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Your data matches 83 different statistics following compositions of up to 3 maps.
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Matching statistic: St000939
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1],[2,3]]
=> [2,2]
=> [2]
=> 1
[[1,1],[3,3]]
=> [2,2]
=> [2]
=> 1
[[1,2],[2,3]]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,3]]
=> [2,2]
=> [2]
=> 1
[[2,2],[3,3]]
=> [2,2]
=> [2]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,1,1],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[2,2],[3,4]]
=> [2,2]
=> [2]
=> 1
[[2,2],[4,4]]
=> [2,2]
=> [2]
=> 1
[[2,3],[3,4]]
=> [2,2]
=> [2]
=> 1
[[2,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[3,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1,1,1],[2,3]]
=> [3,2]
=> [2]
=> 1
[[1,1,1],[3,3]]
=> [3,2]
=> [2]
=> 1
Description
The number of characters of the symmetric group whose value on the partition is positive.
Matching statistic: St001866
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001866: Signed permutations ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 60%
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001866: Signed permutations ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 60%
Values
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[[1],[2],[4]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1],[3],[4]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1 = 2 - 1
[[2],[3],[4]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1,1],[2,3]]
=> [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[[1,1],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[[1,2],[2,3]]
=> [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 0 = 1 - 1
[[1,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[[2,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [2,4,3,1] => [2,4,3,1] => 1 = 2 - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 1 = 2 - 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 0 = 1 - 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[[1],[2],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1],[3],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1],[4],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1 = 2 - 1
[[2],[3],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1 = 2 - 1
[[2],[4],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1 = 2 - 1
[[3],[4],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1,1],[2,4]]
=> [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[[1,1],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[[1,1],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[[1,2],[2,4]]
=> [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 0 = 1 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 0 = 1 - 1
[[1,2],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[[1,3],[3,4]]
=> [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 0 = 1 - 1
[[1,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[[2,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[[2,2],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[[2,3],[3,4]]
=> [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 0 = 1 - 1
[[2,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[[3,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [2,4,3,1] => [2,4,3,1] => 1 = 2 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,3,1] => [2,4,3,1] => 1 = 2 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 1 = 2 - 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 1 = 2 - 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [2,4,3,1] => [2,4,3,1] => 1 = 2 - 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 1 = 2 - 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [2,4,3,1] => [2,4,3,1] => 1 = 2 - 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 2 = 3 - 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 0 = 1 - 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 0 = 1 - 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 0 = 1 - 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1 - 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 1 - 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 2 - 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 2 - 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 2 - 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 1 - 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,5,1,3,2] => [4,5,1,3,2] => ? = 1 - 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 1 - 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [1,2,5,6,3,4] => [1,2,5,6,3,4] => ? = 1 - 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [1,4,5,6,2,3] => [1,4,5,6,2,3] => ? = 1 - 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [3,4,5,6,1,2] => [3,4,5,6,1,2] => ? = 1 - 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? = 2 - 1
[[1,1,4],[2,2]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[[1,1,4],[2,3]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[[1,1,4],[2,4]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[[1,1,4],[3,3]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[[1,1,4],[3,4]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[[1,1,4],[4,4]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[[1,2,2],[2,4]]
=> [2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1 - 1
[[1,2,3],[2,4]]
=> [2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1 - 1
[[1,2,4],[2,3]]
=> [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 1 - 1
[[1,2,4],[2,4]]
=> [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 1 - 1
[[1,2,4],[3,3]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[[1,3,3],[2,4]]
=> [2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1 - 1
[[1,2,4],[3,4]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[[1,3,4],[2,4]]
=> [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 1 - 1
[[1,2,4],[4,4]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[[1,3,3],[3,4]]
=> [2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1 - 1
[[1,3,4],[3,4]]
=> [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 1 - 1
[[1,3,4],[4,4]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[[2,2,4],[3,3]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[[2,2,4],[3,4]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[[2,2,4],[4,4]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[[2,3,3],[3,4]]
=> [2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1 - 1
[[2,3,4],[3,4]]
=> [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 1 - 1
[[2,3,4],[4,4]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[[3,3,4],[4,4]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[[1,2,2],[2],[4]]
=> [5,2,1,3,4] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 2 - 1
[[1,2,3],[2],[4]]
=> [5,2,1,3,4] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 2 - 1
[[1,2,4],[2],[3]]
=> [4,2,1,3,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 2 - 1
[[1,2,4],[2],[4]]
=> [4,2,1,3,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 2 - 1
[[1,3,3],[2],[4]]
=> [5,2,1,3,4] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 2 - 1
[[1,3,4],[2],[3]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 2 - 1
[[1,3,4],[2],[4]]
=> [4,2,1,3,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 2 - 1
Description
The nesting alignments of a signed permutation.
A nesting alignment of a signed permutation $\pi\in\mathfrak H_n$ is a pair $1\leq i, j \leq n$ such that
* $-i < -j < -\pi(j) < -\pi(i)$, or
* $-i < j \leq \pi(j) < -\pi(i)$, or
* $i < j \leq \pi(j) < \pi(i)$.
