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Your data matches 46 different statistics following compositions of up to 3 maps.
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Matching statistic: St000147
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> 2
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> 2
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 2
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 2
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 2
[2,1,4,5,3] => [3,2]
=> [2]
=> 2
[2,1,5,3,4] => [3,2]
=> [2]
=> 2
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 2
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> 1
[2,3,1,5,4] => [3,2]
=> [2]
=> 2
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> 1
[2,4,5,1,3] => [3,2]
=> [2]
=> 2
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> 1
[2,5,4,3,1] => [3,2]
=> [2]
=> 2
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> 1
[3,1,2,5,4] => [3,2]
=> [2]
=> 2
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 2
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 1
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> 2
[3,4,1,5,2] => [3,2]
=> [2]
=> 2
[3,4,5,2,1] => [3,2]
=> [2]
=> 2
Description
The largest part of an integer partition.
Matching statistic: St001280
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,1,1]
=> [3]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> 1
[2,1,4,3] => [2,2]
=> [2,2]
=> 2
[3,2,1,4] => [2,1,1]
=> [3,1]
=> 1
[3,4,1,2] => [2,2]
=> [2,2]
=> 2
[4,2,3,1] => [2,1,1]
=> [3,1]
=> 1
[4,3,2,1] => [2,2]
=> [2,2]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [3,1,1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [3,1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [3,1,1]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [3,1,1]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [3,2]
=> 2
[1,5,2,4,3] => [3,1,1]
=> [3,1,1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [3,1,1]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [4,1]
=> 1
[1,5,4,3,2] => [2,2,1]
=> [3,2]
=> 2
[2,1,3,4,5] => [2,1,1,1]
=> [4,1]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [3,2]
=> 2
[2,1,4,3,5] => [2,2,1]
=> [3,2]
=> 2
[2,1,4,5,3] => [3,2]
=> [2,2,1]
=> 2
[2,1,5,3,4] => [3,2]
=> [2,2,1]
=> 2
[2,1,5,4,3] => [2,2,1]
=> [3,2]
=> 2
[2,3,1,4,5] => [3,1,1]
=> [3,1,1]
=> 1
[2,3,1,5,4] => [3,2]
=> [2,2,1]
=> 2
[2,4,3,1,5] => [3,1,1]
=> [3,1,1]
=> 1
[2,4,5,1,3] => [3,2]
=> [2,2,1]
=> 2
[2,5,3,4,1] => [3,1,1]
=> [3,1,1]
=> 1
[2,5,4,3,1] => [3,2]
=> [2,2,1]
=> 2
[3,1,2,4,5] => [3,1,1]
=> [3,1,1]
=> 1
[3,1,2,5,4] => [3,2]
=> [2,2,1]
=> 2
[3,2,1,4,5] => [2,1,1,1]
=> [4,1]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [3,2]
=> 2
[3,2,4,1,5] => [3,1,1]
=> [3,1,1]
=> 1
[3,2,5,4,1] => [3,1,1]
=> [3,1,1]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [3,2]
=> 2
[3,4,1,5,2] => [3,2]
=> [2,2,1]
=> 2
[3,4,5,2,1] => [3,2]
=> [2,2,1]
=> 2
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000010
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,1,1]
=> [1,1]
=> [2]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> [1,1]
=> 2
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> [1,1]
=> 2
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> [1,1]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[2,1,4,5,3] => [3,2]
=> [2]
=> [1,1]
=> 2
[2,1,5,3,4] => [3,2]
=> [2]
=> [1,1]
=> 2
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[2,3,1,5,4] => [3,2]
=> [2]
=> [1,1]
=> 2
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[2,4,5,1,3] => [3,2]
=> [2]
=> [1,1]
=> 2
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[2,5,4,3,1] => [3,2]
=> [2]
=> [1,1]
=> 2
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[3,1,2,5,4] => [3,2]
=> [2]
=> [1,1]
=> 2
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[3,4,1,5,2] => [3,2]
=> [2]
=> [1,1]
=> 2
[3,4,5,2,1] => [3,2]
=> [2]
=> [1,1]
=> 2
Description
The length of the partition.
