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Your data matches 34 different statistics following compositions of up to 3 maps.
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Matching statistic: St000366
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(load all 13 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000366: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => 0
[[2,2]]
=> [1,2] => 0
[[1],[2]]
=> [2,1] => 0
[[1,3]]
=> [1,2] => 0
[[2,3]]
=> [1,2] => 0
[[3,3]]
=> [1,2] => 0
[[1],[3]]
=> [2,1] => 0
[[2],[3]]
=> [2,1] => 0
[[1,1,2]]
=> [1,2,3] => 0
[[1,2,2]]
=> [1,2,3] => 0
[[2,2,2]]
=> [1,2,3] => 0
[[1,1],[2]]
=> [3,1,2] => 0
[[1,2],[2]]
=> [2,1,3] => 0
[[1,4]]
=> [1,2] => 0
[[2,4]]
=> [1,2] => 0
[[3,4]]
=> [1,2] => 0
[[4,4]]
=> [1,2] => 0
[[1],[4]]
=> [2,1] => 0
[[2],[4]]
=> [2,1] => 0
[[3],[4]]
=> [2,1] => 0
[[1,1,3]]
=> [1,2,3] => 0
[[1,2,3]]
=> [1,2,3] => 0
[[1,3,3]]
=> [1,2,3] => 0
[[2,2,3]]
=> [1,2,3] => 0
[[2,3,3]]
=> [1,2,3] => 0
[[3,3,3]]
=> [1,2,3] => 0
[[1,1],[3]]
=> [3,1,2] => 0
[[1,2],[3]]
=> [3,1,2] => 0
[[1,3],[2]]
=> [2,1,3] => 0
[[1,3],[3]]
=> [2,1,3] => 0
[[2,2],[3]]
=> [3,1,2] => 0
[[2,3],[3]]
=> [2,1,3] => 0
[[1],[2],[3]]
=> [3,2,1] => 1
[[1,1,1,2]]
=> [1,2,3,4] => 0
[[1,1,2,2]]
=> [1,2,3,4] => 0
[[1,2,2,2]]
=> [1,2,3,4] => 0
[[2,2,2,2]]
=> [1,2,3,4] => 0
[[1,1,1],[2]]
=> [4,1,2,3] => 0
[[1,1,2],[2]]
=> [3,1,2,4] => 0
[[1,2,2],[2]]
=> [2,1,3,4] => 0
[[1,1],[2,2]]
=> [3,4,1,2] => 0
[[1,5]]
=> [1,2] => 0
[[2,5]]
=> [1,2] => 0
[[3,5]]
=> [1,2] => 0
[[4,5]]
=> [1,2] => 0
[[5,5]]
=> [1,2] => 0
[[1],[5]]
=> [2,1] => 0
[[2],[5]]
=> [2,1] => 0
[[3],[5]]
=> [2,1] => 0
[[4],[5]]
=> [2,1] => 0
Description
The number of double descents of a permutation.
A double descent of a permutation $\pi$ is a position $i$ such that $\pi(i) > \pi(i+1) > \pi(i+2)$.
Matching statistic: St001172
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001172: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001172: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[[2,2]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[[1],[2]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[[1,3]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[[2,3]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[[3,3]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[[1],[3]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[[2],[3]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[[1,1,2]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[[1,2,2]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[[2,2,2]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[[1,1],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[[1,2],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[[1,4]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[[2,4]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[[3,4]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[[4,4]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[[1],[4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[[2],[4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[[3],[4]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[[1,1,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[[1,2,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[[1,3,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[[2,2,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[[2,3,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[[3,3,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[[1,1],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[[1,2],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[[1,3],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[[1,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[[2,2],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[[2,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[[1,1,1,2]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[[1,1,2,2]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[[1,2,2,2]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[[2,2,2,2]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[[1,1,1],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[[1,1,2],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[[1,2,2],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[[1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[[1,5]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[[2,5]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[[3,5]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[[4,5]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[[5,5]]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[[1],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[[2],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[[3],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[[4],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[[1,1,1,1,1,1]]
=> ?
=> ?
=> ?
=> ? = 0
[[1,1,1,1,2,2,2],[3,3],[4]]
=> [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[[1,1,1,2,2,2,2],[3,3],[4]]
=> [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
Description
The number of 1-rises at odd height of a Dyck path.
Matching statistic: St001167
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[2,2]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[1],[2]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[1,3]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[2,3]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[3,3]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[1],[3]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[2],[3]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[1,1,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[1,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[2,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[1,1],[2]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[[1,2],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[[1,4]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[2,4]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[3,4]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[4,4]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[1],[4]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[2],[4]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[3],[4]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[1,1,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[1,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[2,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[2,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[3,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[1,1],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[[1,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[[1,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[[2,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[[2,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1
[[1,1,1,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,1,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[[2,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,1,1],[2]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 0
[[1,1,2],[2]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 0
[[1,2,2],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 0
[[1,1],[2,2]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[[1,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[2,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[3,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[4,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[5,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[1],[5]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[2],[5]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[3],[5]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[4],[5]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[1,1,1,1,1,1,1,2]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,1,1,2,2]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,1,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,2,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,2,2,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,2,2,2,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[1,2,2,2,2,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[2,2,2,2,2,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,1,1,1],[2]]
=> [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,1,1,2],[2]]
=> [7,1,2,3,4,5,6,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,1,2,2],[2]]
=> [6,1,2,3,4,5,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,2,2,2],[2]]
=> [5,1,2,3,4,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,2,2,2,2,2,2],[2]]
=> [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,1,1],[2,2]]
=> [7,8,1,2,3,4,5,6] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,1,2],[2,2]]
=> [6,7,1,2,3,4,5,8] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[[1,1,2,2,2,2],[2,2]]
=> [3,4,1,2,5,6,7,8] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,1],[2,2,2]]
=> [6,7,8,1,2,3,4,5] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,2],[2,2,2]]
=> [5,6,7,1,2,3,4,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[[1,1,1,2,2],[2,2,2]]
=> [4,5,6,1,2,3,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[[1,1,1,1],[2,2,2,2]]
=> [5,6,7,8,1,2,3,4] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[[1,1,1,1],[2,2,2],[3,3],[4]]
=> [10,8,9,5,6,7,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,2],[2,2,2],[3,3],[4]]
=> [10,8,9,4,5,6,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,1],[2,2,3],[3,3],[4]]
=> [10,7,8,5,6,9,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,2],[2,2,3],[3,3],[4]]
=> [10,7,8,4,5,9,1,2,3,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,3],[2,2,3],[3,3],[4]]
=> [10,6,7,4,5,8,1,2,3,9] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,2,2],[2,2,3],[3,3],[4]]
=> [10,7,8,3,4,9,1,2,5,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,2,3],[2,2,3],[3,3],[4]]
=> [10,6,7,3,4,8,1,2,5,9] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,1],[2,2,2],[3,4],[4]]
=> [9,8,10,5,6,7,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,2],[2,2,2],[3,4],[4]]
=> [9,8,10,4,5,6,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,1],[2,2,3],[3,4],[4]]
=> [9,7,10,5,6,8,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,2],[2,2,3],[3,4],[4]]
=> [9,7,10,4,5,8,1,2,3,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,3],[2,2,3],[3,4],[4]]
=> [9,6,10,4,5,7,1,2,3,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,2,2],[2,2,3],[3,4],[4]]
=> [9,7,10,3,4,8,1,2,5,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,2,3],[2,2,3],[3,4],[4]]
=> [9,6,10,3,4,7,1,2,5,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,1],[2,2,4],[3,4],[4]]
=> [8,7,9,5,6,10,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,2],[2,2,4],[3,4],[4]]
=> [8,7,9,4,5,10,1,2,3,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,3],[2,2,4],[3,4],[4]]
=> [8,6,9,4,5,10,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,4],[2,2,4],[3,4],[4]]
=> [7,6,8,4,5,9,1,2,3,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,2,2],[2,2,4],[3,4],[4]]
=> [8,7,9,3,4,10,1,2,5,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,2,3],[2,2,4],[3,4],[4]]
=> [8,6,9,3,4,10,1,2,5,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,2,4],[2,2,4],[3,4],[4]]
=> [7,6,8,3,4,9,1,2,5,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,1],[2,3,3],[3,4],[4]]
=> [9,6,10,5,7,8,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,2],[2,3,3],[3,4],[4]]
=> [9,6,10,4,7,8,1,2,3,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,3],[2,3,3],[3,4],[4]]
=> [9,5,10,4,6,7,1,2,3,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,2,2],[2,3,3],[3,4],[4]]
=> [9,6,10,3,7,8,1,2,4,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,2,3],[2,3,3],[3,4],[4]]
=> [9,5,10,3,6,7,1,2,4,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,1],[2,3,4],[3,4],[4]]
=> [8,6,9,5,7,10,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,2],[2,3,4],[3,4],[4]]
=> [8,6,9,4,7,10,1,2,3,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,3],[2,3,4],[3,4],[4]]
=> [8,5,9,4,6,10,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,4],[2,3,4],[3,4],[4]]
=> [7,5,8,4,6,9,1,2,3,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
Description
The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra.
