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Your data matches 115 different statistics following compositions of up to 3 maps.
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Matching statistic: St000195
St000195: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,1] => 0
[1,2] => 0
[2,1] => 0
[1,1,1] => 0
[1,1,2] => 0
[1,2,1] => 0
[2,1,1] => 0
[1,1,3] => 0
[1,3,1] => 1
[3,1,1] => 1
[1,2,2] => 0
[2,1,2] => 0
[2,2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,1,1,1] => 0
[1,1,1,2] => 0
[1,1,2,1] => 0
[1,2,1,1] => 0
[2,1,1,1] => 0
[1,1,1,3] => 0
[1,1,3,1] => 1
[1,3,1,1] => 1
[3,1,1,1] => 1
[1,1,1,4] => 0
[1,1,4,1] => 0
[1,4,1,1] => 1
[4,1,1,1] => 1
[1,1,2,2] => 0
[1,2,1,2] => 0
[1,2,2,1] => 0
[2,1,1,2] => 0
[2,1,2,1] => 0
[2,2,1,1] => 0
[1,1,2,3] => 0
[1,1,3,2] => 0
[1,2,1,3] => 0
[1,2,3,1] => 0
[1,3,1,2] => 0
[1,3,2,1] => 0
[2,1,1,3] => 0
[2,1,3,1] => 0
[2,3,1,1] => 0
[3,1,1,2] => 0
[3,1,2,1] => 0
Description
The number of secondary dinversion pairs of the dyck path corresponding to a parking function.
Matching statistic: St000355
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St000355: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St000355: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,1] => [1,2] => [1,2] => [1,2] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 0
[1,1,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,1,1] => [2,1,3] => [2,1,3] => [1,3,2] => 0
[1,1,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,1] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[3,1,1] => [2,3,1] => [2,3,1] => [2,1,3] => 1
[1,2,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,1,2] => [2,1,3] => [2,1,3] => [1,3,2] => 0
[2,2,1] => [3,1,2] => [1,3,2] => [2,3,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[2,1,3] => [2,1,3] => [2,1,3] => [1,3,2] => 0
[2,3,1] => [3,1,2] => [1,3,2] => [2,3,1] => 0
[3,1,2] => [2,3,1] => [2,3,1] => [2,1,3] => 1
[3,2,1] => [3,2,1] => [3,2,1] => [3,2,1] => 0
[1,1,1,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,1,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,1,1] => [2,1,3,4] => [2,1,3,4] => [1,3,2,4] => 1
[1,1,1,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,1,1] => [1,3,2,4] => [3,1,2,4] => [1,3,4,2] => 0
[3,1,1,1] => [2,3,1,4] => [2,3,1,4] => [1,4,2,3] => 0
[1,1,1,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,4,1] => [1,2,4,3] => [4,1,2,3] => [4,1,3,2] => 0
[1,4,1,1] => [1,3,4,2] => [3,1,4,2] => [2,4,1,3] => 0
[4,1,1,1] => [2,3,4,1] => [2,3,4,1] => [2,1,4,3] => 2
[1,1,2,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,1,2] => [2,1,3,4] => [2,1,3,4] => [1,3,2,4] => 1
[2,1,2,1] => [2,1,3,4] => [2,1,3,4] => [1,3,2,4] => 1
[2,2,1,1] => [3,1,2,4] => [1,3,2,4] => [1,2,4,3] => 0
[1,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,3,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,1,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,1,2] => [1,3,2,4] => [3,1,2,4] => [1,3,4,2] => 0
[1,3,2,1] => [1,3,2,4] => [3,1,2,4] => [1,3,4,2] => 0
[2,1,1,3] => [2,1,3,4] => [2,1,3,4] => [1,3,2,4] => 1
[2,1,3,1] => [2,1,3,4] => [2,1,3,4] => [1,3,2,4] => 1
[2,3,1,1] => [3,1,2,4] => [1,3,2,4] => [1,2,4,3] => 0
[3,1,1,2] => [2,3,1,4] => [2,3,1,4] => [1,4,2,3] => 0
[3,1,2,1] => [2,3,1,4] => [2,3,1,4] => [1,4,2,3] => 0
Description
The number of occurrences of the pattern 21-3.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $21\!\!-\!\!3$.
