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Your data matches 129 different statistics following compositions of up to 3 maps.
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Matching statistic: St000936
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Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000936: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000936: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,3,2] => [2,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3] => [2,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,1] => [2,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [2,2]
=> [2]
=> 0
[1,2,4,3] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,3,2,4] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,4,3,2] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,1,3,4] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,3,4,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,4,1,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,1,4,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,2,1,4] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,4,2,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[4,1,2,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[4,2,3,1] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,3,1,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [2,2,1]
=> [2,1]
=> 2
[1,2,3,5,4] => [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,5,3] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,2,5,3,4] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,2,5,4,3] => [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,4,2,5] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,3,4,5,2] => [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,3,5,2,4] => [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,4,2,3,5] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,4,2,5,3] => [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,4,5,3,2] => [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,5,2,3,4] => [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,5,3,4,2] => [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,2,3] => [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,1,3,4,5] => [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,1,4,5,3] => [3,2]
=> [3,1,1]
=> [1,1]
=> 0
[2,1,5,3,4] => [3,2]
=> [3,1,1]
=> [1,1]
=> 0
[2,3,1,4,5] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[2,3,1,5,4] => [3,2]
=> [3,1,1]
=> [1,1]
=> 0
[2,3,4,1,5] => [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,3,5,4,1] => [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,4,1,3,5] => [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,4,3,1,5] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[2,4,3,5,1] => [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,4,5,1,3] => [3,2]
=> [3,1,1]
=> [1,1]
=> 0
[2,5,1,4,3] => [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,5,3,1,4] => [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
Description
The number of even values of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugace class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $4$.
It is shown in [1] that the sum of the values of the statistic over all partitions of a given size is even.
Matching statistic: St001359
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001359: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 9%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001359: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 9%
Values
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 2 = 0 + 2
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 2 = 0 + 2
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 2 = 0 + 2
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 2 = 0 + 2
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 2 = 0 + 2
[2,3,4,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 2 = 0 + 2
[2,4,1,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 2 = 0 + 2
[3,1,4,2] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 2 = 0 + 2
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 2 = 0 + 2
[3,4,2,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 2 = 0 + 2
[4,1,2,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 2 = 0 + 2
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 2 = 0 + 2
[4,3,1,2] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 2 = 0 + 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 4 = 2 + 2
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 2 = 0 + 2
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 2 = 0 + 2
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 2 = 0 + 2
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 2 = 0 + 2
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 2 = 0 + 2
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 2 = 0 + 2
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 2 = 0 + 2
[1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 2 = 0 + 2
[1,3,5,2,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 2 = 0 + 2
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 2 = 0 + 2
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 2 = 0 + 2
[1,4,2,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 2 = 0 + 2
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 2 = 0 + 2
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 2 = 0 + 2
[1,4,5,3,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 2 = 0 + 2
[1,5,2,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 2 = 0 + 2
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 2 = 0 + 2
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 2 = 0 + 2
[1,5,3,4,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 2 = 0 + 2
[1,5,4,2,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 2 = 0 + 2
[2,1,3,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 2 = 0 + 2
[2,1,4,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 2 = 0 + 2
[2,1,5,3,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 2 = 0 + 2
[2,3,1,4,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 2 = 0 + 2
[2,3,1,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 2 = 0 + 2
[2,3,4,1,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 2 = 0 + 2
[2,3,5,4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 2 = 0 + 2
[2,4,1,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 2 = 0 + 2
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 2 = 0 + 2
[2,4,3,5,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 2 = 0 + 2
[2,4,5,1,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 2 = 0 + 2
[2,5,1,4,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 2 = 0 + 2
[2,5,3,1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 2 = 0 + 2
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ? = 4 + 2
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 1 + 2
[1,2,3,5,4,6] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 1 + 2
[1,2,3,5,6,4] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 2 + 2
[1,2,3,6,4,5] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 2 + 2
[1,2,3,6,5,4] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 1 + 2
[1,2,4,3,5,6] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 1 + 2
[1,2,4,5,3,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 2 + 2
[1,2,4,5,6,3] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 2
[1,2,4,6,3,5] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 2
[1,2,4,6,5,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 2 + 2
[1,2,5,3,4,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 2 + 2
[1,2,5,3,6,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 2
[1,2,5,4,3,6] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 1 + 2
[1,2,5,4,6,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 2 + 2
[1,2,5,6,4,3] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 2
[1,2,6,3,4,5] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 2
[1,2,6,3,5,4] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 2 + 2
[1,2,6,4,3,5] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 2 + 2
[1,2,6,4,5,3] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 1 + 2
[1,2,6,5,3,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 2
[1,3,2,4,5,6] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 1 + 2
[1,3,4,2,5,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 2 + 2
[1,3,4,5,2,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 2
[1,3,4,6,5,2] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 2
[1,3,5,2,4,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 2
[1,3,5,4,2,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 2 + 2
[1,3,5,4,6,2] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 2
[1,3,6,2,5,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 2
[1,3,6,4,2,5] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 2
[1,3,6,4,5,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 2 + 2
[1,4,2,3,5,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 2 + 2
[1,4,2,5,3,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 2
[1,4,2,6,5,3] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 2
[1,4,3,2,5,6] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 1 + 2
[1,4,3,5,2,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 2 + 2
[1,4,3,5,6,2] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 2
[1,4,3,6,2,5] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 2
[1,4,3,6,5,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 2 + 2
[1,4,5,3,2,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 2
[1,4,6,3,5,2] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 2
[1,5,2,3,4,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 2
[1,5,2,4,3,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 2 + 2
[1,5,2,4,6,3] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 2
[1,5,3,2,4,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 2 + 2