Matching statistic: St001896
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001896: Signed permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 60%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001896: Signed permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 60%
Values
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[1,1],[2,2]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 1
[[1],[2],[4]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[1],[3],[4]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[2],[3],[4]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[1,1],[2,3]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 1
[[1,1],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 1
[[1,2],[2,3]]
=> [2,4,1,3] => [1,3,4,2] => [1,3,4,2] => 1
[[1,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 1
[[2,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 2
[[1,2],[2],[3]]
=> [4,2,1,3] => [3,2,4,1] => [3,2,4,1] => 2
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1
[[1],[2],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[1],[3],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[1],[4],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[2],[3],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[2],[4],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[3],[4],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[1,1],[2,4]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 1
[[1,1],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 1
[[1,1],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 1
[[1,2],[2,4]]
=> [2,4,1,3] => [1,3,4,2] => [1,3,4,2] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 1
[[1,3],[2,4]]
=> [2,4,1,3] => [1,3,4,2] => [1,3,4,2] => 1
[[1,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 1
[[1,3],[3,4]]
=> [2,4,1,3] => [1,3,4,2] => [1,3,4,2] => 1
[[1,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 1
[[2,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 1
[[2,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 1
[[2,3],[3,4]]
=> [2,4,1,3] => [1,3,4,2] => [1,3,4,2] => 1
[[2,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 1
[[3,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 2
[[1,1],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 2
[[1,2],[2],[4]]
=> [4,2,1,3] => [3,2,4,1] => [3,2,4,1] => 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,2,4,1] => [3,2,4,1] => 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[[1,4],[2],[4]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[[1,3],[3],[4]]
=> [4,2,1,3] => [3,2,4,1] => [3,2,4,1] => 2
[[1,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[[2,2],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 2
[[2,3],[3],[4]]
=> [4,2,1,3] => [3,2,4,1] => [3,2,4,1] => 2
[[2,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [3,1,4,5,2] => [3,1,4,5,2] => ? = 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [1,3,4,5,2] => [1,3,4,5,2] => 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [1,3,4,2,5] => [1,3,4,2,5] => 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 2
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [3,4,2,5,1] => [3,4,2,5,1] => ? = 2
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [3,4,2,1,5] => [3,4,2,1,5] => ? = 2
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 2
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [3,2,4,1,5] => [3,2,4,1,5] => ? = 2
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 2
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [4,2,5,3,1] => [4,2,5,3,1] => ? = 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,2,1,5,3] => [4,2,1,5,3] => ? = 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [3,4,5,1,6,2] => [3,4,5,1,6,2] => ? = 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [3,4,1,5,2,6] => [3,4,1,5,2,6] => ? = 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [3,1,4,2,5,6] => [3,1,4,2,5,6] => ? = 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => [4,1,5,2,6,3] => ? = 2
[[1],[2],[6]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[1],[3],[6]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[1,1,1],[2,4]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 1
[[1,1,1],[3,4]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 1
[[1,1,1],[4,4]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 1
[[1,1,2],[2,4]]
=> [3,5,1,2,4] => [3,1,4,5,2] => [3,1,4,5,2] => ? = 1
[[1,1,4],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1
[[1,1,2],[3,4]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 1
[[1,1,3],[2,4]]
=> [3,5,1,2,4] => [3,1,4,5,2] => [3,1,4,5,2] => ? = 1
[[1,1,4],[2,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1
[[1,1,2],[4,4]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 1
[[1,1,4],[2,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1
[[1,1,3],[3,4]]
=> [3,5,1,2,4] => [3,1,4,5,2] => [3,1,4,5,2] => ? = 1
[[1,1,4],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1
[[1,1,3],[4,4]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 1
[[1,1,4],[3,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1
[[1,1,4],[4,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1
[[1,2,2],[3,4]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 1
[[1,2,2],[4,4]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 1
[[1,2,3],[3,4]]
=> [3,5,1,2,4] => [3,1,4,5,2] => [3,1,4,5,2] => ? = 1
[[1,2,4],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1
[[1,2,3],[4,4]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 1
[[1,2,4],[3,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1
[[1,2,4],[4,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1
[[1,3,3],[4,4]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 1
[[1,3,4],[4,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1
Description
The number of right descents of a signed permutations.
An index is a right descent if it is a left descent of the inverse signed permutation.
Matching statistic: St001946
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001946: Parking functions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 60%
Mp00305: Permutations —parking function⟶ Parking functions
St001946: Parking functions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 60%
Values
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 2
[[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => 1
[[1],[2],[4]]
=> [3,2,1] => [3,2,1] => 2
[[1],[3],[4]]
=> [3,2,1] => [3,2,1] => 2
[[2],[3],[4]]
=> [3,2,1] => [3,2,1] => 2
[[1,1],[2,3]]
=> [3,4,1,2] => [3,4,1,2] => 1
[[1,1],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 1
[[1,2],[2,3]]
=> [2,4,1,3] => [2,4,1,3] => 1
[[1,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 1
[[2,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [4,3,1,2] => 2
[[1,2],[2],[3]]
=> [4,2,1,3] => [4,2,1,3] => 2
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1
[[1],[2],[5]]
=> [3,2,1] => [3,2,1] => 2
[[1],[3],[5]]
=> [3,2,1] => [3,2,1] => 2
[[1],[4],[5]]
=> [3,2,1] => [3,2,1] => 2
[[2],[3],[5]]
=> [3,2,1] => [3,2,1] => 2
[[2],[4],[5]]
=> [3,2,1] => [3,2,1] => 2
[[3],[4],[5]]
=> [3,2,1] => [3,2,1] => 2
[[1,1],[2,4]]
=> [3,4,1,2] => [3,4,1,2] => 1
[[1,1],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => 1
[[1,1],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => 1
[[1,2],[2,4]]
=> [2,4,1,3] => [2,4,1,3] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => 1
[[1,3],[2,4]]
=> [2,4,1,3] => [2,4,1,3] => 1
[[1,2],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => 1
[[1,3],[3,4]]
=> [2,4,1,3] => [2,4,1,3] => 1
[[1,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => 1
[[2,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => 1
[[2,2],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => 1
[[2,3],[3,4]]
=> [2,4,1,3] => [2,4,1,3] => 1
[[2,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => 1
[[3,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [4,3,1,2] => 2