Matching statistic: St000676
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,4,5,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,5,3,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,3,1,5,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,4,5,1,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,5,4,3,1] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,1,2,5,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,4,1,5,2] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[3,4,5,2,1] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St000734
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> [[1,2]]
=> 2
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> [[1,2]]
=> 2
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> [[1,2]]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[2,1,4,5,3] => [3,2]
=> [2]
=> [[1,2]]
=> 2
[2,1,5,3,4] => [3,2]
=> [2]
=> [[1,2]]
=> 2
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[2,3,1,5,4] => [3,2]
=> [2]
=> [[1,2]]
=> 2
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[2,4,5,1,3] => [3,2]
=> [2]
=> [[1,2]]
=> 2
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[2,5,4,3,1] => [3,2]
=> [2]
=> [[1,2]]
=> 2
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[3,1,2,5,4] => [3,2]
=> [2]
=> [[1,2]]
=> 2
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[3,4,1,5,2] => [3,2]
=> [2]
=> [[1,2]]
=> 2
[3,4,5,2,1] => [3,2]
=> [2]
=> [[1,2]]
=> 2
Description
The last entry in the first row of a standard tableau.
Matching statistic: St001039
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,4,5,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,5,3,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,3,1,5,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,4,5,1,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,5,4,3,1] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,1,2,5,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,4,1,5,2] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[3,4,5,2,1] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St001291
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 98%●distinct values known / distinct values provided: 83%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 98%●distinct values known / distinct values provided: 83%
Values
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,4,3] => [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[3,4,1,2] => [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[2,1,4,5,3] => [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,1,5,3,4] => [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,3,1,5,4] => [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,4,5,1,3] => [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,5,4,3,1] => [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,2,5,4] => [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[3,4,1,5,2] => [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[3,4,5,2,1] => [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[8,2,3,4,5,6,7,1] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[7,2,3,4,5,6,1,8] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[6,2,3,4,5,1,7,8] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[5,2,3,4,1,6,7,8] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[4,2,3,1,5,6,7,8] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[3,2,1,4,5,6,7,8] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[2,1,3,4,5,6,7,8] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,2,3,4,5,8,7,6] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,2,3,4,5,6,8,7] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,2,3,4,7,6,5,8] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,2,3,4,5,7,6,8] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,2,3,6,5,4,7,8] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,2,3,4,6,5,7,8] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,2,5,4,3,6,7,8] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,2,3,5,4,6,7,8] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,4,3,2,5,6,7,8] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,2,4,3,5,6,7,8] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,3,2,4,5,6,7,8] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,2,3,4,8,6,7,5] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,2,3,7,5,6,4,8] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,2,3,8,5,6,7,4] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,2,6,4,5,3,7,8] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,2,8,4,5,6,7,3] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,5,3,4,2,6,7,8] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,6,3,4,5,2,7,8] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,7,3,4,5,6,2,8] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,8,3,4,5,6,7,2] => [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[2,1,4,3,6,5,8,7,10,9,12,11] => [2,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[6,5,4,3,2,1,12,11,10,9,8,7] => [2,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[12,11,10,9,8,7,6,5,4,3,2,1] => [2,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,3,4,5,6,7,9,8] => [2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[2,1,3,4,5,6,7,8,9] => [2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,2,3,4,5,6,8,9,7] => [3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,2,3,4,5,6,9,8,7] => [2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,2,3,4,5,6,7,9,10,8] => [3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,2,3,4,5,8,7,9,6] => [3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,2,3,4,5,6,8,9,10,7] => [4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[2,4,7,8,10,12,1,3,5,6,9,11] => [6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 + 1
[3,4,6,9,11,12,1,2,5,7,8,10] => [6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 + 1
[3,4,9,10,11,12,1,2,5,6,7,8] => [6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 + 1
[3,5,6,8,11,12,1,2,4,7,9,10] => [6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 + 1
[4,6,8,10,11,12,1,2,3,5,7,9] => [6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 + 1
[5,6,7,8,11,12,1,2,3,4,9,10] => [6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 + 1
[7,8,9,10,11,12,1,2,3,4,5,6] => [2,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,3,4,5,6,9,7,8] => [3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[3,5,2,6,4,1,9,11,8,12,10,7] => [6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 + 1
[4,6,9,3,10,5,11,12,8,7,2,1] => [6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 + 1
[5,6,8,10,4,3,11,7,12,9,2,1] => [6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 + 1
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
Matching statistic: St000028