The top of a module is the cokernel of the inclusion of the radical of the module into the module.
For Nakayama algebras with at most 8 simple modules, the statistic also coincides with the number of simple modules with projective dimension at least 3 in the corresponding Nakayama algebra.
Matching statistic: St001253
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001253: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001253: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[2,2]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[1],[2]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[1,3]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[2,3]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[3,3]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[1],[3]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[2],[3]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[1,1,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[1,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[2,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[1,1],[2]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[[1,2],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[[1,4]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[2,4]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[3,4]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[4,4]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[1],[4]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[2],[4]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[3],[4]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[1,1,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[1,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[2,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[2,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[3,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[1,1],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[[1,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[[1,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[[2,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[[2,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1
[[1,1,1,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,1,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[[2,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,1,1],[2]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 0
[[1,1,2],[2]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 0
[[1,2,2],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 0
[[1,1],[2,2]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[[1,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[2,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[3,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[4,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[5,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[1],[5]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[2],[5]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[3],[5]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[4],[5]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[1,1,1,1,1,1,1,2]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,1,1,2,2]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,1,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,2,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,2,2,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,2,2,2,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[1,2,2,2,2,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[2,2,2,2,2,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,1,1,1],[2]]
=> [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,1,1,2],[2]]
=> [7,1,2,3,4,5,6,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,1,2,2],[2]]
=> [6,1,2,3,4,5,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,2,2,2],[2]]
=> [5,1,2,3,4,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,2,2,2,2,2,2],[2]]
=> [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,1,1],[2,2]]
=> [7,8,1,2,3,4,5,6] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,1,2],[2,2]]
=> [6,7,1,2,3,4,5,8] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[[1,1,2,2,2,2],[2,2]]
=> [3,4,1,2,5,6,7,8] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,1],[2,2,2]]
=> [6,7,8,1,2,3,4,5] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,2],[2,2,2]]
=> [5,6,7,1,2,3,4,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[[1,1,1,2,2],[2,2,2]]
=> [4,5,6,1,2,3,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[[1,1,1,1],[2,2,2,2]]
=> [5,6,7,8,1,2,3,4] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[[1,1,1,1],[2,2,2],[3,3],[4]]
=> [10,8,9,5,6,7,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,2],[2,2,2],[3,3],[4]]
=> [10,8,9,4,5,6,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,1],[2,2,3],[3,3],[4]]
=> [10,7,8,5,6,9,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,2],[2,2,3],[3,3],[4]]
=> [10,7,8,4,5,9,1,2,3,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,3],[2,2,3],[3,3],[4]]
=> [10,6,7,4,5,8,1,2,3,9] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,2,2],[2,2,3],[3,3],[4]]
=> [10,7,8,3,4,9,1,2,5,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,2,3],[2,2,3],[3,3],[4]]
=> [10,6,7,3,4,8,1,2,5,9] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,1],[2,2,2],[3,4],[4]]
=> [9,8,10,5,6,7,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,2],[2,2,2],[3,4],[4]]
=> [9,8,10,4,5,6,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,1],[2,2,3],[3,4],[4]]
=> [9,7,10,5,6,8,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,2],[2,2,3],[3,4],[4]]
=> [9,7,10,4,5,8,1,2,3,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,3],[2,2,3],[3,4],[4]]
=> [9,6,10,4,5,7,1,2,3,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,2,2],[2,2,3],[3,4],[4]]
=> [9,7,10,3,4,8,1,2,5,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,2,3],[2,2,3],[3,4],[4]]
=> [9,6,10,3,4,7,1,2,5,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,1],[2,2,4],[3,4],[4]]
=> [8,7,9,5,6,10,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,2],[2,2,4],[3,4],[4]]
=> [8,7,9,4,5,10,1,2,3,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,3],[2,2,4],[3,4],[4]]
=> [8,6,9,4,5,10,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,4],[2,2,4],[3,4],[4]]
=> [7,6,8,4,5,9,1,2,3,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,2,2],[2,2,4],[3,4],[4]]
=> [8,7,9,3,4,10,1,2,5,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,2,3],[2,2,4],[3,4],[4]]
=> [8,6,9,3,4,10,1,2,5,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,2,4],[2,2,4],[3,4],[4]]
=> [7,6,8,3,4,9,1,2,5,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,1],[2,3,3],[3,4],[4]]
=> [9,6,10,5,7,8,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,2],[2,3,3],[3,4],[4]]
=> [9,6,10,4,7,8,1,2,3,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,3],[2,3,3],[3,4],[4]]
=> [9,5,10,4,6,7,1,2,3,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,2,2],[2,3,3],[3,4],[4]]
=> [9,6,10,3,7,8,1,2,4,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,2,3],[2,3,3],[3,4],[4]]
=> [9,5,10,3,6,7,1,2,4,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,1],[2,3,4],[3,4],[4]]
=> [8,6,9,5,7,10,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,2],[2,3,4],[3,4],[4]]
=> [8,6,9,4,7,10,1,2,3,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,3],[2,3,4],[3,4],[4]]
=> [8,5,9,4,6,10,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
[[1,1,1,4],[2,3,4],[3,4],[4]]
=> [7,5,8,4,6,9,1,2,3,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0
Description
The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra.
For the first 196 values the statistic coincides also with the number of fixed points of $\tau \Omega^2$ composed with its inverse, see theorem 5.8. in the reference for more details.
The number of Dyck paths of length n where the statistics returns zero seems to be 2^(n-1).