Matching statistic: St000358
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,1] => [1,2] => [1,2] => [1,2] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 0
[1,1,1] => [1,2,3] => [1,3,2] => [1,3,2] => 0
[1,1,2] => [1,2,3] => [1,3,2] => [1,3,2] => 0
[1,2,1] => [1,2,3] => [1,3,2] => [1,3,2] => 0
[2,1,1] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[1,1,3] => [1,2,3] => [1,3,2] => [1,3,2] => 0
[1,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[3,1,1] => [2,3,1] => [2,3,1] => [3,2,1] => 0
[1,2,2] => [1,2,3] => [1,3,2] => [1,3,2] => 0
[2,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,2,1] => [3,1,2] => [3,1,2] => [3,1,2] => 1
[1,2,3] => [1,2,3] => [1,3,2] => [1,3,2] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,3,1] => [3,1,2] => [3,1,2] => [3,1,2] => 1
[3,1,2] => [2,3,1] => [2,3,1] => [3,2,1] => 0
[3,2,1] => [3,2,1] => [3,2,1] => [2,3,1] => 0
[1,1,1,1] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[1,1,1,2] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[1,1,2,1] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[1,2,1,1] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[2,1,1,1] => [2,1,3,4] => [2,1,4,3] => [2,1,4,3] => 0
[1,1,1,3] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[1,1,3,1] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[1,3,1,1] => [1,3,2,4] => [1,4,3,2] => [1,3,4,2] => 0
[3,1,1,1] => [2,3,1,4] => [2,4,1,3] => [4,2,1,3] => 1
[1,1,1,4] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[1,1,4,1] => [1,2,4,3] => [1,4,3,2] => [1,3,4,2] => 0
[1,4,1,1] => [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 0
[4,1,1,1] => [2,3,4,1] => [2,4,3,1] => [3,2,4,1] => 0
[1,1,2,2] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[1,2,1,2] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[1,2,2,1] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[2,1,1,2] => [2,1,3,4] => [2,1,4,3] => [2,1,4,3] => 0
[2,1,2,1] => [2,1,3,4] => [2,1,4,3] => [2,1,4,3] => 0
[2,2,1,1] => [3,1,2,4] => [3,1,4,2] => [4,1,3,2] => 2
[1,1,2,3] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[1,1,3,2] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[1,2,1,3] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[1,2,3,1] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[1,3,1,2] => [1,3,2,4] => [1,4,3,2] => [1,3,4,2] => 0
[1,3,2,1] => [1,3,2,4] => [1,4,3,2] => [1,3,4,2] => 0
[2,1,1,3] => [2,1,3,4] => [2,1,4,3] => [2,1,4,3] => 0
[2,1,3,1] => [2,1,3,4] => [2,1,4,3] => [2,1,4,3] => 0
[2,3,1,1] => [3,1,2,4] => [3,1,4,2] => [4,1,3,2] => 2
[3,1,1,2] => [2,3,1,4] => [2,4,1,3] => [4,2,1,3] => 1
[3,1,2,1] => [2,3,1,4] => [2,4,1,3] => [4,2,1,3] => 1
Description
The number of occurrences of the pattern 31-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $31\!\!-\!\!2$.