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000028: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000028: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,2,4,5,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,2,5,3,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,2,5,4,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,3,2,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[1,3,4,2,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,3,5,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,2,3,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,3,2,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,4,3,5,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,4,5,2,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[1,5,2,4,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,5,3,2,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,5,3,4,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,5,4,3,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[2,1,3,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[2,1,3,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[2,1,4,3,5] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[2,1,4,5,3] => [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2
[2,1,5,3,4] => [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2
[2,1,5,4,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[2,3,1,4,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[2,3,1,5,4] => [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2
[2,4,3,1,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[2,4,5,1,3] => [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2
[2,5,3,4,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[2,5,4,3,1] => [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2
[3,1,2,4,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[3,1,2,5,4] => [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2
[3,2,1,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[3,2,1,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[3,2,4,1,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[3,2,5,4,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[3,4,1,2,5] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[3,4,1,5,2] => [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2
[3,4,5,2,1] => [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2
[2,1,4,3,6,5,8,7,10,9,12,11] => [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 2
[6,5,4,3,2,1,12,11,10,9,8,7] => [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 2
[12,11,10,9,8,7,6,5,4,3,2,1] => [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 2
[11,1,2,3,4,5,6,10,7,8,9] => [9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? => ? = 2
[10,1,2,3,4,5,11,6,7,8,9] => [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? => ? = 3
[11,1,2,3,4,10,5,6,7,8,9] => [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? => ? = 3
[11,10,1,2,3,4,5,6,7,8,9] => [6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? => ? = 5
[10,9,8,7,6,5,4,3,2,1,11] => [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? = 2
[1,11,10,9,8,7,6,5,4,3,2] => [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? = 2
[2,4,6,8,11,12,1,3,5,7,9,10] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 3
[2,4,6,9,10,12,1,3,5,7,8,11] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 5
[2,4,6,10,11,12,1,3,5,7,8,9] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 5
[2,4,7,8,11,12,1,3,5,6,9,10] => [6,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 3
[2,4,7,9,11,12,1,3,5,6,8,10] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 3
[2,4,8,9,10,12,1,3,5,6,7,11] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 2
[2,4,8,9,11,12,1,3,5,6,7,10] => [7,3,2]
=> [[1,2,3,4,5,6,7],[8,9,10],[11,12]]
=> ? => ? = 3
[2,4,8,10,11,12,1,3,5,6,7,9] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 2
[2,5,6,8,10,12,1,3,4,7,9,11] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 5
[2,5,6,9,10,12,1,3,4,7,8,11] => [5,5,2]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12]]
=> ? => ? = 5
[2,5,6,9,11,12,1,3,4,7,8,10] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 2
[2,5,7,8,11,12,1,3,4,6,9,10] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 3
[2,5,7,9,10,12,1,3,4,6,8,11] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 2
[2,5,7,9,11,12,1,3,4,6,8,10] => [7,3,2]
=> [[1,2,3,4,5,6,7],[8,9,10],[11,12]]
=> ? => ? = 3
[2,5,7,10,11,12,1,3,4,6,8,9] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 5
[2,5,8,9,10,12,1,3,4,6,7,11] => [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? => ? = 2
[2,5,8,9,11,12,1,3,4,6,7,10] => [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [11,12,9,10,6,7,8,1,2,3,4,5] => ? = 3
[2,5,8,10,11,12,1,3,4,6,7,9] => [5,5,2]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12]]
=> ? => ? = 5
[2,5,9,10,11,12,1,3,4,6,7,8] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 5
[2,6,7,8,10,12,1,3,4,5,9,11] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 2
[2,6,7,9,10,12,1,3,4,5,8,11] => [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? => ? = 2
[2,6,7,9,11,12,1,3,4,5,8,10] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 2
[2,6,8,9,10,12,1,3,4,5,7,11] => [6,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12]]
=> ? => ? = 2
[2,6,8,9,11,12,1,3,4,5,7,10] => [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? => ? = 2
[2,6,8,10,11,12,1,3,4,5,7,9] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 2
[2,7,8,9,10,12,1,3,4,5,6,11] => [3,3,2,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11,12]]
=> ? => ? = 3
[2,7,8,9,11,12,1,3,4,5,6,10] => [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [11,12,9,10,6,7,8,1,2,3,4,5] => ? = 3
[2,7,8,10,11,12,1,3,4,5,6,9] => [7,3,2]
=> [[1,2,3,4,5,6,7],[8,9,10],[11,12]]
=> ? => ? = 3
[2,7,9,10,11,12,1,3,4,5,6,8] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 3
[3,4,6,8,10,12,1,2,5,7,9,11] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 3
[3,4,6,8,11,12,1,2,5,7,9,10] => [6,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 3
[3,4,7,8,10,12,1,2,5,6,9,11] => [6,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 3
[3,4,7,8,11,12,1,2,5,6,9,10] => [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 3
[3,4,7,9,10,12,1,2,5,6,8,11] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 3
[3,4,7,9,11,12,1,2,5,6,8,10] => [6,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 3
[3,4,7,10,11,12,1,2,5,6,8,9] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 3
[3,4,8,9,11,12,1,2,5,6,7,10] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 3
[3,5,6,9,10,12,1,2,4,7,8,11] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 2
[3,5,6,9,11,12,1,2,4,7,8,10] => [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [11,12,7,8,9,10,1,2,3,4,5,6] => ? = 4
[3,5,6,10,11,12,1,2,4,7,8,9] => [8,4]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12]]
=> ? => ? = 4
[3,5,7,8,10,12,1,2,4,6,9,11] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 3
Description
The number of stack-sorts needed to sort a permutation.
