 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => ? = 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => [2,5,4,3,1] => ? = 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => [5,2,4,3,1] => ? = 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => [5,2,4,3,1] => ? = 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => [4,2,5,3,1] => ? = 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => [5,4,2,3,1] => ? = 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => [5,4,2,3,1] => ? = 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => [4,2,5,3,1] => ? = 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => [4,5,2,3,1] => ? = 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => [5,4,2,3,1] => ? = 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => ? = 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => [3,2,5,4,1] => ? = 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => [5,3,2,4,1] => ? = 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,4,2,3,5] => [5,3,2,4,1] => ? = 1
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [1,5,2,3,4] => [4,3,2,5,1] => ? = 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [1,5,2,3,4] => [4,3,2,5,1] => ? = 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,2,3,5,4] => [4,5,3,2,1] => ? = 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [1,4,5,2,3] => [3,2,5,4,1] => ? = 1
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [1,4,2,5,3] => [3,5,2,4,1] => ? = 2
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [1,4,2,3,5] => [5,3,2,4,1] => ? = 2
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [1,4,2,3,5] => [5,3,2,4,1] => ? = 1
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [1,5,2,4,3] => [3,4,2,5,1] => ? = 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,2,4,5,3] => [3,5,4,2,1] => ? = 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,2,4,3,5] => [5,3,4,2,1] => ? = 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => ? = 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [1,5,2,3,4] => [4,3,2,5,1] => ? = 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,2,5,3,4] => [4,3,5,2,1] => ? = 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,5,2,3,4] => [4,3,2,5,1] => ? = 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,2,5,3,4] => [4,3,5,2,1] => ? = 1
Description
The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
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