Matching statistic: St001066
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[[2,2]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[[1],[2]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[[1,3]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[[2,3]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[[3,3]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[[1],[3]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[[2],[3]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[[1,1,2]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[[1,2,2]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[[2,2,2]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[[1,1],[2]]
=> [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[[1,2],[2]]
=> [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
=> 1 = 0 + 1
[[1,4]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[[2,4]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[[3,4]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[[4,4]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[[1],[4]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[[2],[4]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[[3],[4]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[[1,1,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[[1,2,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[[1,3,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[[2,2,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[[2,3,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[[3,3,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[[1,1],[3]]
=> [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[[1,3],[2]]
=> [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
=> 1 = 0 + 1
[[1,3],[3]]
=> [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
=> 1 = 0 + 1
[[2,2],[3]]
=> [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[[2,3],[3]]
=> [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
=> 1 = 0 + 1
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[[1,1,1,2]]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[1,1,2,2]]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[1,2,2,2]]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[2,2,2,2]]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[1,1,1],[2]]
=> [4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[[1,1,2],[2]]
=> [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[[1,2,2],[2]]
=> [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[[1,5]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[[2,5]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[[3,5]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[[4,5]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[[5,5]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[[1],[5]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[[2],[5]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[[3],[5]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[[4],[5]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[[1,1,1,1,1,1,1,2]]
=> [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1,1,1,2,2]]
=> [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1,1,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1,2,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,2,2,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,2,2,2,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,2,2,2,2,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[2,2,2,2,2,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1,1,1,1],[2]]
=> [8,1,2,3,4,5,6,7] => [1,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1,1,1,2],[2]]
=> [7,1,2,3,4,5,6,8] => [2,8,7,6,5,4,3,1] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1,1,2,2],[2]]
=> [6,1,2,3,4,5,7,8] => [3,8,7,6,5,4,2,1] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1,2,2,2],[2]]
=> [5,1,2,3,4,6,7,8] => [4,8,7,6,5,3,2,1] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,2,2,2,2,2,2],[2]]
=> [2,1,3,4,5,6,7,8] => [7,8,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1,1,1],[2,2]]
=> [7,8,1,2,3,4,5,6] => [2,1,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1,1,2],[2,2]]
=> [6,7,1,2,3,4,5,8] => [3,2,8,7,6,5,4,1] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,2,2,2,2],[2,2]]
=> [3,4,1,2,5,6,7,8] => [6,5,8,7,4,3,2,1] => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1,1],[2,2,2]]
=> [6,7,8,1,2,3,4,5] => [3,2,1,8,7,6,5,4] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1,2],[2,2,2]]
=> [5,6,7,1,2,3,4,8] => [4,3,2,8,7,6,5,1] => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,2,2],[2,2,2]]
=> [4,5,6,1,2,3,7,8] => [5,4,3,8,7,6,2,1] => [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1],[2,2,2,2]]
=> [5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1],[2,2,2],[3,3],[4]]
=> [10,8,9,5,6,7,1,2,3,4] => [1,3,2,6,5,4,10,9,8,7] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,2],[2,2,2],[3,3],[4]]
=> [10,8,9,4,5,6,1,2,3,7] => [1,3,2,7,6,5,10,9,8,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1],[2,2,3],[3,3],[4]]
=> [10,7,8,5,6,9,1,2,3,4] => [1,4,3,6,5,2,10,9,8,7] => [1,0,1,1,1,0,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,2],[2,2,3],[3,3],[4]]
=> [10,7,8,4,5,9,1,2,3,6] => [1,4,3,7,6,2,10,9,8,5] => [1,0,1,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,3],[2,2,3],[3,3],[4]]
=> [10,6,7,4,5,8,1,2,3,9] => [1,5,4,7,6,3,10,9,8,2] => [1,0,1,1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,2,2],[2,2,3],[3,3],[4]]
=> [10,7,8,3,4,9,1,2,5,6] => [1,4,3,8,7,2,10,9,6,5] => [1,0,1,1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,2,3],[2,2,3],[3,3],[4]]
=> [10,6,7,3,4,8,1,2,5,9] => [1,5,4,8,7,3,10,9,6,2] => [1,0,1,1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1],[2,2,2],[3,4],[4]]
=> [9,8,10,5,6,7,1,2,3,4] => [2,3,1,6,5,4,10,9,8,7] => [1,1,0,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,2],[2,2,2],[3,4],[4]]
=> [9,8,10,4,5,6,1,2,3,7] => [2,3,1,7,6,5,10,9,8,4] => [1,1,0,1,0,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1],[2,2,3],[3,4],[4]]
=> [9,7,10,5,6,8,1,2,3,4] => [2,4,1,6,5,3,10,9,8,7] => [1,1,0,1,1,0,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,2],[2,2,3],[3,4],[4]]
=> [9,7,10,4,5,8,1,2,3,6] => [2,4,1,7,6,3,10,9,8,5] => [1,1,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,3],[2,2,3],[3,4],[4]]
=> [9,6,10,4,5,7,1,2,3,8] => [2,5,1,7,6,4,10,9,8,3] => [1,1,0,1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,2,2],[2,2,3],[3,4],[4]]
=> [9,7,10,3,4,8,1,2,5,6] => [2,4,1,8,7,3,10,9,6,5] => [1,1,0,1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,2,3],[2,2,3],[3,4],[4]]
=> [9,6,10,3,4,7,1,2,5,8] => [2,5,1,8,7,4,10,9,6,3] => [1,1,0,1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1],[2,2,4],[3,4],[4]]
=> [8,7,9,5,6,10,1,2,3,4] => [3,4,2,6,5,1,10,9,8,7] => [1,1,1,0,1,0,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,2],[2,2,4],[3,4],[4]]
=> [8,7,9,4,5,10,1,2,3,6] => [3,4,2,7,6,1,10,9,8,5] => [1,1,1,0,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,3],[2,2,4],[3,4],[4]]
=> [8,6,9,4,5,10,1,2,3,7] => [3,5,2,7,6,1,10,9,8,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,4],[2,2,4],[3,4],[4]]
=> [7,6,8,4,5,9,1,2,3,10] => [4,5,3,7,6,2,10,9,8,1] => [1,1,1,1,0,1,0,0,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,2,2],[2,2,4],[3,4],[4]]
=> [8,7,9,3,4,10,1,2,5,6] => [3,4,2,8,7,1,10,9,6,5] => [1,1,1,0,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,2,3],[2,2,4],[3,4],[4]]
=> [8,6,9,3,4,10,1,2,5,7] => [3,5,2,8,7,1,10,9,6,4] => [1,1,1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,2,4],[2,2,4],[3,4],[4]]
=> [7,6,8,3,4,9,1,2,5,10] => [4,5,3,8,7,2,10,9,6,1] => [1,1,1,1,0,1,0,0,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1],[2,3,3],[3,4],[4]]
=> [9,6,10,5,7,8,1,2,3,4] => [2,5,1,6,4,3,10,9,8,7] => [1,1,0,1,1,1,0,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,2],[2,3,3],[3,4],[4]]
=> [9,6,10,4,7,8,1,2,3,5] => [2,5,1,7,4,3,10,9,8,6] => [1,1,0,1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,3],[2,3,3],[3,4],[4]]
=> [9,5,10,4,6,7,1,2,3,8] => [2,6,1,7,5,4,10,9,8,3] => [1,1,0,1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,2,2],[2,3,3],[3,4],[4]]
=> [9,6,10,3,7,8,1,2,4,5] => [2,5,1,8,4,3,10,9,7,6] => [1,1,0,1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,2,3],[2,3,3],[3,4],[4]]
=> [9,5,10,3,6,7,1,2,4,8] => [2,6,1,8,5,4,10,9,7,3] => [1,1,0,1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1],[2,3,4],[3,4],[4]]
=> [8,6,9,5,7,10,1,2,3,4] => [3,5,2,6,4,1,10,9,8,7] => [1,1,1,0,1,1,0,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,2],[2,3,4],[3,4],[4]]
=> [8,6,9,4,7,10,1,2,3,5] => [3,5,2,7,4,1,10,9,8,6] => [1,1,1,0,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,3],[2,3,4],[3,4],[4]]
=> [8,5,9,4,6,10,1,2,3,7] => [3,6,2,7,5,1,10,9,8,4] => [1,1,1,0,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,4],[2,3,4],[3,4],[4]]
=> [7,5,8,4,6,9,1,2,3,10] => [4,6,3,7,5,2,10,9,8,1] => [1,1,1,1,0,1,1,0,0,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
Description
The number of simple reflexive modules in the corresponding Nakayama algebra.
