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Your data matches 28 different statistics following compositions of up to 3 maps.
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Matching statistic: St000021
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
=> [1,0]
=> [2,1] => 1
[1,-2] => [1]
=> [1,0]
=> [2,1] => 1
[-1,2] => [1]
=> [1,0]
=> [2,1] => 1
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[2,-1] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[-2,1] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,2,-3] => [1]
=> [1,0]
=> [2,1] => 1
[1,-2,3] => [1]
=> [1,0]
=> [2,1] => 1
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[-1,2,3] => [1]
=> [1,0]
=> [2,1] => 1
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 2
[1,3,-2] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,-3,2] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[-1,3,2] => [1]
=> [1,0]
=> [2,1] => 1
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-1,-3,-2] => [1]
=> [1,0]
=> [2,1] => 1
[2,1,-3] => [1]
=> [1,0]
=> [2,1] => 1
[2,-1,3] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-2,1,3] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-2,-1,-3] => [1]
=> [1,0]
=> [2,1] => 1
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[3,2,-1] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[3,-2,1] => [1]
=> [1,0]
=> [2,1] => 1
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-3,2,1] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-3,-2,-1] => [1]
=> [1,0]
=> [2,1] => 1
[1,2,3,-4] => [1]
=> [1,0]
=> [2,1] => 1
[1,2,-3,4] => [1]
=> [1,0]
=> [2,1] => 1
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[1,-2,3,4] => [1]
=> [1,0]
=> [2,1] => 1
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 2
[-1,2,3,4] => [1]
=> [1,0]
=> [2,1] => 1
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 2
Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000092
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
=> [1,0]
=> [1] => 1
[1,-2] => [1]
=> [1,0]
=> [1] => 1
[-1,2] => [1]
=> [1,0]
=> [1] => 1
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[2,-1] => [2]
=> [1,0,1,0]
=> [2,1] => 1
[-2,1] => [2]
=> [1,0,1,0]
=> [2,1] => 1
[1,2,-3] => [1]
=> [1,0]
=> [1] => 1
[1,-2,3] => [1]
=> [1,0]
=> [1] => 1
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[-1,2,3] => [1]
=> [1,0]
=> [1] => 1
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [3,1,2] => 2
[1,3,-2] => [2]
=> [1,0,1,0]
=> [2,1] => 1
[1,-3,2] => [2]
=> [1,0,1,0]
=> [2,1] => 1
[-1,3,2] => [1]
=> [1,0]
=> [1] => 1
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 2
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 2
[-1,-3,-2] => [1]
=> [1,0]
=> [1] => 1
[2,1,-3] => [1]
=> [1,0]
=> [1] => 1
[2,-1,3] => [2]
=> [1,0,1,0]
=> [2,1] => 1
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 2
[-2,1,3] => [2]
=> [1,0,1,0]
=> [2,1] => 1
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 2
[-2,-1,-3] => [1]
=> [1,0]
=> [1] => 1
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1
[3,2,-1] => [2]
=> [1,0,1,0]
=> [2,1] => 1
[3,-2,1] => [1]
=> [1,0]
=> [1] => 1
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 2
[-3,2,1] => [2]
=> [1,0,1,0]
=> [2,1] => 1
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 2
[-3,-2,-1] => [1]
=> [1,0]
=> [1] => 1
[1,2,3,-4] => [1]
=> [1,0]
=> [1] => 1
[1,2,-3,4] => [1]
=> [1,0]
=> [1] => 1
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[1,-2,3,4] => [1]
=> [1,0]
=> [1] => 1
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [3,1,2] => 2
[-1,2,3,4] => [1]
=> [1,0]
=> [1] => 1
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [3,1,2] => 2
Description
The number of outer peaks of a permutation.
An outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$ or $n$ if $w_{n} > w_{n-1}$.
In other words, it is a peak in the word $[0,w_1,..., w_n,0]$.