A permutation is (West) $t$-stack sortable if it is sortable using $t$ stacks in series.
Let $W_t(n,k)$ be the number of permutations of size $n$
with $k$ descents which are $t$-stack sortable. Then the polynomials $W_{n,t}(x) = \sum_{k=0}^n W_t(n,k)x^k$
are symmetric and unimodal.
We have $W_{n,1}(x) = A_n(x)$, the Eulerian polynomials. One can show that $W_{n,1}(x)$ and $W_{n,2}(x)$ are real-rooted.
Precisely the permutations that avoid the pattern $231$ have statistic at most $1$, see [3]. These are counted by $\frac{1}{n+1}\binom{2n}{n}$ ([[OEIS:A000108]]). Precisely the permutations that avoid the pattern $2341$ and the barred pattern $3\bar 5241$ have statistic at most $2$, see [4]. These are counted by $\frac{2(3n)!}{(n+1)!(2n+1)!}$ ([[OEIS:A000139]]).
Matching statistic: St000374
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[1,3,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[2,1,3,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,2,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[4,2,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[1,2,3,5,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[1,2,4,5,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,2,5,3,4] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,2,5,4,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[1,3,2,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2
[1,3,4,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,4,2,3,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,4,3,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,4,5,2,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2
[1,5,2,4,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,5,3,2,4] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,5,3,4,2] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[1,5,4,3,2] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2
[2,1,3,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[2,1,3,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2
[2,1,4,3,5] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2
[2,1,4,5,3] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[2,1,5,3,4] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[2,1,5,4,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2
[2,3,1,4,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[2,3,1,5,4] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[2,4,3,1,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[2,4,5,1,3] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[2,5,3,4,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[2,5,4,3,1] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[3,1,2,4,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[3,1,2,5,4] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[3,2,1,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[3,2,1,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2
[3,2,4,1,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[3,2,5,4,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[3,4,1,2,5] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2
[3,4,1,5,2] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[3,4,5,2,1] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[2,1,4,3,6,5,8,7,10,9,12,11] => [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 2
[6,5,4,3,2,1,12,11,10,9,8,7] => [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 2
[12,11,10,9,8,7,6,5,4,3,2,1] => [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 2
[11,1,2,3,4,5,6,10,7,8,9] => [9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> ? => ? = 2
[10,1,2,3,4,5,11,6,7,8,9] => [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? => ? = 3
[11,1,2,3,4,10,5,6,7,8,9] => [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? => ? = 3
[11,10,1,2,3,4,5,6,7,8,9] => [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? => ? = 5
[10,9,8,7,6,5,4,3,2,1,11] => [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 2
[1,11,10,9,8,7,6,5,4,3,2] => [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 2
[2,4,6,8,11,12,1,3,5,7,9,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 3
[2,4,6,9,10,12,1,3,5,7,8,11] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 5
[2,4,6,10,11,12,1,3,5,7,8,9] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 5
[2,4,7,8,11,12,1,3,5,6,9,10] => [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? => ? = 3
[2,4,7,9,11,12,1,3,5,6,8,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 3
[2,4,8,9,10,12,1,3,5,6,7,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 2
[2,4,8,9,11,12,1,3,5,6,7,10] => [7,3,2]
=> [[1,2,5,9,10,11,12],[3,4,8],[6,7]]
=> ? => ? = 3
[2,4,8,10,11,12,1,3,5,6,7,9] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 2
[2,5,6,8,10,12,1,3,4,7,9,11] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 5
[2,5,6,9,10,12,1,3,4,7,8,11] => [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? => ? = 5
[2,5,6,9,11,12,1,3,4,7,8,10] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 2
[2,5,7,8,11,12,1,3,4,6,9,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 3
[2,5,7,9,10,12,1,3,4,6,8,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 2
[2,5,7,9,11,12,1,3,4,6,8,10] => [7,3,2]
=> [[1,2,5,9,10,11,12],[3,4,8],[6,7]]
=> ? => ? = 3