Matching statistic: St001483
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1 = 0 + 1
[[2,2]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1 = 0 + 1
[[1],[2]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1 = 0 + 1
[[1,3]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1 = 0 + 1
[[2,3]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1 = 0 + 1
[[3,3]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1 = 0 + 1
[[1],[3]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1 = 0 + 1
[[2],[3]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1 = 0 + 1
[[1,1,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[[1,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[[2,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[[1,1],[2]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[[1,2],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[[1,4]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1 = 0 + 1
[[2,4]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1 = 0 + 1
[[3,4]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1 = 0 + 1
[[4,4]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1 = 0 + 1
[[1],[4]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1 = 0 + 1
[[2],[4]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1 = 0 + 1
[[3],[4]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1 = 0 + 1
[[1,1,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[[1,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[[2,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[[2,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[[3,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[[1,1],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[[1,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[[1,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[[2,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[[2,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[[1,1,1,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[1,1,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[1,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[2,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[1,1,1],[2]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[[1,1,2],[2]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[[1,2,2],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[[1,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1 = 0 + 1
[[2,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1 = 0 + 1
[[3,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1 = 0 + 1
[[4,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1 = 0 + 1
[[5,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1 = 0 + 1
[[1],[5]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1 = 0 + 1
[[2],[5]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1 = 0 + 1
[[3],[5]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1 = 0 + 1
[[4],[5]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1 = 0 + 1
[[1,1,1,1,1,1,1,2]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1,1,1,2,2]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1,1,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1,2,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,2,2,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,2,2,2,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,2,2,2,2,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[2,2,2,2,2,2,2,2]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1,1,1,1],[2]]
=> [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1,1,1,2],[2]]
=> [7,1,2,3,4,5,6,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1,1,2,2],[2]]
=> [6,1,2,3,4,5,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1,2,2,2],[2]]
=> [5,1,2,3,4,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,2,2,2,2,2,2],[2]]
=> [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1,1,1],[2,2]]
=> [7,8,1,2,3,4,5,6] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1,1,2],[2,2]]
=> [6,7,1,2,3,4,5,8] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,2,2,2,2],[2,2]]
=> [3,4,1,2,5,6,7,8] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1,1],[2,2,2]]
=> [6,7,8,1,2,3,4,5] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1,2],[2,2,2]]
=> [5,6,7,1,2,3,4,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,2,2],[2,2,2]]
=> [4,5,6,1,2,3,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1],[2,2,2,2]]
=> [5,6,7,8,1,2,3,4] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[[1,1,1,1],[2,2,2],[3,3],[4]]
=> [10,8,9,5,6,7,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,1,2],[2,2,2],[3,3],[4]]
=> [10,8,9,4,5,6,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,1,1],[2,2,3],[3,3],[4]]
=> [10,7,8,5,6,9,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,1,2],[2,2,3],[3,3],[4]]
=> [10,7,8,4,5,9,1,2,3,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,1,3],[2,2,3],[3,3],[4]]
=> [10,6,7,4,5,8,1,2,3,9] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,2,2],[2,2,3],[3,3],[4]]
=> [10,7,8,3,4,9,1,2,5,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,2,3],[2,2,3],[3,3],[4]]
=> [10,6,7,3,4,8,1,2,5,9] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,1,1],[2,2,2],[3,4],[4]]
=> [9,8,10,5,6,7,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,1,2],[2,2,2],[3,4],[4]]
=> [9,8,10,4,5,6,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,1,1],[2,2,3],[3,4],[4]]
=> [9,7,10,5,6,8,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,1,2],[2,2,3],[3,4],[4]]
=> [9,7,10,4,5,8,1,2,3,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,1,3],[2,2,3],[3,4],[4]]
=> [9,6,10,4,5,7,1,2,3,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,2,2],[2,2,3],[3,4],[4]]
=> [9,7,10,3,4,8,1,2,5,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,2,3],[2,2,3],[3,4],[4]]
=> [9,6,10,3,4,7,1,2,5,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,1,1],[2,2,4],[3,4],[4]]
=> [8,7,9,5,6,10,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,1,2],[2,2,4],[3,4],[4]]
=> [8,7,9,4,5,10,1,2,3,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,1,3],[2,2,4],[3,4],[4]]
=> [8,6,9,4,5,10,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,1,4],[2,2,4],[3,4],[4]]
=> [7,6,8,4,5,9,1,2,3,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,2,2],[2,2,4],[3,4],[4]]
=> [8,7,9,3,4,10,1,2,5,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,2,3],[2,2,4],[3,4],[4]]
=> [8,6,9,3,4,10,1,2,5,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,2,4],[2,2,4],[3,4],[4]]
=> [7,6,8,3,4,9,1,2,5,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,1,1],[2,3,3],[3,4],[4]]
=> [9,6,10,5,7,8,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,1,2],[2,3,3],[3,4],[4]]
=> [9,6,10,4,7,8,1,2,3,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,1,3],[2,3,3],[3,4],[4]]
=> [9,5,10,4,6,7,1,2,3,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,2,2],[2,3,3],[3,4],[4]]
=> [9,6,10,3,7,8,1,2,4,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,2,3],[2,3,3],[3,4],[4]]
=> [9,5,10,3,6,7,1,2,4,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,1,1],[2,3,4],[3,4],[4]]
=> [8,6,9,5,7,10,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,1,2],[2,3,4],[3,4],[4]]
=> [8,6,9,4,7,10,1,2,3,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,1,3],[2,3,4],[3,4],[4]]
=> [8,5,9,4,6,10,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
[[1,1,1,4],[2,3,4],[3,4],[4]]
=> [7,5,8,4,6,9,1,2,3,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 0 + 1
Description
The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module.