Matching statistic: St000099
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Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
=> [1,0]
=> [1] => 1
[1,-2] => [1]
=> [1,0]
=> [1] => 1
[-1,2] => [1]
=> [1,0]
=> [1] => 1
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[2,-1] => [2]
=> [1,0,1,0]
=> [1,2] => 1
[-2,1] => [2]
=> [1,0,1,0]
=> [1,2] => 1
[1,2,-3] => [1]
=> [1,0]
=> [1] => 1
[1,-2,3] => [1]
=> [1,0]
=> [1] => 1
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[-1,2,3] => [1]
=> [1,0]
=> [1] => 1
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
[1,3,-2] => [2]
=> [1,0,1,0]
=> [1,2] => 1
[1,-3,2] => [2]
=> [1,0,1,0]
=> [1,2] => 1
[-1,3,2] => [1]
=> [1,0]
=> [1] => 1
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[-1,-3,-2] => [1]
=> [1,0]
=> [1] => 1
[2,1,-3] => [1]
=> [1,0]
=> [1] => 1
[2,-1,3] => [2]
=> [1,0,1,0]
=> [1,2] => 1
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[-2,1,3] => [2]
=> [1,0,1,0]
=> [1,2] => 1
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[-2,-1,-3] => [1]
=> [1,0]
=> [1] => 1
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[3,2,-1] => [2]
=> [1,0,1,0]
=> [1,2] => 1
[3,-2,1] => [1]
=> [1,0]
=> [1] => 1
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[-3,2,1] => [2]
=> [1,0,1,0]
=> [1,2] => 1
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[-3,-2,-1] => [1]
=> [1,0]
=> [1] => 1
[1,2,3,-4] => [1]
=> [1,0]
=> [1] => 1
[1,2,-3,4] => [1]
=> [1,0]
=> [1] => 1
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[1,-2,3,4] => [1]
=> [1,0]
=> [1] => 1
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
[-1,2,3,4] => [1]
=> [1,0]
=> [1] => 1
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
Description
The number of valleys of a permutation, including the boundary.
The number of valleys excluding the boundary is [[St000353]].
Matching statistic: St000333
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000333: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000333: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
=> [1,0]
=> [2,1] => 1
[1,-2] => [1]
=> [1,0]
=> [2,1] => 1
[-1,2] => [1]
=> [1,0]
=> [2,1] => 1
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[2,-1] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[-2,1] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,2,-3] => [1]
=> [1,0]
=> [2,1] => 1
[1,-2,3] => [1]
=> [1,0]
=> [2,1] => 1
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[-1,2,3] => [1]
=> [1,0]
=> [2,1] => 1
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 2
[1,3,-2] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,-3,2] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[-1,3,2] => [1]
=> [1,0]
=> [2,1] => 1
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-1,-3,-2] => [1]
=> [1,0]
=> [2,1] => 1
[2,1,-3] => [1]
=> [1,0]
=> [2,1] => 1
[2,-1,3] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-2,1,3] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-2,-1,-3] => [1]
=> [1,0]
=> [2,1] => 1
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[3,2,-1] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[3,-2,1] => [1]
=> [1,0]
=> [2,1] => 1
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-3,2,1] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-3,-2,-1] => [1]
=> [1,0]
=> [2,1] => 1
[1,2,3,-4] => [1]
=> [1,0]
=> [2,1] => 1
[1,2,-3,4] => [1]
=> [1,0]
=> [2,1] => 1
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[1,-2,3,4] => [1]
=> [1,0]
=> [2,1] => 1
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 2
[-1,2,3,4] => [1]
=> [1,0]
=> [2,1] => 1
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 2
Description
The dez statistic, the number of descents of a permutation after replacing fixed points by zeros.
This descent set is denoted by $\operatorname{ZDer}(\sigma)$ in [1].
Matching statistic: St000903
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000903: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000903: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
=> [1,0]
=> [1] => 1
[1,-2] => [1]
=> [1,0]
=> [1] => 1
[-1,2] => [1]
=> [1,0]
=> [1] => 1
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [2] => 1
[2,-1] => [2]
=> [1,0,1,0]
=> [1,1] => 1
[-2,1] => [2]
=> [1,0,1,0]
=> [1,1] => 1
[1,2,-3] => [1]
=> [1,0]
=> [1] => 1
[1,-2,3] => [1]
=> [1,0]
=> [1] => 1
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [2] => 1
[-1,2,3] => [1]
=> [1,0]
=> [1] => 1
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [2] => 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [2] => 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1] => 2
[1,3,-2] => [2]
=> [1,0,1,0]
=> [1,1] => 1
[1,-3,2] => [2]
=> [1,0,1,0]
=> [1,1] => 1
[-1,3,2] => [1]
=> [1,0]
=> [1] => 1
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[-1,-3,-2] => [1]
=> [1,0]
=> [1] => 1
[2,1,-3] => [1]
=> [1,0]
=> [1] => 1
[2,-1,3] => [2]
=> [1,0,1,0]
=> [1,1] => 1
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[-2,1,3] => [2]
=> [1,0,1,0]
=> [1,1] => 1
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[-2,-1,-3] => [1]
=> [1,0]
=> [1] => 1
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[3,2,-1] => [2]
=> [1,0,1,0]
=> [1,1] => 1
[3,-2,1] => [1]
=> [1,0]
=> [1] => 1
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[-3,2,1] => [2]
=> [1,0,1,0]
=> [1,1] => 1
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[-3,-2,-1] => [1]
=> [1,0]
=> [1] => 1
[1,2,3,-4] => [1]
=> [1,0]
=> [1] => 1
[1,2,-3,4] => [1]
=> [1,0]
=> [1] => 1
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [2] => 1
[1,-2,3,4] => [1]
=> [1,0]
=> [1] => 1
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [2] => 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [2] => 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1] => 2
[-1,2,3,4] => [1]
=> [1,0]
=> [1] => 1
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [2] => 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [2] => 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1] => 2
Description
The number of different parts of an integer composition.