[1,5,3,2,6,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 2
[1,5,3,4,2,6] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 1 + 2
[1,5,3,4,6,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 2 + 2
[1,5,3,6,4,2] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 2
[1,5,4,2,3,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 2
Description
The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles.
In other words, this is $2^k$ where $k$ is the number of cycles of length at least three ([[St000486]]) in its cycle decomposition.
The generating function for the number of equivalence classes, $f(n)$, is
$$\sum_{n\geq 0} f(n)\frac{x^n}{n!} = \frac{e(\frac{x}{2} + \frac{x^2}{4})}{\sqrt{1-x}}.$$
Matching statistic: St001001
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 4%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 4%
Values
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[3,2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0
[2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0
[3,1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[3,4,2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0
[4,1,2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[4,3,1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,3,4,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[1,3,5,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,4,2,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,4,5,3,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[1,5,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,5,3,4,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
[1,5,4,2,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[2,1,3,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[2,3,1,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[2,3,4,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[2,3,5,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[2,4,1,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[2,4,3,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[2,4,5,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[2,5,1,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[2,5,3,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[2,5,3,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[2,5,4,3,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[3,1,2,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[3,1,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[3,1,4,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[3,1,5,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[3,2,1,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[3,2,4,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[3,2,5,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[3,2,5,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[3,4,1,5,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[3,4,2,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[3,4,5,2,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[3,5,1,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[3,5,2,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[3,5,4,1,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[4,1,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[4,1,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[4,1,3,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[4,1,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[4,2,1,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[4,2,1,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[4,2,3,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
[4,2,3,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[4,2,5,3,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[4,3,1,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[4,3,2,5,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[4,3,5,1,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[4,5,1,3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[4,5,2,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[4,5,3,2,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[5,1,2,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[5,1,3,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[5,1,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[5,1,4,3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[5,2,1,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[5,2,1,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[5,2,3,1,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[5,2,3,4,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
[5,2,4,1,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[5,3,1,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[5,3,2,1,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[5,3,4,2,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[5,4,1,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[5,4,2,3,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[5,4,3,1,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 4
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[1,3,2,5,6,4] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
Description
The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001371
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 4%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 4%
Values
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0
[3,2,1] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0
[1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 0
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
[2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
[3,1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[3,4,2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
[4,1,2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[4,3,1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 2
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,3,4,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[1,3,5,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,4,2,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,4,5,3,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[1,5,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,5,3,4,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[1,5,4,2,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[2,1,3,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[2,3,1,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[2,3,4,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[2,3,5,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[2,4,1,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[2,4,3,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[2,4,5,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[2,5,1,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[2,5,3,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[2,5,3,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[2,5,4,3,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[3,1,2,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[3,1,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[3,1,4,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[3,1,5,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[3,2,1,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[3,2,4,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[3,2,5,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[3,2,5,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[3,4,1,5,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[3,4,2,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[3,4,5,2,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[3,5,1,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[3,5,2,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[3,5,4,1,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[4,1,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[4,1,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[4,1,3,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[4,1,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[4,2,1,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[4,2,1,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[4,2,3,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[4,2,3,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[4,2,5,3,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[4,3,1,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[4,3,2,5,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[4,3,5,1,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[4,5,1,3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[4,5,2,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[4,5,3,2,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[5,1,2,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[5,1,3,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[5,1,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[5,1,4,3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[5,2,1,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[5,2,1,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[5,2,3,1,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[5,2,3,4,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[5,2,4,1,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[5,3,1,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[5,3,2,1,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[5,3,4,2,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[5,4,1,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[5,4,2,3,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[5,4,3,1,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => ? = 4
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 1
[1,3,2,5,6,4] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
Description
The length of the longest Yamanouchi prefix of a binary word.