Matching statistic: St000731
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,2] => 0
[[2,2]]
=> [1,2] => [1,2] => 0
[[1],[2]]
=> [2,1] => [2,1] => 0
[[1,3]]
=> [1,2] => [1,2] => 0
[[2,3]]
=> [1,2] => [1,2] => 0
[[3,3]]
=> [1,2] => [1,2] => 0
[[1],[3]]
=> [2,1] => [2,1] => 0
[[2],[3]]
=> [2,1] => [2,1] => 0
[[1,1,2]]
=> [1,2,3] => [1,2,3] => 0
[[1,2,2]]
=> [1,2,3] => [1,2,3] => 0
[[2,2,2]]
=> [1,2,3] => [1,2,3] => 0
[[1,1],[2]]
=> [3,1,2] => [3,1,2] => 0
[[1,2],[2]]
=> [2,1,3] => [2,1,3] => 0
[[1,4]]
=> [1,2] => [1,2] => 0
[[2,4]]
=> [1,2] => [1,2] => 0
[[3,4]]
=> [1,2] => [1,2] => 0
[[4,4]]
=> [1,2] => [1,2] => 0
[[1],[4]]
=> [2,1] => [2,1] => 0
[[2],[4]]
=> [2,1] => [2,1] => 0
[[3],[4]]
=> [2,1] => [2,1] => 0
[[1,1,3]]
=> [1,2,3] => [1,2,3] => 0
[[1,2,3]]
=> [1,2,3] => [1,2,3] => 0
[[1,3,3]]
=> [1,2,3] => [1,2,3] => 0
[[2,2,3]]
=> [1,2,3] => [1,2,3] => 0
[[2,3,3]]
=> [1,2,3] => [1,2,3] => 0
[[3,3,3]]
=> [1,2,3] => [1,2,3] => 0
[[1,1],[3]]
=> [3,1,2] => [3,1,2] => 0
[[1,2],[3]]
=> [3,1,2] => [3,1,2] => 0
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => 0
[[1,3],[3]]
=> [2,1,3] => [2,1,3] => 0
[[2,2],[3]]
=> [3,1,2] => [3,1,2] => 0
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => 0
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => 0
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => 0
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => 0
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => 0
[[1,1,1],[2]]
=> [4,1,2,3] => [4,1,2,3] => 0
[[1,1,2],[2]]
=> [3,1,2,4] => [3,1,2,4] => 0
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => 0
[[1,1],[2,2]]
=> [3,4,1,2] => [4,1,3,2] => 0
[[1,5]]
=> [1,2] => [1,2] => 0
[[2,5]]
=> [1,2] => [1,2] => 0
[[3,5]]
=> [1,2] => [1,2] => 0
[[4,5]]
=> [1,2] => [1,2] => 0
[[5,5]]
=> [1,2] => [1,2] => 0
[[1],[5]]
=> [2,1] => [2,1] => 0
[[2],[5]]
=> [2,1] => [2,1] => 0
[[3],[5]]
=> [2,1] => [2,1] => 0
[[4],[5]]
=> [2,1] => [2,1] => 0
[[1,1,1,1,1],[2,2]]
=> [6,7,1,2,3,4,5] => [7,1,2,3,4,6,5] => ? = 0
[[1,1,1,1,2],[2,2]]
=> [5,6,1,2,3,4,7] => [6,1,2,3,5,4,7] => ? = 0
[[1,1,2,2,2],[2,2]]
=> [3,4,1,2,5,6,7] => [4,1,3,2,5,6,7] => ? = 0
[[1,1,1,1],[2,2,2]]
=> [5,6,7,1,2,3,4] => [7,1,2,3,5,6,4] => ? = 0
[[1,1,1,2],[2,2,2]]
=> [4,5,6,1,2,3,7] => [6,1,2,4,5,3,7] => ? = 0
[[1,1,1,1,1],[2,3]]
=> [6,7,1,2,3,4,5] => [7,1,2,3,4,6,5] => ? = 0
[[1,1,1,1,1],[3,3]]
=> [6,7,1,2,3,4,5] => [7,1,2,3,4,6,5] => ? = 0
[[1,1,1,1,3],[2,2]]
=> [5,6,1,2,3,4,7] => [6,1,2,3,5,4,7] => ? = 0
[[1,1,1,1,2],[3,3]]
=> [6,7,1,2,3,4,5] => [7,1,2,3,4,6,5] => ? = 0
[[1,1,1,1,3],[2,3]]
=> [5,6,1,2,3,4,7] => [6,1,2,3,5,4,7] => ? = 0
[[1,1,1,1,3],[3,3]]
=> [5,6,1,2,3,4,7] => [6,1,2,3,5,4,7] => ? = 0
[[1,1,1,2,2],[3,3]]
=> [6,7,1,2,3,4,5] => [7,1,2,3,4,6,5] => ? = 0
[[1,1,1,2,3],[2,3]]
=> [4,6,1,2,3,5,7] => [6,1,2,4,3,5,7] => ? = 0
[[1,1,1,2,3],[3,3]]
=> [5,6,1,2,3,4,7] => [6,1,2,3,5,4,7] => ? = 0
[[1,1,2,2,3],[2,2]]
=> [3,4,1,2,5,6,7] => [4,1,3,2,5,6,7] => ? = 0
[[1,1,2,2,2],[3,3]]
=> [6,7,1,2,3,4,5] => [7,1,2,3,4,6,5] => ? = 0
[[1,1,2,2,3],[2,3]]
=> [3,6,1,2,4,5,7] => [6,1,3,2,4,5,7] => ? = 0
[[1,1,2,3,3],[2,2]]
=> [3,4,1,2,5,6,7] => [4,1,3,2,5,6,7] => ? = 0
[[1,1,2,2,3],[3,3]]
=> [5,6,1,2,3,4,7] => [6,1,2,3,5,4,7] => ? = 0
[[1,1,3,3,3],[2,2]]
=> [3,4,1,2,5,6,7] => [4,1,3,2,5,6,7] => ? = 0
[[1,1,3,3,3],[2,3]]
=> [3,4,1,2,5,6,7] => [4,1,3,2,5,6,7] => ? = 0
[[1,1,3,3,3],[3,3]]
=> [3,4,1,2,5,6,7] => [4,1,3,2,5,6,7] => ? = 0
[[1,2,2,2,2],[3,3]]
=> [6,7,1,2,3,4,5] => [7,1,2,3,4,6,5] => ? = 0
[[1,2,2,2,3],[2,3]]
=> [2,6,1,3,4,5,7] => [6,2,1,3,4,5,7] => ? = 0
[[1,2,2,2,3],[3,3]]
=> [5,6,1,2,3,4,7] => [6,1,2,3,5,4,7] => ? = 0
[[1,2,2,3,3],[2,3]]
=> [2,5,1,3,4,6,7] => [5,2,1,3,4,6,7] => ? = 0
[[1,2,3,3,3],[2,3]]
=> [2,4,1,3,5,6,7] => [4,2,1,3,5,6,7] => ? = 0
[[1,2,3,3,3],[3,3]]
=> [3,4,1,2,5,6,7] => [4,1,3,2,5,6,7] => ? = 0
[[2,2,2,2,2],[3,3]]
=> [6,7,1,2,3,4,5] => [7,1,2,3,4,6,5] => ? = 0
[[2,2,2,2,3],[3,3]]
=> [5,6,1,2,3,4,7] => [6,1,2,3,5,4,7] => ? = 0
[[2,2,3,3,3],[3,3]]
=> [3,4,1,2,5,6,7] => [4,1,3,2,5,6,7] => ? = 0
[[1,1,1,2,2],[2],[3]]
=> [7,4,1,2,3,5,6] => [4,1,2,7,3,5,6] => ? = 1
[[1,1,2,2,2],[2],[3]]
=> [7,3,1,2,4,5,6] => [3,1,7,2,4,5,6] => ? = 1
[[1,1,1,1],[2,2,3]]
=> [5,6,7,1,2,3,4] => [7,1,2,3,5,6,4] => ? = 0