[[1,1],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => 2
[[1,2],[2],[4]]
=> [4,2,1,3] => [4,2,1,3] => 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [4,2,1,3] => 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 2
[[1,4],[2],[4]]
=> [3,2,1,4] => [3,2,1,4] => 2
[[1,3],[3],[4]]
=> [4,2,1,3] => [4,2,1,3] => 2
[[1,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => 2
[[2,2],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => 2
[[2,3],[3],[4]]
=> [4,2,1,3] => [4,2,1,3] => 2
[[2,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 3
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [3,5,1,2,4] => ? = 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [2,5,1,3,4] => ? = 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [2,4,1,3,5] => ? = 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [5,4,1,2,3] => ? = 2
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [5,3,1,2,4] => ? = 2
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [4,3,1,2,5] => ? = 2
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [5,2,1,3,4] => ? = 2
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [4,2,1,3,5] => ? = 2
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => ? = 2
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [5,3,4,1,2] => ? = 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,3,5,1,2] => ? = 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [4,2,5,1,3] => ? = 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [5,6,1,2,3,4] => ? = 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => ? = 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? = 2
[[1],[2],[6]]
=> [3,2,1] => [3,2,1] => 2
[[1],[3],[6]]
=> [3,2,1] => [3,2,1] => 2
[[1],[4],[6]]
=> [3,2,1] => [3,2,1] => 2
[[1],[5],[6]]
=> [3,2,1] => [3,2,1] => 2
[[1,1,1],[2,4]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 1
[[1,1,1],[3,4]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 1
[[1,1,1],[4,4]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 1
[[1,1,2],[2,4]]
=> [3,5,1,2,4] => [3,5,1,2,4] => ? = 1
[[1,1,4],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1
[[1,1,2],[3,4]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 1
[[1,1,3],[2,4]]
=> [3,5,1,2,4] => [3,5,1,2,4] => ? = 1
[[1,1,4],[2,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1
[[1,1,2],[4,4]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 1
[[1,1,4],[2,4]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1
[[1,1,3],[3,4]]
=> [3,5,1,2,4] => [3,5,1,2,4] => ? = 1
[[1,1,4],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1
[[1,1,3],[4,4]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 1
[[1,1,4],[3,4]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1
[[1,1,4],[4,4]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1
[[1,2,2],[2,4]]
=> [2,5,1,3,4] => [2,5,1,3,4] => ? = 1
[[1,2,2],[3,4]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 1
[[1,2,3],[2,4]]
=> [2,5,1,3,4] => [2,5,1,3,4] => ? = 1
[[1,2,4],[2,3]]
=> [2,4,1,3,5] => [2,4,1,3,5] => ? = 1
[[1,2,2],[4,4]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 1
[[1,2,4],[2,4]]
=> [2,4,1,3,5] => [2,4,1,3,5] => ? = 1
[[1,2,3],[3,4]]
=> [3,5,1,2,4] => [3,5,1,2,4] => ? = 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: St001207
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 60%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 60%
Values
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[1,1],[2,2]]
=> [3,4,1,2] => [2,4,1,3] => [2,1,4,3] => 1
[[1],[2],[4]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[1],[3],[4]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[2],[3],[4]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[1,1],[2,3]]
=> [3,4,1,2] => [2,4,1,3] => [2,1,4,3] => 1
[[1,1],[3,3]]
=> [3,4,1,2] => [2,4,1,3] => [2,1,4,3] => 1
[[1,2],[2,3]]
=> [2,4,1,3] => [2,4,1,3] => [2,1,4,3] => 1
[[1,2],[3,3]]
=> [3,4,1,2] => [2,4,1,3] => [2,1,4,3] => 1
[[2,2],[3,3]]
=> [3,4,1,2] => [2,4,1,3] => [2,1,4,3] => 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [4,3,1,2] => [1,4,3,2] => 2
[[1,2],[2],[3]]
=> [4,2,1,3] => [4,2,1,3] => [2,4,1,3] => 2
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [1,3,2,5,4] => ? = 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,1,4,3,5] => ? = 1
[[1],[2],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[1],[3],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[1],[4],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[2],[3],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[2],[4],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[3],[4],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[1,1],[2,4]]
=> [3,4,1,2] => [2,4,1,3] => [2,1,4,3] => 1
[[1,1],[3,4]]
=> [3,4,1,2] => [2,4,1,3] => [2,1,4,3] => 1
[[1,1],[4,4]]
=> [3,4,1,2] => [2,4,1,3] => [2,1,4,3] => 1
[[1,2],[2,4]]
=> [2,4,1,3] => [2,4,1,3] => [2,1,4,3] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [2,4,1,3] => [2,1,4,3] => 1
[[1,3],[2,4]]
=> [2,4,1,3] => [2,4,1,3] => [2,1,4,3] => 1
[[1,2],[4,4]]
=> [3,4,1,2] => [2,4,1,3] => [2,1,4,3] => 1
[[1,3],[3,4]]
=> [2,4,1,3] => [2,4,1,3] => [2,1,4,3] => 1
[[1,3],[4,4]]
=> [3,4,1,2] => [2,4,1,3] => [2,1,4,3] => 1
[[2,2],[3,4]]
=> [3,4,1,2] => [2,4,1,3] => [2,1,4,3] => 1
[[2,2],[4,4]]
=> [3,4,1,2] => [2,4,1,3] => [2,1,4,3] => 1
[[2,3],[3,4]]
=> [2,4,1,3] => [2,4,1,3] => [2,1,4,3] => 1
[[2,3],[4,4]]
=> [3,4,1,2] => [2,4,1,3] => [2,1,4,3] => 1
[[3,3],[4,4]]
=> [3,4,1,2] => [2,4,1,3] => [2,1,4,3] => 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [4,3,1,2] => [1,4,3,2] => 2
[[1,1],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => [1,4,3,2] => 2
[[1,2],[2],[4]]
=> [4,2,1,3] => [4,2,1,3] => [2,4,1,3] => 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => [1,4,3,2] => 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [4,2,1,3] => [2,4,1,3] => 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[[1,4],[2],[4]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[[1,3],[3],[4]]
=> [4,2,1,3] => [4,2,1,3] => [2,4,1,3] => 2
[[1,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[[2,2],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => [1,4,3,2] => 2
[[2,3],[3],[4]]
=> [4,2,1,3] => [4,2,1,3] => [2,4,1,3] => 2
[[2,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [1,3,2,5,4] => ? = 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [1,3,2,5,4] => ? = 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [3,5,1,2,4] => [1,3,2,5,4] => ? = 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,1,4,3,5] => ? = 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [1,3,2,5,4] => ? = 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,1,4,3,5] => ? = 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,1,4,3,5] => ? = 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [2,5,1,3,4] => [2,1,3,5,4] => ? = 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [1,3,2,5,4] => ? = 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [2,4,1,3,5] => [2,1,4,3,5] => ? = 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,1,4,3,5] => ? = 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [1,3,2,5,4] => ? = 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,1,4,3,5] => ? = 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [5,4,1,2,3] => [1,2,5,4,3] => ? = 2
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [5,3,1,2,4] => [1,3,5,2,4] => ? = 2
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [1,4,3,2,5] => ? = 2
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [5,2,1,3,4] => [2,1,5,3,4] => ? = 2
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [2,4,1,3,5] => ? = 2
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 2