Matching statistic: St000955
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000955: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000955: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
=> [1,0]
=> [1,0]
=> 1
[1,-2] => [1]
=> [1,0]
=> [1,0]
=> 1
[-1,2] => [1]
=> [1,0]
=> [1,0]
=> 1
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2,-1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[-2,1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,2,-3] => [1]
=> [1,0]
=> [1,0]
=> 1
[1,-2,3] => [1]
=> [1,0]
=> [1,0]
=> 1
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[-1,2,3] => [1]
=> [1,0]
=> [1,0]
=> 1
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,3,-2] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,-3,2] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[-1,3,2] => [1]
=> [1,0]
=> [1,0]
=> 1
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[-1,-3,-2] => [1]
=> [1,0]
=> [1,0]
=> 1
[2,1,-3] => [1]
=> [1,0]
=> [1,0]
=> 1
[2,-1,3] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[-2,1,3] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[-2,-1,-3] => [1]
=> [1,0]
=> [1,0]
=> 1
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[3,2,-1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[3,-2,1] => [1]
=> [1,0]
=> [1,0]
=> 1
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[-3,2,1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[-3,-2,-1] => [1]
=> [1,0]
=> [1,0]
=> 1
[1,2,3,-4] => [1]
=> [1,0]
=> [1,0]
=> 1
[1,2,-3,4] => [1]
=> [1,0]
=> [1,0]
=> 1
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,-2,3,4] => [1]
=> [1,0]
=> [1,0]
=> 1
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[-1,2,3,4] => [1]
=> [1,0]
=> [1,0]
=> 1
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
Description
Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra.
Matching statistic: St001269
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001269: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001269: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
=> [1,0]
=> [2,1] => 1
[1,-2] => [1]
=> [1,0]
=> [2,1] => 1
[-1,2] => [1]
=> [1,0]
=> [2,1] => 1
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[2,-1] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[-2,1] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,2,-3] => [1]
=> [1,0]
=> [2,1] => 1
[1,-2,3] => [1]
=> [1,0]
=> [2,1] => 1
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[-1,2,3] => [1]
=> [1,0]
=> [2,1] => 1
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 2
[1,3,-2] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,-3,2] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[-1,3,2] => [1]
=> [1,0]
=> [2,1] => 1
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-1,-3,-2] => [1]
=> [1,0]
=> [2,1] => 1
[2,1,-3] => [1]
=> [1,0]
=> [2,1] => 1
[2,-1,3] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-2,1,3] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-2,-1,-3] => [1]
=> [1,0]
=> [2,1] => 1
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[3,2,-1] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[3,-2,1] => [1]
=> [1,0]
=> [2,1] => 1
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-3,2,1] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-3,-2,-1] => [1]
=> [1,0]
=> [2,1] => 1
[1,2,3,-4] => [1]
=> [1,0]
=> [2,1] => 1
[1,2,-3,4] => [1]
=> [1,0]
=> [2,1] => 1
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[1,-2,3,4] => [1]
=> [1,0]
=> [2,1] => 1
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 2
[-1,2,3,4] => [1]
=> [1,0]
=> [2,1] => 1
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 2
Description
The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation.
Matching statistic: St001569
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001569: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001569: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
=> [1,0]
=> [2,1] => 1
[1,-2] => [1]
=> [1,0]
=> [2,1] => 1
[-1,2] => [1]
=> [1,0]
=> [2,1] => 1
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[2,-1] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[-2,1] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,2,-3] => [1]
=> [1,0]
=> [2,1] => 1
[1,-2,3] => [1]
=> [1,0]
=> [2,1] => 1
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[-1,2,3] => [1]
=> [1,0]
=> [2,1] => 1
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 2
[1,3,-2] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,-3,2] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[-1,3,2] => [1]
=> [1,0]
=> [2,1] => 1
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-1,-3,-2] => [1]
=> [1,0]
=> [2,1] => 1
[2,1,-3] => [1]
=> [1,0]
=> [2,1] => 1
[2,-1,3] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-2,1,3] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-2,-1,-3] => [1]
=> [1,0]
=> [2,1] => 1
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[3,2,-1] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[3,-2,1] => [1]
=> [1,0]
=> [2,1] => 1
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-3,2,1] => [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[-3,-2,-1] => [1]
=> [1,0]
=> [2,1] => 1
[1,2,3,-4] => [1]
=> [1,0]
=> [2,1] => 1
[1,2,-3,4] => [1]
=> [1,0]
=> [2,1] => 1
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[1,-2,3,4] => [1]
=> [1,0]
=> [2,1] => 1
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 2
[-1,2,3,4] => [1]
=> [1,0]
=> [2,1] => 1
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 2
Description
The maximal modular displacement of a permutation.