This is the largest index $i$ such that in each of the prefixes $w_1$, $w_1w_2$, $w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.
Matching statistic: St001730
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 4%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 4%
Values
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0
[3,2,1] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0
[1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 0
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
[2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
[3,1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[3,4,2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
[4,1,2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[4,3,1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 2
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,3,4,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[1,3,5,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,4,2,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,4,5,3,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[1,5,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,5,3,4,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[1,5,4,2,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[2,1,3,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[2,3,1,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[2,3,4,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[2,3,5,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[2,4,1,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[2,4,3,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[2,4,5,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[2,5,1,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[2,5,3,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[2,5,3,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[2,5,4,3,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[3,1,2,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[3,1,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[3,1,4,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[3,1,5,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[3,2,1,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[3,2,4,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[3,2,5,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[3,2,5,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[3,4,1,5,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[3,4,2,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[3,4,5,2,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[3,5,1,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[3,5,2,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[3,5,4,1,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[4,1,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[4,1,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[4,1,3,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[4,1,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[4,2,1,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[4,2,1,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[4,2,3,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[4,2,3,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[4,2,5,3,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[4,3,1,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[4,3,2,5,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[4,3,5,1,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[4,5,1,3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[4,5,2,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[4,5,3,2,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[5,1,2,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[5,1,3,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[5,1,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[5,1,4,3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[5,2,1,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[5,2,1,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[5,2,3,1,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[5,2,3,4,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[5,2,4,1,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[5,3,1,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[5,3,2,1,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[5,3,4,2,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[5,4,1,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[5,4,2,3,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[5,4,3,1,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => ? = 4
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 1
[1,3,2,5,6,4] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
Description
The number of times the path corresponding to a binary word crosses the base line.
Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.
Matching statistic: St001803
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 4%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 4%
Values
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[3,2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
[2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0
[2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0
[3,1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
[3,4,2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0
[4,1,2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
[4,3,1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 2
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[1,3,4,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[1,3,5,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[1,4,2,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[1,4,5,3,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[1,5,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[1,5,3,4,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
[1,5,4,2,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[2,1,3,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[2,3,1,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[2,3,4,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[2,3,5,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[2,4,1,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[2,4,3,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[2,4,5,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[2,5,1,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[2,5,3,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[2,5,3,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[2,5,4,3,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[3,1,2,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[3,1,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[3,1,4,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[3,1,5,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[3,2,1,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[3,2,4,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[3,2,5,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[3,2,5,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[3,4,1,5,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[3,4,2,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[3,4,5,2,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[3,5,1,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[3,5,2,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[3,5,4,1,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[4,1,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[4,1,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[4,1,3,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[4,1,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[4,2,1,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[4,2,1,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[4,2,3,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
[4,2,3,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[4,2,5,3,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[4,3,1,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[4,3,2,5,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[4,3,5,1,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[4,5,1,3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[4,5,2,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[4,5,3,2,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[5,1,2,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[5,1,3,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[5,1,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[5,1,4,3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[5,2,1,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[5,2,1,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[5,2,3,1,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[5,2,3,4,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
[5,2,4,1,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[5,3,1,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[5,3,2,1,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[5,3,4,2,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[5,4,1,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[5,4,2,3,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[5,4,3,1,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> ? = 4
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 1
[1,3,2,5,6,4] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
Description
The maximal overlap of the cylindrical tableau associated with a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
In particular, the statistic equals $0$, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
Matching statistic: St001195
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 4%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 4%
Values
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0 + 1
[2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0 + 1
[3,1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0 + 1
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[3,4,2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0 + 1
[4,1,2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0 + 1