[2,5,7,10,11,12,1,3,4,6,8,9] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 5
[2,5,8,9,10,12,1,3,4,6,7,11] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? => ? = 2
[2,5,8,9,11,12,1,3,4,6,7,10] => [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> [8,9,5,6,3,4,10,1,2,7,11,12] => ? = 3
[2,5,8,10,11,12,1,3,4,6,7,9] => [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? => ? = 5
[2,5,9,10,11,12,1,3,4,6,7,8] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 5
[2,6,7,8,10,12,1,3,4,5,9,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 2
[2,6,7,9,10,12,1,3,4,5,8,11] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? => ? = 2
[2,6,7,9,11,12,1,3,4,5,8,10] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 2
[2,6,8,9,10,12,1,3,4,5,7,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? => ? = 2
[2,6,8,9,11,12,1,3,4,5,7,10] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? => ? = 2
[2,6,8,10,11,12,1,3,4,5,7,9] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 2
[2,7,8,9,10,12,1,3,4,5,6,11] => [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> ? => ? = 3
[2,7,8,9,11,12,1,3,4,5,6,10] => [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> [8,9,5,6,3,4,10,1,2,7,11,12] => ? = 3
[2,7,8,10,11,12,1,3,4,5,6,9] => [7,3,2]
=> [[1,2,5,9,10,11,12],[3,4,8],[6,7]]
=> ? => ? = 3
[2,7,9,10,11,12,1,3,4,5,6,8] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 3
[3,4,6,8,10,12,1,2,5,7,9,11] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 3
[3,4,6,8,11,12,1,2,5,7,9,10] => [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? => ? = 3
[3,4,7,8,10,12,1,2,5,6,9,11] => [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? => ? = 3
[3,4,7,8,11,12,1,2,5,6,9,10] => [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 3
[3,4,7,9,10,12,1,2,5,6,8,11] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 3
[3,4,7,9,11,12,1,2,5,6,8,10] => [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? => ? = 3
[3,4,7,10,11,12,1,2,5,6,8,9] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 3
[3,4,8,9,11,12,1,2,5,6,7,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 3
[3,5,6,9,10,12,1,2,4,7,8,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 2
[3,5,6,9,11,12,1,2,4,7,8,10] => [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? = 4
[3,5,6,10,11,12,1,2,4,7,8,9] => [8,4]
=> [[1,2,3,4,9,10,11,12],[5,6,7,8]]
=> ? => ? = 4
[3,5,7,8,10,12,1,2,4,6,9,11] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 3
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000996
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[1,3,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[2,1,3,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,2,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[4,2,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[1,2,3,5,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[1,2,4,5,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,2,5,3,4] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,2,5,4,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[1,3,2,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2
[1,3,4,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,4,2,3,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,4,3,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,4,5,2,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2
[1,5,2,4,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,5,3,2,4] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,5,3,4,2] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[1,5,4,3,2] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2
[2,1,3,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[2,1,3,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2
[2,1,4,3,5] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2
[2,1,4,5,3] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[2,1,5,3,4] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[2,1,5,4,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2
[2,3,1,4,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[2,3,1,5,4] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[2,4,3,1,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[2,4,5,1,3] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[2,5,3,4,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[2,5,4,3,1] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[3,1,2,4,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[3,1,2,5,4] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[3,2,1,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[3,2,1,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2
[3,2,4,1,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[3,2,5,4,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[3,4,1,2,5] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2
[3,4,1,5,2] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[3,4,5,2,1] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[2,1,4,3,6,5,8,7,10,9,12,11] => [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 2
[6,5,4,3,2,1,12,11,10,9,8,7] => [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 2
[12,11,10,9,8,7,6,5,4,3,2,1] => [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 2
[11,1,2,3,4,5,6,10,7,8,9] => [9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> ? => ? = 2