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [5,2,4,1,3] => [2,1,5,4,3] => ? = 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [3,2,5,1,4] => [3,2,1,5,4] => ? = 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [3,2,5,1,4] => [3,2,1,5,4] => ? = 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [4,6,1,2,3,5] => [1,2,4,3,6,5] => ? = 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [3,5,1,2,4,6] => [1,3,2,5,4,6] => ? = 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [2,4,1,3,5,6] => [2,1,4,3,5,6] => ? = 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [2,4,6,1,3,5] => [2,1,4,3,6,5] => ? = 2
[[1],[2],[6]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[1],[3],[6]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[1],[4],[6]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[1],[5],[6]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[1,1,1],[2,4]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [1,3,2,5,4] => ? = 1
[[1,1,1],[3,4]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [1,3,2,5,4] => ? = 1
[[1,1,1],[4,4]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [1,3,2,5,4] => ? = 1
[[1,1,2],[2,4]]
=> [3,5,1,2,4] => [3,5,1,2,4] => [1,3,2,5,4] => ? = 1
[[1,1,4],[2,2]]
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,1,4,3,5] => ? = 1
[[1,1,2],[3,4]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [1,3,2,5,4] => ? = 1
[[1,1,3],[2,4]]
=> [3,5,1,2,4] => [3,5,1,2,4] => [1,3,2,5,4] => ? = 1
[[1,1,4],[2,3]]
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,1,4,3,5] => ? = 1
[[1,1,2],[4,4]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [1,3,2,5,4] => ? = 1
[[1,1,4],[2,4]]
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,1,4,3,5] => ? = 1
[[1,1,3],[3,4]]
=> [3,5,1,2,4] => [3,5,1,2,4] => [1,3,2,5,4] => ? = 1
[[1,1,4],[3,3]]
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,1,4,3,5] => ? = 1
[[1,1,3],[4,4]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [1,3,2,5,4] => ? = 1
[[1,1,4],[3,4]]
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,1,4,3,5] => ? = 1
[[1,1,4],[4,4]]
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,1,4,3,5] => ? = 1
[[1,2,2],[2,4]]
=> [2,5,1,3,4] => [2,5,1,3,4] => [2,1,3,5,4] => ? = 1
[[1,2,2],[3,4]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [1,3,2,5,4] => ? = 1
[[1,2,3],[2,4]]
=> [2,5,1,3,4] => [2,5,1,3,4] => [2,1,3,5,4] => ? = 1
[[1,2,4],[2,3]]
=> [2,4,1,3,5] => [2,4,1,3,5] => [2,1,4,3,5] => ? = 1
[[1,2,2],[4,4]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [1,3,2,5,4] => ? = 1
[[1,2,4],[2,4]]
=> [2,4,1,3,5] => [2,4,1,3,5] => [2,1,4,3,5] => ? = 1
[[1,2,3],[3,4]]
=> [3,5,1,2,4] => [3,5,1,2,4] => [1,3,2,5,4] => ? = 1
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St001864
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001864: Signed permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 60%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001864: Signed permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 60%
Values
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1,1],[2,2]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1],[2],[4]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[3],[4]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[2],[3],[4]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1,1],[2,3]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,1],[3,3]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,2],[2,3]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[1,2],[3,3]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[2,2],[3,3]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 2
[[1,2],[2],[3]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[[1,3],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1],[2],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[3],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[4],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[2],[3],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[2],[4],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[3],[4],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1,1],[2,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,1],[3,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,1],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,2],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,3],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[1,2],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,3],[3,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[1,3],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[2,2],[3,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[2,2],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[2,3],[3,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[2,3],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[3,3],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 2
[[1,1],[3],[4]]
=> [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 2
[[1,2],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[[1,4],[2],[4]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[[1,3],[3],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[[1,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[[2,2],[3],[4]]
=> [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 2
[[2,3],[3],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[[2,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [2,3,4,1] => [2,3,4,1] => 3
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 2
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 2
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 2
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 2
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 2
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [4,1,5,3,2] => [4,1,5,3,2] => ? = 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [5,4,2,1,3] => [5,4,2,1,3] => ? = 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [6,1,2,3,5,4] => [6,1,2,3,5,4] => ? = 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [5,1,2,4,3,6] => [5,1,2,4,3,6] => ? = 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [4,1,3,2,5,6] => [4,1,3,2,5,6] => ? = 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [6,1,2,4,5,3] => [6,1,2,4,5,3] => ? = 2
[[1],[2],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[3],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[4],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[5],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1,1,1],[2,4]]
=> [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1
[[1,1,1],[3,4]]
=> [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1
[[1,1,1],[4,4]]
=> [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1
[[1,1,2],[2,4]]
=> [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1
[[1,1,4],[2,2]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,1,2],[3,4]]
=> [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1
[[1,1,3],[2,4]]
=> [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1
[[1,1,4],[2,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,1,2],[4,4]]
=> [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1
[[1,1,4],[2,4]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,1,3],[3,4]]
=> [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1
[[1,1,4],[3,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,1,3],[4,4]]
=> [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1
[[1,1,4],[3,4]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,1,4],[4,4]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,2,2],[2,4]]
=> [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1
[[1,2,2],[3,4]]
=> [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1
[[1,2,3],[2,4]]
=> [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1
[[1,2,4],[2,3]]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1
[[1,2,2],[4,4]]
=> [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1
[[1,2,4],[2,4]]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1
[[1,2,3],[3,4]]
=> [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1
Description
The number of excedances of a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) > i\}\rvert$.