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[4,3,1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,3,4,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[1,3,5,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,4,2,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,4,5,3,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[1,5,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[1,5,4,2,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,3,1,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,3,4,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[2,3,5,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[2,4,1,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[2,4,3,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[2,4,5,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,5,1,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[2,5,3,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[2,5,3,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[2,5,4,3,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,1,2,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[3,1,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,1,4,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[3,1,5,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[3,2,1,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[3,2,4,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[3,2,5,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[3,2,5,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[3,4,1,5,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,4,2,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[3,4,5,2,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,5,1,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,5,2,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[3,5,4,1,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[4,1,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[4,1,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[4,1,3,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[4,1,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[4,2,1,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[4,2,1,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[4,2,3,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[4,2,3,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[4,2,5,3,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[4,3,1,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[4,3,2,5,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[4,3,5,1,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[4,5,1,3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[4,5,2,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[4,5,3,2,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[5,1,2,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[5,1,3,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[5,1,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[5,1,4,3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[5,2,1,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[5,2,1,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[5,2,3,1,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[5,2,3,4,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[5,2,4,1,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[5,3,1,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[5,3,2,1,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[5,3,4,2,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[5,4,1,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[5,4,2,3,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[5,4,3,1,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 4 + 1
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 + 1
[1,3,2,5,6,4] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Matching statistic: St001208
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 4%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 4%
Values
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
[3,2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 0 + 1
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0 + 1
[2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0 + 1
[3,1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0 + 1
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[3,4,2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0 + 1
[4,1,2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0 + 1
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[4,3,1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 2 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[1,3,4,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[1,3,5,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[1,4,2,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[1,4,5,3,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[1,5,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[1,5,4,2,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[2,3,1,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[2,3,4,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[2,3,5,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[2,4,1,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[2,4,3,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[2,4,5,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[2,5,1,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[2,5,3,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[2,5,3,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[2,5,4,3,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[3,1,2,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[3,1,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[3,1,4,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[3,1,5,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[3,2,1,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[3,2,4,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[3,2,5,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[3,2,5,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[3,4,1,5,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[3,4,2,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[3,4,5,2,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[3,5,1,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[3,5,2,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[3,5,4,1,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[4,1,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[4,1,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[4,1,3,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[4,1,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[4,2,1,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[4,2,1,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[4,2,3,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[4,2,3,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[4,2,5,3,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[4,3,1,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[4,3,2,5,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[4,3,5,1,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[4,5,1,3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[4,5,2,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[4,5,3,2,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[5,1,2,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[5,1,3,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[5,1,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[5,1,4,3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[5,2,1,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[5,2,1,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[5,2,3,1,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[5,2,3,4,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[5,2,4,1,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[5,3,1,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[5,3,2,1,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[5,3,4,2,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[5,4,1,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[5,4,2,3,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[5,4,3,1,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? = 4 + 1
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 1 + 1
[1,3,2,5,6,4] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
Description
The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$.
Matching statistic: St001520
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001520: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 4%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001520: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 4%
Values
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
[3,2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 0 + 1
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0 + 1
[2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0 + 1
[3,1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0 + 1
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[3,4,2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0 + 1
[4,1,2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0 + 1
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[4,3,1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 2 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[1,3,4,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[1,3,5,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[1,4,2,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[1,4,5,3,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[1,5,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[1,5,4,2,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[2,3,1,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[2,3,4,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[2,3,5,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[2,4,1,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[2,4,3,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[2,4,5,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[2,5,1,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[2,5,3,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[2,5,3,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[2,5,4,3,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[3,1,2,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[3,1,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[3,1,4,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[3,1,5,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[3,2,1,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[3,2,4,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[3,2,5,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[3,2,5,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[3,4,1,5,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[3,4,2,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[3,4,5,2,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[3,5,1,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[3,5,2,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[3,5,4,1,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[4,1,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[4,1,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[4,1,3,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[4,1,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[4,2,1,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[4,2,1,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[4,2,3,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[4,2,3,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[4,2,5,3,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[4,3,1,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[4,3,2,5,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[4,3,5,1,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[4,5,1,3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[4,5,2,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[4,5,3,2,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[5,1,2,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[5,1,3,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[5,1,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[5,1,4,3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[5,2,1,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[5,2,1,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[5,2,3,1,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[5,2,3,4,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[5,2,4,1,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[5,3,1,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[5,3,2,1,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[5,3,4,2,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[5,4,1,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[5,4,2,3,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[5,4,3,1,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? = 4 + 1
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 1 + 1
[1,3,2,5,6,4] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
Description
The number of strict 3-descents.