[[1,1,1,1],[2,3,3]]
=> [5,6,7,1,2,3,4] => [7,1,2,3,5,6,4] => ? = 0
[[1,1,1,1],[3,3,3]]
=> [5,6,7,1,2,3,4] => [7,1,2,3,5,6,4] => ? = 0
[[1,1,1,3],[2,2,2]]
=> [4,5,6,1,2,3,7] => [6,1,2,4,5,3,7] => ? = 0
[[1,1,1,3],[2,2,3]]
=> [4,5,6,1,2,3,7] => [6,1,2,4,5,3,7] => ? = 0
[[1,1,1,2],[3,3,3]]
=> [5,6,7,1,2,3,4] => [7,1,2,3,5,6,4] => ? = 0
[[1,1,1,3],[2,3,3]]
=> [4,5,6,1,2,3,7] => [6,1,2,4,5,3,7] => ? = 0
[[1,1,1,3],[3,3,3]]
=> [4,5,6,1,2,3,7] => [6,1,2,4,5,3,7] => ? = 0
[[1,1,2,2],[3,3,3]]
=> [5,6,7,1,2,3,4] => [7,1,2,3,5,6,4] => ? = 0
[[1,1,2,3],[2,3,3]]
=> [3,5,6,1,2,4,7] => [6,1,3,2,5,4,7] => ? = 0
[[1,1,2,3],[3,3,3]]
=> [4,5,6,1,2,3,7] => [6,1,2,4,5,3,7] => ? = 0
[[1,2,2,2],[2,3,3]]
=> [2,6,7,1,3,4,5] => [7,2,1,3,4,6,5] => ? = 0
[[1,2,2,2],[3,3,3]]
=> [5,6,7,1,2,3,4] => [7,1,2,3,5,6,4] => ? = 0
[[1,2,2,3],[3,3,3]]
=> [4,5,6,1,2,3,7] => [6,1,2,4,5,3,7] => ? = 0
[[2,2,2,2],[3,3,3]]
=> [5,6,7,1,2,3,4] => [7,1,2,3,5,6,4] => ? = 0
[[2,2,2,3],[3,3,3]]
=> [4,5,6,1,2,3,7] => [6,1,2,4,5,3,7] => ? = 0
[[1,1,1,1],[2,2],[3]]
=> [7,5,6,1,2,3,4] => [6,1,2,3,7,5,4] => ? = 0
Description
The number of double exceedences of a permutation.
A double exceedence is an index $\sigma(i)$ such that $i < \sigma(i) < \sigma(\sigma(i))$.
Matching statistic: St000732
(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
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000732: Permutations ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000732: Permutations ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,2] => [1,2] => 0
[[2,2]]
=> [1,2] => [1,2] => [1,2] => 0
[[1],[2]]
=> [2,1] => [2,1] => [2,1] => 0
[[1,3]]
=> [1,2] => [1,2] => [1,2] => 0
[[2,3]]
=> [1,2] => [1,2] => [1,2] => 0
[[3,3]]
=> [1,2] => [1,2] => [1,2] => 0
[[1],[3]]
=> [2,1] => [2,1] => [2,1] => 0
[[2],[3]]
=> [2,1] => [2,1] => [2,1] => 0
[[1,1,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[[1,2,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[[2,2,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[[1,1],[2]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 0
[[1,2],[2]]
=> [2,1,3] => [2,3,1] => [3,2,1] => 0
[[1,4]]
=> [1,2] => [1,2] => [1,2] => 0
[[2,4]]
=> [1,2] => [1,2] => [1,2] => 0
[[3,4]]
=> [1,2] => [1,2] => [1,2] => 0
[[4,4]]
=> [1,2] => [1,2] => [1,2] => 0
[[1],[4]]
=> [2,1] => [2,1] => [2,1] => 0
[[2],[4]]
=> [2,1] => [2,1] => [2,1] => 0
[[3],[4]]
=> [2,1] => [2,1] => [2,1] => 0
[[1,1,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[[1,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[[2,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[[2,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[[3,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[[1,1],[3]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 0
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 0
[[1,3],[2]]
=> [2,1,3] => [2,3,1] => [3,2,1] => 0
[[1,3],[3]]
=> [2,1,3] => [2,3,1] => [3,2,1] => 0
[[2,2],[3]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 0
[[2,3],[3]]
=> [2,1,3] => [2,3,1] => [3,2,1] => 0
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [3,1,2] => 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[[1,1,1],[2]]
=> [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 0
[[1,1,2],[2]]
=> [3,1,2,4] => [1,3,4,2] => [1,4,3,2] => 0
[[1,2,2],[2]]
=> [2,1,3,4] => [2,3,4,1] => [4,2,3,1] => 0
[[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => [2,4,3,1] => 0
[[1,5]]
=> [1,2] => [1,2] => [1,2] => 0
[[2,5]]
=> [1,2] => [1,2] => [1,2] => 0
[[3,5]]
=> [1,2] => [1,2] => [1,2] => 0
[[4,5]]
=> [1,2] => [1,2] => [1,2] => 0
[[5,5]]
=> [1,2] => [1,2] => [1,2] => 0
[[1],[5]]
=> [2,1] => [2,1] => [2,1] => 0
[[2],[5]]
=> [2,1] => [2,1] => [2,1] => 0
[[3],[5]]
=> [2,1] => [2,1] => [2,1] => 0
[[4],[5]]
=> [2,1] => [2,1] => [2,1] => 0
[[1,1,2,2,2,2],[2]]
=> [3,1,2,4,5,6,7] => [1,3,4,5,6,7,2] => [1,7,3,4,5,6,2] => ? = 0
[[1,2,2,2,2,2],[2]]
=> [2,1,3,4,5,6,7] => [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => ? = 0
[[1,1,2,2,2],[2,2]]
=> [3,4,1,2,5,6,7] => [3,4,5,6,7,1,2] => [2,7,3,4,5,6,1] => ? = 0
[[1,1,1,2],[2,2,2]]
=> [4,5,6,1,2,3,7] => [4,5,6,7,1,2,3] => [2,3,7,4,5,6,1] => ? = 0
[[1,1,2,2,2,3],[2]]
=> [3,1,2,4,5,6,7] => [1,3,4,5,6,7,2] => [1,7,3,4,5,6,2] => ? = 0