Matching statistic: St001905
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001905: Parking functions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 60%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001905: Parking functions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 60%
Values
[[1],[2],[3]]
=> [3,2,1] => [3,1,2] => [3,1,2] => 2
[[1,1],[2,2]]
=> [3,4,1,2] => [2,4,3,1] => [2,4,3,1] => 1
[[1],[2],[4]]
=> [3,2,1] => [3,1,2] => [3,1,2] => 2
[[1],[3],[4]]
=> [3,2,1] => [3,1,2] => [3,1,2] => 2
[[2],[3],[4]]
=> [3,2,1] => [3,1,2] => [3,1,2] => 2
[[1,1],[2,3]]
=> [3,4,1,2] => [2,4,3,1] => [2,4,3,1] => 1
[[1,1],[3,3]]
=> [3,4,1,2] => [2,4,3,1] => [2,4,3,1] => 1
[[1,2],[2,3]]
=> [2,4,1,3] => [3,2,4,1] => [3,2,4,1] => 1
[[1,2],[3,3]]
=> [3,4,1,2] => [2,4,3,1] => [2,4,3,1] => 1
[[2,2],[3,3]]
=> [3,4,1,2] => [2,4,3,1] => [2,4,3,1] => 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [2,4,1,3] => [2,4,1,3] => 2
[[1,2],[2],[3]]
=> [4,2,1,3] => [3,1,4,2] => [3,1,4,2] => 2
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,1,2,4] => [3,1,2,4] => 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 1
[[1],[2],[5]]
=> [3,2,1] => [3,1,2] => [3,1,2] => 2
[[1],[3],[5]]
=> [3,2,1] => [3,1,2] => [3,1,2] => 2
[[1],[4],[5]]
=> [3,2,1] => [3,1,2] => [3,1,2] => 2
[[2],[3],[5]]
=> [3,2,1] => [3,1,2] => [3,1,2] => 2
[[2],[4],[5]]
=> [3,2,1] => [3,1,2] => [3,1,2] => 2
[[3],[4],[5]]
=> [3,2,1] => [3,1,2] => [3,1,2] => 2
[[1,1],[2,4]]
=> [3,4,1,2] => [2,4,3,1] => [2,4,3,1] => 1
[[1,1],[3,4]]
=> [3,4,1,2] => [2,4,3,1] => [2,4,3,1] => 1
[[1,1],[4,4]]
=> [3,4,1,2] => [2,4,3,1] => [2,4,3,1] => 1
[[1,2],[2,4]]
=> [2,4,1,3] => [3,2,4,1] => [3,2,4,1] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [2,4,3,1] => [2,4,3,1] => 1
[[1,3],[2,4]]
=> [2,4,1,3] => [3,2,4,1] => [3,2,4,1] => 1
[[1,2],[4,4]]
=> [3,4,1,2] => [2,4,3,1] => [2,4,3,1] => 1
[[1,3],[3,4]]
=> [2,4,1,3] => [3,2,4,1] => [3,2,4,1] => 1
[[1,3],[4,4]]
=> [3,4,1,2] => [2,4,3,1] => [2,4,3,1] => 1
[[2,2],[3,4]]
=> [3,4,1,2] => [2,4,3,1] => [2,4,3,1] => 1
[[2,2],[4,4]]
=> [3,4,1,2] => [2,4,3,1] => [2,4,3,1] => 1
[[2,3],[3,4]]
=> [2,4,1,3] => [3,2,4,1] => [3,2,4,1] => 1
[[2,3],[4,4]]
=> [3,4,1,2] => [2,4,3,1] => [2,4,3,1] => 1
[[3,3],[4,4]]
=> [3,4,1,2] => [2,4,3,1] => [2,4,3,1] => 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [2,4,1,3] => [2,4,1,3] => 2
[[1,1],[3],[4]]
=> [4,3,1,2] => [2,4,1,3] => [2,4,1,3] => 2
[[1,2],[2],[4]]
=> [4,2,1,3] => [3,1,4,2] => [3,1,4,2] => 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [2,4,1,3] => [2,4,1,3] => 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,1,4,2] => [3,1,4,2] => 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,1,2,4] => [3,1,2,4] => 2
[[1,4],[2],[4]]
=> [3,2,1,4] => [3,1,2,4] => [3,1,2,4] => 2
[[1,3],[3],[4]]
=> [4,2,1,3] => [3,1,4,2] => [3,1,4,2] => 2
[[1,4],[3],[4]]
=> [3,2,1,4] => [3,1,2,4] => [3,1,2,4] => 2
[[2,2],[3],[4]]
=> [4,3,1,2] => [2,4,1,3] => [2,4,1,3] => 2
[[2,3],[3],[4]]
=> [4,2,1,3] => [3,1,4,2] => [3,1,4,2] => 2
[[2,4],[3],[4]]
=> [3,2,1,4] => [3,1,2,4] => [3,1,2,4] => 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 3
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [3,2,4,1,5] => [3,2,4,1,5] => ? = 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 2
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 2
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 2
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [3,1,4,5,2] => [3,1,4,5,2] => ? = 2
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 2
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [2,3,4,6,5,1] => [2,3,4,6,5,1] => ? = 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [2,3,5,4,1,6] => [2,3,5,4,1,6] => ? = 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [2,4,3,1,5,6] => [2,4,3,1,5,6] => ? = 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [2,3,6,4,5,1] => [2,3,6,4,5,1] => ? = 2
[[1],[2],[6]]
=> [3,2,1] => [3,1,2] => [3,1,2] => 2
[[1],[3],[6]]
=> [3,2,1] => [3,1,2] => [3,1,2] => 2
[[1],[4],[6]]
=> [3,2,1] => [3,1,2] => [3,1,2] => 2
[[1],[5],[6]]
=> [3,2,1] => [3,1,2] => [3,1,2] => 2
[[1,1,1],[2,4]]
=> [4,5,1,2,3] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 1
[[1,1,1],[3,4]]
=> [4,5,1,2,3] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 1
[[1,1,1],[4,4]]
=> [4,5,1,2,3] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 1
[[1,1,2],[2,4]]
=> [3,5,1,2,4] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 1
[[1,1,4],[2,2]]
=> [3,4,1,2,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 1
[[1,1,2],[3,4]]
=> [4,5,1,2,3] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 1
[[1,1,3],[2,4]]
=> [3,5,1,2,4] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 1
[[1,1,4],[2,3]]
=> [3,4,1,2,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 1
[[1,1,2],[4,4]]
=> [4,5,1,2,3] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 1
[[1,1,4],[2,4]]
=> [3,4,1,2,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 1
[[1,1,3],[3,4]]
=> [3,5,1,2,4] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 1
[[1,1,4],[3,3]]
=> [3,4,1,2,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 1
[[1,1,3],[4,4]]
=> [4,5,1,2,3] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 1
[[1,1,4],[3,4]]
=> [3,4,1,2,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 1
[[1,1,4],[4,4]]
=> [3,4,1,2,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 1
[[1,2,2],[2,4]]
=> [2,5,1,3,4] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 1
[[1,2,2],[3,4]]
=> [4,5,1,2,3] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 1
[[1,2,3],[2,4]]
=> [2,5,1,3,4] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 1
[[1,2,4],[2,3]]
=> [2,4,1,3,5] => [3,2,4,1,5] => [3,2,4,1,5] => ? = 1
[[1,2,2],[4,4]]
=> [4,5,1,2,3] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 1
[[1,2,4],[2,4]]
=> [2,4,1,3,5] => [3,2,4,1,5] => [3,2,4,1,5] => ? = 1
[[1,2,3],[3,4]]
=> [3,5,1,2,4] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 1
Description
The number of preferred parking spots in a parking function less than the index of the car.