[10,1,2,3,4,5,11,6,7,8,9] => [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? => ? = 3
[11,1,2,3,4,10,5,6,7,8,9] => [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? => ? = 3
[11,10,1,2,3,4,5,6,7,8,9] => [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? => ? = 5
[10,9,8,7,6,5,4,3,2,1,11] => [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 2
[1,11,10,9,8,7,6,5,4,3,2] => [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 2
[2,4,6,8,11,12,1,3,5,7,9,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 3
[2,4,6,9,10,12,1,3,5,7,8,11] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 5
[2,4,6,10,11,12,1,3,5,7,8,9] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 5
[2,4,7,8,11,12,1,3,5,6,9,10] => [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? => ? = 3
[2,4,7,9,11,12,1,3,5,6,8,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 3
[2,4,8,9,10,12,1,3,5,6,7,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 2
[2,4,8,9,11,12,1,3,5,6,7,10] => [7,3,2]
=> [[1,2,5,9,10,11,12],[3,4,8],[6,7]]
=> ? => ? = 3
[2,4,8,10,11,12,1,3,5,6,7,9] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 2
[2,5,6,8,10,12,1,3,4,7,9,11] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 5
[2,5,6,9,10,12,1,3,4,7,8,11] => [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? => ? = 5
[2,5,6,9,11,12,1,3,4,7,8,10] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 2
[2,5,7,8,11,12,1,3,4,6,9,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 3
[2,5,7,9,10,12,1,3,4,6,8,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 2
[2,5,7,9,11,12,1,3,4,6,8,10] => [7,3,2]
=> [[1,2,5,9,10,11,12],[3,4,8],[6,7]]
=> ? => ? = 3
[2,5,7,10,11,12,1,3,4,6,8,9] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 5
[2,5,8,9,10,12,1,3,4,6,7,11] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? => ? = 2
[2,5,8,9,11,12,1,3,4,6,7,10] => [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> [8,9,5,6,3,4,10,1,2,7,11,12] => ? = 3
[2,5,8,10,11,12,1,3,4,6,7,9] => [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? => ? = 5
[2,5,9,10,11,12,1,3,4,6,7,8] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 5
[2,6,7,8,10,12,1,3,4,5,9,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 2
[2,6,7,9,10,12,1,3,4,5,8,11] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? => ? = 2
[2,6,7,9,11,12,1,3,4,5,8,10] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 2
[2,6,8,9,10,12,1,3,4,5,7,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? => ? = 2
[2,6,8,9,11,12,1,3,4,5,7,10] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? => ? = 2
[2,6,8,10,11,12,1,3,4,5,7,9] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 2
[2,7,8,9,10,12,1,3,4,5,6,11] => [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> ? => ? = 3
[2,7,8,9,11,12,1,3,4,5,6,10] => [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> [8,9,5,6,3,4,10,1,2,7,11,12] => ? = 3
[2,7,8,10,11,12,1,3,4,5,6,9] => [7,3,2]
=> [[1,2,5,9,10,11,12],[3,4,8],[6,7]]
=> ? => ? = 3
[2,7,9,10,11,12,1,3,4,5,6,8] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 3
[3,4,6,8,10,12,1,2,5,7,9,11] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 3
[3,4,6,8,11,12,1,2,5,7,9,10] => [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? => ? = 3
[3,4,7,8,10,12,1,2,5,6,9,11] => [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? => ? = 3
[3,4,7,8,11,12,1,2,5,6,9,10] => [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 3
[3,4,7,9,10,12,1,2,5,6,8,11] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 3
[3,4,7,9,11,12,1,2,5,6,8,10] => [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? => ? = 3
[3,4,7,10,11,12,1,2,5,6,8,9] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 3
[3,4,8,9,11,12,1,2,5,6,7,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 3
[3,5,6,9,10,12,1,2,4,7,8,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 2
[3,5,6,9,11,12,1,2,4,7,8,10] => [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? = 4
[3,5,6,10,11,12,1,2,4,7,8,9] => [8,4]
=> [[1,2,3,4,9,10,11,12],[5,6,7,8]]
=> ? => ? = 4
[3,5,7,8,10,12,1,2,4,6,9,11] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 3
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
The following 36 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000451The length of the longest pattern of the form k 1 2. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001557The number of inversions of the second entry of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000455The second largest eigenvalue of a graph if it is integral. St001593This is the number of standard Young tableaux of the given shifted shape. St001625The Möbius invariant of a lattice. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001875The number of simple modules with projective dimension at most 1. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001964The interval resolution global dimension of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001779The order of promotion on the set of linear extensions of a poset. St000632The jump number of the poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001866The nesting alignments of a signed permutation. 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.
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