[[1,1,2,2,3,3],[2]]
=> [3,1,2,4,5,6,7] => [1,3,4,5,6,7,2] => [1,7,3,4,5,6,2] => ? = 0
[[1,1,2,3,3,3],[2]]
=> [3,1,2,4,5,6,7] => [1,3,4,5,6,7,2] => [1,7,3,4,5,6,2] => ? = 0
[[1,1,3,3,3,3],[2]]
=> [3,1,2,4,5,6,7] => [1,3,4,5,6,7,2] => [1,7,3,4,5,6,2] => ? = 0
[[1,1,3,3,3,3],[3]]
=> [3,1,2,4,5,6,7] => [1,3,4,5,6,7,2] => [1,7,3,4,5,6,2] => ? = 0
[[1,2,2,2,2,3],[2]]
=> [2,1,3,4,5,6,7] => [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => ? = 0
[[1,2,2,2,3,3],[2]]
=> [2,1,3,4,5,6,7] => [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => ? = 0
[[1,2,2,3,3,3],[2]]
=> [2,1,3,4,5,6,7] => [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => ? = 0
[[1,2,3,3,3,3],[2]]
=> [2,1,3,4,5,6,7] => [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => ? = 0
[[1,2,3,3,3,3],[3]]
=> [3,1,2,4,5,6,7] => [1,3,4,5,6,7,2] => [1,7,3,4,5,6,2] => ? = 0
[[1,3,3,3,3,3],[2]]
=> [2,1,3,4,5,6,7] => [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => ? = 0
[[1,3,3,3,3,3],[3]]
=> [2,1,3,4,5,6,7] => [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => ? = 0
[[2,2,3,3,3,3],[3]]
=> [3,1,2,4,5,6,7] => [1,3,4,5,6,7,2] => [1,7,3,4,5,6,2] => ? = 0
[[2,3,3,3,3,3],[3]]
=> [2,1,3,4,5,6,7] => [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => ? = 0
[[1,1,2,2,3],[2,2]]
=> [3,4,1,2,5,6,7] => [3,4,5,6,7,1,2] => [2,7,3,4,5,6,1] => ? = 0
[[1,1,2,2,3],[2,3]]
=> [3,6,1,2,4,5,7] => [1,3,4,6,7,2,5] => [1,5,3,4,7,6,2] => ? = 0
[[1,1,2,3,3],[2,2]]
=> [3,4,1,2,5,6,7] => [3,4,5,6,7,1,2] => [2,7,3,4,5,6,1] => ? = 0
[[1,1,3,3,3],[2,2]]
=> [3,4,1,2,5,6,7] => [3,4,5,6,7,1,2] => [2,7,3,4,5,6,1] => ? = 0
[[1,1,3,3,3],[2,3]]
=> [3,4,1,2,5,6,7] => [3,4,5,6,7,1,2] => [2,7,3,4,5,6,1] => ? = 0
[[1,1,3,3,3],[3,3]]
=> [3,4,1,2,5,6,7] => [3,4,5,6,7,1,2] => [2,7,3,4,5,6,1] => ? = 0
[[1,2,2,2,2],[2,3]]
=> [2,7,1,3,4,5,6] => [2,3,4,5,7,1,6] => [6,2,3,4,5,7,1] => ? = 0
[[1,2,2,2,3],[2,3]]
=> [2,6,1,3,4,5,7] => [2,3,4,6,7,1,5] => [5,2,3,4,7,6,1] => ? = 0
[[1,2,2,3,3],[2,3]]
=> [2,5,1,3,4,6,7] => [2,3,5,6,7,1,4] => [4,2,3,7,5,6,1] => ? = 0
[[1,2,3,3,3],[2,3]]
=> [2,4,1,3,5,6,7] => [2,4,5,6,7,1,3] => [3,2,7,4,5,6,1] => ? = 0
[[1,2,3,3,3],[3,3]]
=> [3,4,1,2,5,6,7] => [3,4,5,6,7,1,2] => [2,7,3,4,5,6,1] => ? = 0
[[2,2,3,3,3],[3,3]]
=> [3,4,1,2,5,6,7] => [3,4,5,6,7,1,2] => [2,7,3,4,5,6,1] => ? = 0
[[1,1,2,2,2],[2],[3]]
=> [7,3,1,2,4,5,6] => [1,3,4,5,7,6,2] => [1,7,3,4,5,2,6] => ? = 1
[[1,2,2,2,2],[2],[3]]
=> [7,2,1,3,4,5,6] => [2,3,4,5,7,6,1] => [7,2,3,4,5,1,6] => ? = 1
[[1,3,3,3,3],[2],[3]]
=> [3,2,1,4,5,6,7] => [3,4,5,6,7,2,1] => [7,1,3,4,5,6,2] => ? = 1
[[1,1,1,3],[2,2,2]]
=> [4,5,6,1,2,3,7] => [4,5,6,7,1,2,3] => [2,3,7,4,5,6,1] => ? = 0
[[1,1,1,3],[2,2,3]]
=> [4,5,6,1,2,3,7] => [4,5,6,7,1,2,3] => [2,3,7,4,5,6,1] => ? = 0
[[1,1,1,3],[2,3,3]]
=> [4,5,6,1,2,3,7] => [4,5,6,7,1,2,3] => [2,3,7,4,5,6,1] => ? = 0
[[1,1,1,3],[3,3,3]]
=> [4,5,6,1,2,3,7] => [4,5,6,7,1,2,3] => [2,3,7,4,5,6,1] => ? = 0
[[1,1,2,3],[2,3,3]]
=> [3,5,6,1,2,4,7] => [3,5,6,7,1,2,4] => [2,4,3,7,5,6,1] => ? = 0
[[1,1,2,3],[3,3,3]]
=> [4,5,6,1,2,3,7] => [4,5,6,7,1,2,3] => [2,3,7,4,5,6,1] => ? = 0
[[1,2,2,2],[2,3,3]]
=> [2,6,7,1,3,4,5] => [2,3,6,7,1,4,5] => [4,2,3,5,7,6,1] => ? = 0
[[1,2,2,3],[3,3,3]]
=> [4,5,6,1,2,3,7] => [4,5,6,7,1,2,3] => [2,3,7,4,5,6,1] => ? = 0
[[2,2,2,3],[3,3,3]]
=> [4,5,6,1,2,3,7] => [4,5,6,7,1,2,3] => [2,3,7,4,5,6,1] => ? = 0
[[1,1,2,2],[2,2],[3]]
=> [7,3,4,1,2,5,6] => [3,4,5,7,1,6,2] => [6,7,3,4,5,2,1] => ? = 0
[[1,2,3,3],[2,3],[3]]
=> [4,2,5,1,3,6,7] => [4,5,6,7,2,3,1] => [7,3,1,4,5,6,2] => ? = 0
[[1,1,1],[2,2,2],[3]]
=> [7,4,5,6,1,2,3] => [4,5,7,1,2,6,3] => [2,6,7,4,5,3,1] => ? = 0
[[1,1,1],[2,2,3],[3]]
=> [6,4,5,7,1,2,3] => [4,6,7,1,2,5,3] => [2,5,7,4,3,6,1] => ? = 0
[[1,1,1],[2,3,3],[3]]
=> [5,4,6,7,1,2,3] => [5,6,7,1,2,4,3] => [2,4,7,3,5,6,1] => ? = 0
[[1,1,2],[2,2,3],[3]]
=> [6,3,4,7,1,2,5] => [3,6,7,1,4,5,2] => [4,7,3,5,2,6,1] => ? = 0
[[1,1,2],[2,3,3],[3]]
=> [5,3,6,7,1,2,4] => [5,6,7,1,3,4,2] => [3,7,4,2,5,6,1] => ? = 0
[[1,2,2],[2,3,3],[3]]
=> [5,2,6,7,1,3,4] => [5,6,7,2,3,4,1] => [7,3,4,1,5,6,2] => ? = 0
Description
The number of double deficiencies of a permutation.
A double deficiency is an index $\sigma(i)$ such that $i > \sigma(i) > \sigma(\sigma(i))$.