Let $(a_1,\dots,a_n)$ be a parking function. Then this statistic returns the number of indices $1\leq i\leq n$ such that $a_i < i$.
Matching statistic: St001935
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001935: Parking functions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 60%
Mp00064: Permutations —reverse⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001935: Parking functions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 60%
Values
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 2
[[1,1],[2,2]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 1
[[1],[2],[4]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 2
[[1],[3],[4]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 2
[[2],[3],[4]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 2
[[1,1],[2,3]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 1
[[1,1],[3,3]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 1
[[1,2],[2,3]]
=> [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 1
[[1,2],[3,3]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 1
[[2,2],[3,3]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 2
[[1,2],[2],[3]]
=> [4,2,1,3] => [3,1,2,4] => [3,1,2,4] => 2
[[1,3],[2],[3]]
=> [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 1
[[1],[2],[5]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 2
[[1],[3],[5]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 2
[[1],[4],[5]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 2
[[2],[3],[5]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 2
[[2],[4],[5]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 2
[[3],[4],[5]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 2
[[1,1],[2,4]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 1
[[1,1],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 1
[[1,1],[4,4]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 1
[[1,2],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 1
[[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 1
[[1,2],[4,4]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 1
[[1,3],[3,4]]
=> [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 1
[[1,3],[4,4]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 1
[[2,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 1
[[2,2],[4,4]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 1
[[2,3],[3,4]]
=> [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 1
[[2,3],[4,4]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 1
[[3,3],[4,4]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 2
[[1,1],[3],[4]]
=> [4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 2
[[1,2],[2],[4]]
=> [4,2,1,3] => [3,1,2,4] => [3,1,2,4] => 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,1,2,4] => [3,1,2,4] => 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 2
[[1,4],[2],[4]]
=> [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 2
[[1,3],[3],[4]]
=> [4,2,1,3] => [3,1,2,4] => [3,1,2,4] => 2
[[1,4],[3],[4]]
=> [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 2
[[2,2],[3],[4]]
=> [4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 2
[[2,3],[3],[4]]
=> [4,2,1,3] => [3,1,2,4] => [3,1,2,4] => 2
[[2,4],[3],[4]]
=> [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 3
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [4,2,1,5,3] => [4,2,1,5,3] => ? = 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [4,3,1,5,2] => [4,3,1,5,2] => ? = 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 2
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 2
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 2
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 2
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 2
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 2
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [4,3,2,1,6,5] => [4,3,2,1,6,5] => ? = 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [6,3,2,1,5,4] => [6,3,2,1,5,4] => ? = 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [6,5,2,1,4,3] => [6,5,2,1,4,3] => ? = 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => [3,2,1,6,5,4] => ? = 2
[[1],[2],[6]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 2
[[1],[3],[6]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 2
[[1],[4],[6]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 2
[[1],[5],[6]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 2
[[1,1,1],[2,4]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[[1,1,1],[3,4]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[[1,1,1],[4,4]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[[1,1,2],[2,4]]
=> [3,5,1,2,4] => [4,2,1,5,3] => [4,2,1,5,3] => ? = 1
[[1,1,4],[2,2]]
=> [3,4,1,2,5] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 1
[[1,1,2],[3,4]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[[1,1,3],[2,4]]
=> [3,5,1,2,4] => [4,2,1,5,3] => [4,2,1,5,3] => ? = 1
[[1,1,4],[2,3]]
=> [3,4,1,2,5] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 1
[[1,1,2],[4,4]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[[1,1,4],[2,4]]
=> [3,4,1,2,5] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 1
[[1,1,3],[3,4]]
=> [3,5,1,2,4] => [4,2,1,5,3] => [4,2,1,5,3] => ? = 1
[[1,1,4],[3,3]]
=> [3,4,1,2,5] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 1
[[1,1,3],[4,4]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[[1,1,4],[3,4]]
=> [3,4,1,2,5] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 1
[[1,1,4],[4,4]]
=> [3,4,1,2,5] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 1
[[1,2,2],[2,4]]
=> [2,5,1,3,4] => [4,3,1,5,2] => [4,3,1,5,2] => ? = 1
[[1,2,2],[3,4]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[[1,2,3],[2,4]]
=> [2,5,1,3,4] => [4,3,1,5,2] => [4,3,1,5,2] => ? = 1
[[1,2,4],[2,3]]
=> [2,4,1,3,5] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 1
[[1,2,2],[4,4]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[[1,2,4],[2,4]]
=> [2,4,1,3,5] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 1
[[1,2,3],[3,4]]
=> [3,5,1,2,4] => [4,2,1,5,3] => [4,2,1,5,3] => ? = 1
Description
The number of ascents in a parking function.