A '''strict 3-descent''' of a permutation $\pi$ of $\{1,2, \dots ,n \}$ is a pair $(i,i+3)$ with $ i+3 \leq n$ and $\pi(i) > \pi(i+3)$.
Matching statistic: St001804
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001804: Standard tableaux ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 4%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001804: Standard tableaux ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 4%
Values
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 0 + 2
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 0 + 2
[3,2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 0 + 2
[1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0 + 2
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 0 + 2
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 0 + 2
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 0 + 2
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 0 + 2
[2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0 + 2
[2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0 + 2
[3,1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0 + 2
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 0 + 2
[3,4,2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0 + 2
[4,1,2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0 + 2
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 0 + 2
[4,3,1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0 + 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 2 + 2
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[1,3,4,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[1,3,5,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[1,4,2,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[1,4,5,3,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[1,5,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[1,5,3,4,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
[1,5,4,2,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[2,1,3,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[2,3,1,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[2,3,4,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[2,3,5,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[2,4,1,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[2,4,3,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[2,4,5,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[2,5,1,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[2,5,3,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[2,5,3,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[2,5,4,3,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[3,1,2,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[3,1,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[3,1,4,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[3,1,5,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[3,2,1,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[3,2,4,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[3,2,5,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[3,2,5,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[3,4,1,5,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[3,4,2,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[3,4,5,2,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[3,5,1,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[3,5,2,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[3,5,4,1,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[4,1,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[4,1,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[4,1,3,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[4,1,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[4,2,1,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[4,2,1,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[4,2,3,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
[4,2,3,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[4,2,5,3,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[4,3,1,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[4,3,2,5,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[4,3,5,1,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[4,5,1,3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[4,5,2,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[4,5,3,2,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[5,1,2,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[5,1,3,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[5,1,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[5,1,4,3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[5,2,1,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[5,2,1,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[5,2,3,1,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[5,2,3,4,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
[5,2,4,1,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[5,3,1,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[5,3,2,1,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[5,3,4,2,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[5,4,1,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[5,4,2,3,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[5,4,3,1,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> ? = 4 + 2
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 1 + 2
[1,3,2,5,6,4] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 0 + 2
Description
The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
This statistic equals $\max_C\big(\ell(C) - \ell(T)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
The following 119 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000879The number of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000929The constant term of the character polynomial of an integer partition. St000667The greatest common divisor of the parts of the partition. St000993The multiplicity of the largest part of an integer partition. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001881The number of factors of a lattice as a Cartesian product of lattices. St000137The Grundy value of an integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000928The sum of the coefficients of the character polynomial of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001249Sum of the odd parts of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001383The BG-rank of an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001561The value of the elementary symmetric function evaluated at 1. St001586The number of odd parts smaller than the largest even part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000284The Plancherel distribution on integer partitions. St000618The number of self-evacuating tableaux of given shape. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001280The number of parts of an integer partition that are at least two. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001568The smallest positive integer that does not appear twice in the partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001780The order of promotion on the set of standard tableaux of given shape. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. 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. St001889The size of the connectivity set of a signed permutation. St001624The breadth of a lattice. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation. St000068The number of minimal elements in a poset. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001625The Möbius invariant of a lattice. St001964The interval resolution global dimension of a poset. St001429The number of negative entries in a signed permutation. St001621The number of atoms of a lattice. St000627The exponent of a binary word. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000878The number of ones minus the number of zeros of a binary word. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001868The number of alignments of type NE of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation.
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