Matching statistic: St000365
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000365: Permutations ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000365: Permutations ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[[2,2]]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[[1],[2]]
=> [2,1] => [[.,.],.]
=> [1,2] => 0
[[1,3]]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[[2,3]]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[[3,3]]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[[1],[3]]
=> [2,1] => [[.,.],.]
=> [1,2] => 0
[[2],[3]]
=> [2,1] => [[.,.],.]
=> [1,2] => 0
[[1,1,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 0
[[1,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 0
[[2,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 0
[[1,1],[2]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => 0
[[1,2],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 0
[[1,4]]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[[2,4]]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[[3,4]]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[[4,4]]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[[1],[4]]
=> [2,1] => [[.,.],.]
=> [1,2] => 0
[[2],[4]]
=> [2,1] => [[.,.],.]
=> [1,2] => 0
[[3],[4]]
=> [2,1] => [[.,.],.]
=> [1,2] => 0
[[1,1,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 0
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 0
[[1,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 0
[[2,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 0
[[2,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 0
[[3,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 0
[[1,1],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => 0
[[1,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => 0
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 0
[[1,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 0
[[2,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => 0
[[2,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 0
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 1
[[1,1,1,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[[1,1,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[[1,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[[2,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[[1,1,1],[2]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 0
[[1,1,2],[2]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 0
[[1,2,2],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 0
[[1,1],[2,2]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0
[[1,5]]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[[2,5]]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[[3,5]]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[[4,5]]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[[5,5]]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[[1],[5]]
=> [2,1] => [[.,.],.]
=> [1,2] => 0
[[2],[5]]
=> [2,1] => [[.,.],.]
=> [1,2] => 0
[[3],[5]]
=> [2,1] => [[.,.],.]
=> [1,2] => 0
[[4],[5]]
=> [2,1] => [[.,.],.]
=> [1,2] => 0
[[1,1,1,1,1,1,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,1,1,1,1,2,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,1,1,1,2,2,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,1,1,2,2,2,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,1,2,2,2,2,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,2,2,2,2,2,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[2,2,2,2,2,2,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,1,1,1,1,1],[2]]
=> [7,1,2,3,4,5,6] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ? = 0
[[1,1,1,1,1,2],[2]]
=> [6,1,2,3,4,5,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ? = 0
[[1,1,1,1,2,2],[2]]
=> [5,1,2,3,4,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ? = 0
[[1,1,1,2,2,2],[2]]
=> [4,1,2,3,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ? = 0
[[1,1,2,2,2,2],[2]]
=> [3,1,2,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ? = 0
[[1,2,2,2,2,2],[2]]
=> [2,1,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ? = 0
[[1,1,1,1,1],[2,2]]
=> [6,7,1,2,3,4,5] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ? = 0
[[1,1,1,1,2],[2,2]]
=> [5,6,1,2,3,4,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ? = 0
[[1,1,2,2,2],[2,2]]
=> [3,4,1,2,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ? = 0
[[1,1,1,1],[2,2,2]]
=> [5,6,7,1,2,3,4] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 0
[[1,1,1,2],[2,2,2]]
=> [4,5,6,1,2,3,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 0
[[1,1,1,1,1,1,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,1,1,1,1,2,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,1,1,1,1,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,1,1,1,2,2,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,1,1,1,2,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,1,1,1,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,1,1,2,2,2,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,1,1,2,2,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,1,1,2,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,1,1,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,1,2,2,2,2,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,1,2,2,2,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,1,2,2,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,1,2,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,1,3,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,2,2,2,2,2,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,2,2,2,2,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,2,2,2,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,2,2,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,2,3,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,3,3,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[2,2,2,2,2,2,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[2,2,2,2,2,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[2,2,2,2,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[2,2,2,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[2,2,3,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[2,3,3,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[3,3,3,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,1,1,1,1,1],[3]]
=> [7,1,2,3,4,5,6] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ? = 0
[[1,1,1,1,1,2],[3]]
=> [7,1,2,3,4,5,6] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ? = 0
[[1,1,1,1,1,3],[2]]
=> [6,1,2,3,4,5,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ? = 0
[[1,1,1,1,1,3],[3]]
=> [6,1,2,3,4,5,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ? = 0
Description
The number of double ascents of a permutation.
A double ascent of a permutation $\pi$ is a position $i$ such that $\pi(i) < \pi(i+1) < \pi(i+2)$.
Matching statistic: St001163
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001163: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001163: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[2,2]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[1],[2]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[1,3]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[2,3]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[3,3]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[1],[3]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[2],[3]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[1,1,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[1,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[2,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[1,1],[2]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[[1,2],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[[1,4]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[2,4]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[3,4]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[4,4]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[1],[4]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[2],[4]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[3],[4]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[1,1,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[1,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[2,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[2,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[3,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[1,1],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[[1,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[[1,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[[2,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[[2,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1
[[1,1,1,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,1,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[[2,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,1,1],[2]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 0
[[1,1,2],[2]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 0
[[1,2,2],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 0
[[1,1],[2,2]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[[1,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[2,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[3,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[4,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[5,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[1],[5]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[2],[5]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[3],[5]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[4],[5]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[[1,1,1,1,1,1,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,1,2,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,2,2,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,2,2,2,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,2,2,2,2,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,2,2,2,2,2,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[2,2,2,2,2,2,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,1,1],[2]]
=> [7,1,2,3,4,5,6] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,1,2],[2]]
=> [6,1,2,3,4,5,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,2,2],[2]]
=> [5,1,2,3,4,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,2,2,2],[2]]
=> [4,1,2,3,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[[1,1,2,2,2,2],[2]]
=> [3,1,2,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[[1,2,2,2,2,2],[2]]
=> [2,1,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,1],[2,2]]
=> [6,7,1,2,3,4,5] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,2],[2,2]]
=> [5,6,1,2,3,4,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[[1,1,2,2,2],[2,2]]
=> [3,4,1,2,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[[1,1,1,1],[2,2,2]]
=> [5,6,7,1,2,3,4] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[[1,1,1,2],[2,2,2]]
=> [4,5,6,1,2,3,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[[1,1,1,1,1,1,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,1,2,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,1,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,2,2,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,2,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,2,2,2,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,2,2,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,2,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,2,2,2,2,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,2,2,2,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,2,2,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,2,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,3,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,2,2,2,2,2,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,2,2,2,2,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,2,2,2,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,2,2,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,2,3,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,3,3,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[2,2,2,2,2,2,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[2,2,2,2,2,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[2,2,2,2,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[2,2,2,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[2,2,3,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[2,3,3,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[3,3,3,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,1,1],[3]]
=> [7,1,2,3,4,5,6] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,1,2],[3]]
=> [7,1,2,3,4,5,6] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,1,3],[2]]
=> [6,1,2,3,4,5,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[[1,1,1,1,1,3],[3]]
=> [6,1,2,3,4,5,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
Description
The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra.
The following 24 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001130The number of two successive successions in a permutation. St001866The nesting alignments of a signed permutation. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001896The number of right descents of a signed permutations. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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. St001857The number of edges in the reduced word graph of a signed permutation. St000782The indicator function of whether a given perfect matching is an L & P matching. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000102The charge of a semistandard tableau. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001408The number of maximal entries in a semistandard tableau. St001410The minimal entry of a semistandard tableau. St001409The maximal entry of a semistandard tableau. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
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