This is the number of indices $i$ such that $p_i > p_{i+1}$.
Matching statistic: St000942
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St000942: Parking functions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 60%
Mp00069: Permutations —complement⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St000942: Parking functions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 60%
Values
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[[1],[2],[4]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[[1],[3],[4]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[[2],[3],[4]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[[1,1],[2,3]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[[1,1],[3,3]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[[1,2],[2,3]]
=> [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 2 = 1 + 1
[[1,2],[3,3]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[[2,2],[3,3]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [1,3,4,2] => [1,3,4,2] => 3 = 2 + 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [2,3,4,1] => [2,3,4,1] => 3 = 2 + 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1 + 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 1 + 1
[[1],[2],[5]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[[1],[3],[5]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[[1],[4],[5]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[[2],[3],[5]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[[2],[4],[5]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[[3],[4],[5]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[[1,1],[2,4]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[[1,1],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[[1,1],[4,4]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[[1,2],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 2 = 1 + 1
[[1,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 2 = 1 + 1
[[1,2],[4,4]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[[1,3],[3,4]]
=> [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 2 = 1 + 1
[[1,3],[4,4]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[[2,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[[2,2],[4,4]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[[2,3],[3,4]]
=> [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 2 = 1 + 1
[[2,3],[4,4]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[[3,3],[4,4]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [1,3,4,2] => [1,3,4,2] => 3 = 2 + 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,3,4,2] => [1,3,4,2] => 3 = 2 + 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,4,1] => [2,3,4,1] => 3 = 2 + 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [2,3,4,1] => [2,3,4,1] => 3 = 2 + 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [1,3,4,2] => [1,3,4,2] => 3 = 2 + 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [2,3,4,1] => [2,3,4,1] => 3 = 2 + 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [1,3,4,2] => [1,3,4,2] => 3 = 2 + 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [2,3,4,1] => [2,3,4,1] => 3 = 2 + 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1 + 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1 + 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [3,1,5,4,2] => [3,1,5,4,2] => ? = 1 + 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 1 + 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1 + 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 1 + 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 1 + 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [4,1,5,3,2] => [4,1,5,3,2] => ? = 1 + 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1 + 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [4,2,5,3,1] => [4,2,5,3,1] => ? = 1 + 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 1 + 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1 + 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 1 + 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 2 + 1
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [1,3,5,4,2] => [1,3,5,4,2] => ? = 2 + 1
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 2 + 1
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [1,4,5,3,2] => [1,4,5,3,2] => ? = 2 + 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 2 + 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 2 + 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [1,3,2,5,4] => [1,3,2,5,4] => ? = 1 + 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 1 + 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 1 + 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [2,1,6,5,4,3] => [2,1,6,5,4,3] => ? = 1 + 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [3,2,6,5,4,1] => [3,2,6,5,4,1] => ? = 1 + 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [4,3,6,5,2,1] => [4,3,6,5,2,1] => ? = 1 + 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => [3,2,1,6,5,4] => ? = 2 + 1
[[1],[2],[6]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[[1],[3],[6]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[[1],[4],[6]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[[1],[5],[6]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[[1,1,1],[2,4]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1 + 1
[[1,1,1],[3,4]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1 + 1
[[1,1,1],[4,4]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1 + 1
[[1,1,2],[2,4]]
=> [3,5,1,2,4] => [3,1,5,4,2] => [3,1,5,4,2] => ? = 1 + 1
[[1,1,4],[2,2]]
=> [3,4,1,2,5] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 1 + 1
[[1,1,2],[3,4]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1 + 1
[[1,1,3],[2,4]]
=> [3,5,1,2,4] => [3,1,5,4,2] => [3,1,5,4,2] => ? = 1 + 1
[[1,1,4],[2,3]]
=> [3,4,1,2,5] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 1 + 1
[[1,1,2],[4,4]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1 + 1
[[1,1,4],[2,4]]
=> [3,4,1,2,5] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 1 + 1
[[1,1,3],[3,4]]
=> [3,5,1,2,4] => [3,1,5,4,2] => [3,1,5,4,2] => ? = 1 + 1
[[1,1,4],[3,3]]
=> [3,4,1,2,5] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 1 + 1
[[1,1,3],[4,4]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1 + 1
[[1,1,4],[3,4]]
=> [3,4,1,2,5] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 1 + 1
[[1,1,4],[4,4]]
=> [3,4,1,2,5] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 1 + 1
[[1,2,2],[2,4]]
=> [2,5,1,3,4] => [4,1,5,3,2] => [4,1,5,3,2] => ? = 1 + 1
[[1,2,2],[3,4]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1 + 1
[[1,2,3],[2,4]]
=> [2,5,1,3,4] => [4,1,5,3,2] => [4,1,5,3,2] => ? = 1 + 1
[[1,2,4],[2,3]]
=> [2,4,1,3,5] => [4,2,5,3,1] => [4,2,5,3,1] => ? = 1 + 1
[[1,2,2],[4,4]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1 + 1
[[1,2,4],[2,4]]
=> [2,4,1,3,5] => [4,2,5,3,1] => [4,2,5,3,1] => ? = 1 + 1
[[1,2,3],[3,4]]
=> [3,5,1,2,4] => [3,1,5,4,2] => [3,1,5,4,2] => ? = 1 + 1
Description
The number of critical left to right maxima of the parking functions.