This is $\max_{1\leq i \leq n} \left(\min(\pi(i)-i\pmod n, i-\pi(i)\pmod n)\right)$ for a permutation $\pi$ of $\{1,\dots,n\}$.
Matching statistic: St001741
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001741: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001741: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
=> [1,0]
=> [1] => 1
[1,-2] => [1]
=> [1,0]
=> [1] => 1
[-1,2] => [1]
=> [1,0]
=> [1] => 1
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[2,-1] => [2]
=> [1,0,1,0]
=> [1,2] => 1
[-2,1] => [2]
=> [1,0,1,0]
=> [1,2] => 1
[1,2,-3] => [1]
=> [1,0]
=> [1] => 1
[1,-2,3] => [1]
=> [1,0]
=> [1] => 1
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[-1,2,3] => [1]
=> [1,0]
=> [1] => 1
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
[1,3,-2] => [2]
=> [1,0,1,0]
=> [1,2] => 1
[1,-3,2] => [2]
=> [1,0,1,0]
=> [1,2] => 1
[-1,3,2] => [1]
=> [1,0]
=> [1] => 1
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[-1,-3,-2] => [1]
=> [1,0]
=> [1] => 1
[2,1,-3] => [1]
=> [1,0]
=> [1] => 1
[2,-1,3] => [2]
=> [1,0,1,0]
=> [1,2] => 1
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[-2,1,3] => [2]
=> [1,0,1,0]
=> [1,2] => 1
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[-2,-1,-3] => [1]
=> [1,0]
=> [1] => 1
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[3,2,-1] => [2]
=> [1,0,1,0]
=> [1,2] => 1
[3,-2,1] => [1]
=> [1,0]
=> [1] => 1
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[-3,2,1] => [2]
=> [1,0,1,0]
=> [1,2] => 1
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[-3,-2,-1] => [1]
=> [1,0]
=> [1] => 1
[1,2,3,-4] => [1]
=> [1,0]
=> [1] => 1
[1,2,-3,4] => [1]
=> [1,0]
=> [1] => 1
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[1,-2,3,4] => [1]
=> [1,0]
=> [1] => 1
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
[-1,2,3,4] => [1]
=> [1,0]
=> [1] => 1
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
Description
The largest integer such that all patterns of this size are contained in the permutation.
Matching statistic: St000023
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
=> [1,0]
=> [1] => 0 = 1 - 1
[1,-2] => [1]
=> [1,0]
=> [1] => 0 = 1 - 1
[-1,2] => [1]
=> [1,0]
=> [1] => 0 = 1 - 1
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[2,-1] => [2]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[-2,1] => [2]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[1,2,-3] => [1]
=> [1,0]
=> [1] => 0 = 1 - 1
[1,-2,3] => [1]
=> [1,0]
=> [1] => 0 = 1 - 1
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[-1,2,3] => [1]
=> [1,0]
=> [1] => 0 = 1 - 1
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[1,3,-2] => [2]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[1,-3,2] => [2]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[-1,3,2] => [1]
=> [1,0]
=> [1] => 0 = 1 - 1
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1 = 2 - 1
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1 = 2 - 1
[-1,-3,-2] => [1]
=> [1,0]
=> [1] => 0 = 1 - 1
[2,1,-3] => [1]
=> [1,0]
=> [1] => 0 = 1 - 1
[2,-1,3] => [2]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1 = 2 - 1
[-2,1,3] => [2]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1 = 2 - 1
[-2,-1,-3] => [1]
=> [1,0]
=> [1] => 0 = 1 - 1
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[3,2,-1] => [2]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[3,-2,1] => [1]
=> [1,0]
=> [1] => 0 = 1 - 1
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1 = 2 - 1
[-3,2,1] => [2]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1 = 2 - 1
[-3,-2,-1] => [1]
=> [1,0]
=> [1] => 0 = 1 - 1
[1,2,3,-4] => [1]
=> [1,0]
=> [1] => 0 = 1 - 1
[1,2,-3,4] => [1]
=> [1,0]
=> [1] => 0 = 1 - 1
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[1,-2,3,4] => [1]
=> [1,0]
=> [1] => 0 = 1 - 1
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[-1,2,3,4] => [1]
=> [1,0]
=> [1] => 0 = 1 - 1
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1 = 2 - 1
Description
The number of inner peaks of a permutation.
The number of peaks including the boundary is [[St000092]].
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000568The hook number of a binary tree. St000353The number of inner valleys of a permutation. St000624The normalized sum of the minimal distances to a greater element. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. 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. 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.
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