An entry $p$ in a parking function is critical, if there are exactly $p-1$ entries smaller than $p$ and $n-p$ entries larger than $p$. It is a left to right maximum, if there are no larger entries before it.
This statistic allows the computation of the Tutte polynomial of the complete graph $K_{n+1}$, via
$$
\sum_{P} x^{st(P)}y^{\binom{n+1}{2}-\sum P},
$$
where the sum is over all parking functions of length $n$, see [1, thm.13.5.16].
Matching statistic: St001862
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001862: Signed permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 60%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001862: Signed permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 60%
Values
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[[1],[2],[4]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[3],[4]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[2],[3],[4]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1,1],[2,3]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[[1,1],[3,3]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[[1,2],[2,3]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[1,2],[3,3]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[[2,2],[3,3]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 1 = 2 - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1 - 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 - 1
[[1],[2],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[3],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[4],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[2],[3],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[2],[4],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[3],[4],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1,1],[2,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[[1,1],[3,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[[1,1],[4,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[[1,2],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[1,2],[4,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[[1,3],[3,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[1,3],[4,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[[2,2],[3,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[[2,2],[4,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[[2,3],[3,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[2,3],[4,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[[3,3],[4,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 1 = 2 - 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 1 = 2 - 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 1 = 2 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 1 = 2 - 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [3,4,1,2] => [3,4,1,2] => 2 = 3 - 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1 - 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1 - 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 1 - 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 - 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1 - 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 - 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 - 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1 - 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1 - 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1 - 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 - 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1 - 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 - 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 2 - 1
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [3,5,2,1,4] => [3,5,2,1,4] => ? = 2 - 1
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [3,4,2,1,5] => [3,4,2,1,5] => ? = 2 - 1
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 2 - 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 2 - 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 2 - 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 1 - 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 1 - 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 1 - 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [6,5,2,3,4,1] => [6,5,2,3,4,1] => ? = 1 - 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [5,4,2,3,1,6] => [5,4,2,3,1,6] => ? = 1 - 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [4,3,2,1,5,6] => [4,3,2,1,5,6] => ? = 1 - 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ? = 2 - 1
[[1],[2],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[3],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[4],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[5],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1,1,1],[2,4]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1 - 1
[[1,1,1],[3,4]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1 - 1
[[1,1,1],[4,4]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1 - 1
[[1,1,2],[2,4]]
=> [3,5,1,2,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 1 - 1
[[1,1,4],[2,2]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 - 1
[[1,1,2],[3,4]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1 - 1
[[1,1,3],[2,4]]
=> [3,5,1,2,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 1 - 1
[[1,1,4],[2,3]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 - 1
[[1,1,2],[4,4]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1 - 1
[[1,1,4],[2,4]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 - 1
[[1,1,3],[3,4]]
=> [3,5,1,2,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 1 - 1
[[1,1,4],[3,3]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 - 1
[[1,1,3],[4,4]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1 - 1
[[1,1,4],[3,4]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 - 1
[[1,1,4],[4,4]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 - 1
[[1,2,2],[2,4]]
=> [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1 - 1
[[1,2,2],[3,4]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1 - 1
[[1,2,3],[2,4]]
=> [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1 - 1
[[1,2,4],[2,3]]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1 - 1
[[1,2,2],[4,4]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1 - 1
[[1,2,4],[2,4]]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1 - 1
[[1,2,3],[3,4]]
=> [3,5,1,2,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 1 - 1
Description
The number of crossings of a signed permutation.
A crossing of a signed permutation $\pi$ is a pair $(i, j)$ of indices such that
* $i < j \leq \pi(i) < \pi(j)$, or
* $-i < j \leq -\pi(i) < \pi(j)$, or
* $i > j > \pi(i) > \pi(j)$.
The following 73 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001722The number of minimal chains with small intervals between a binary word and the top element. St001754The number of tolerances of a finite lattice. St000640The rank of the largest boolean interval in a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000528The height of a poset. St000906The length of the shortest maximal chain in a poset. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001626The number of maximal proper sublattices of a lattice. St001875The number of simple modules with projective dimension at most 1. St000080The rank of the poset. St001857The number of edges in the reduced word graph of a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St000084The number of subtrees. St000168The number of internal nodes of an ordered tree. St000328The maximum number of child nodes in a tree. St000417The size of the automorphism group of the ordered tree. St000679The pruning number of an ordered tree. St001058The breadth of the ordered tree. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion. St000166The depth minus 1 of an ordered tree. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000181The number of connected components of the Hasse diagram for the poset. St000522The number of 1-protected nodes of a rooted tree. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000782The indicator function of whether a given perfect matching is an L & P matching. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001625The Möbius invariant of a lattice. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001890The maximum magnitude of the Möbius function of a poset. St000094The depth of an ordered tree. St000116The major index of a semistandard tableau obtained by standardizing. St000327The number of cover relations in a poset. St000413The number of ordered trees with the same underlying unordered tree. St000521The number of distinct subtrees of an ordered tree. St000635The number of strictly order preserving maps of a poset into itself. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001645The pebbling number of a connected graph. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001877Number of indecomposable injective modules with projective dimension 2. St001964The interval resolution global dimension of a poset. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St000189The number of elements in the poset. St000415The size of the automorphism group of the rooted tree underlying the ordered tree. St000307The number of rowmotion orbits of a poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000400The path length of an ordered tree. St000529The number of permutations whose descent word is the given binary word. St000180The number of chains of a poset. St000416The number of inequivalent increasing trees of an ordered tree. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000100The number of linear extensions of a poset. St001909The number of interval-closed sets of a poset. St000410The tree factorial of an ordered tree. St000634The number of endomorphisms of a poset. St000454The largest eigenvalue of a graph if it is integral. St000422The energy of a graph, if it is integral.
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