Matching statistic: St000359
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000359: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000359: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,1] => [1,2] => [1,2] => [1,2] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 0
[1,1,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,1,1] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[1,1,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,1] => [1,3,2] => [2,3,1] => [3,1,2] => 0
[3,1,1] => [2,3,1] => [3,1,2] => [1,3,2] => 0
[1,2,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,2,1] => [3,1,2] => [1,3,2] => [2,3,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [2,3,1] => [3,1,2] => 0
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,3,1] => [3,1,2] => [1,3,2] => [2,3,1] => 1
[3,1,2] => [2,3,1] => [3,1,2] => [1,3,2] => 0
[3,2,1] => [3,2,1] => [3,2,1] => [3,2,1] => 0
[1,1,1,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,1,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,1,1] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[1,1,1,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,1,1] => [1,3,2,4] => [2,3,1,4] => [3,1,2,4] => 0
[3,1,1,1] => [2,3,1,4] => [3,1,2,4] => [1,3,2,4] => 0
[1,1,1,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,4,1] => [1,2,4,3] => [2,3,4,1] => [4,1,2,3] => 0
[1,4,1,1] => [1,3,4,2] => [2,4,1,3] => [1,3,4,2] => 1
[4,1,1,1] => [2,3,4,1] => [4,1,2,3] => [1,2,4,3] => 0
[1,1,2,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,1,2] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,2,1] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[2,2,1,1] => [3,1,2,4] => [1,3,2,4] => [2,3,1,4] => 1
[1,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,3,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,1,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,1,2] => [1,3,2,4] => [2,3,1,4] => [3,1,2,4] => 0
[1,3,2,1] => [1,3,2,4] => [2,3,1,4] => [3,1,2,4] => 0
[2,1,1,3] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,3,1] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[2,3,1,1] => [3,1,2,4] => [1,3,2,4] => [2,3,1,4] => 1
[3,1,1,2] => [2,3,1,4] => [3,1,2,4] => [1,3,2,4] => 0
[3,1,2,1] => [2,3,1,4] => [3,1,2,4] => [1,3,2,4] => 0
Description
The number of occurrences of the pattern 23-1.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $23\!\!-\!\!1$.
Matching statistic: St000360
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000360: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000360: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,1] => [1,2] => [1,2] => [1,2] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 0
[1,1,1] => [1,2,3] => [1,3,2] => [1,3,2] => 0
[1,1,2] => [1,2,3] => [1,3,2] => [1,3,2] => 0
[1,2,1] => [1,2,3] => [1,3,2] => [1,3,2] => 0
[2,1,1] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[1,1,3] => [1,2,3] => [1,3,2] => [1,3,2] => 0
[1,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[3,1,1] => [2,3,1] => [2,3,1] => [3,2,1] => 1
[1,2,2] => [1,2,3] => [1,3,2] => [1,3,2] => 0
[2,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,2,1] => [3,1,2] => [3,1,2] => [3,1,2] => 0
[1,2,3] => [1,2,3] => [1,3,2] => [1,3,2] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,3,1] => [3,1,2] => [3,1,2] => [3,1,2] => 0
[3,1,2] => [2,3,1] => [2,3,1] => [3,2,1] => 1
[3,2,1] => [3,2,1] => [3,2,1] => [2,3,1] => 0
[1,1,1,1] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[1,1,1,2] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[1,1,2,1] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[1,2,1,1] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[2,1,1,1] => [2,1,3,4] => [2,1,4,3] => [2,1,4,3] => 0
[1,1,1,3] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[1,1,3,1] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[1,3,1,1] => [1,3,2,4] => [1,4,3,2] => [1,3,4,2] => 0
[3,1,1,1] => [2,3,1,4] => [2,4,1,3] => [4,2,1,3] => 1
[1,1,1,4] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[1,1,4,1] => [1,2,4,3] => [1,4,3,2] => [1,3,4,2] => 0
[1,4,1,1] => [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 0
[4,1,1,1] => [2,3,4,1] => [2,4,3,1] => [3,4,2,1] => 1
[1,1,2,2] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[1,2,1,2] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[1,2,2,1] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[2,1,1,2] => [2,1,3,4] => [2,1,4,3] => [2,1,4,3] => 0
[2,1,2,1] => [2,1,3,4] => [2,1,4,3] => [2,1,4,3] => 0
[2,2,1,1] => [3,1,2,4] => [3,1,4,2] => [4,3,1,2] => 2
[1,1,2,3] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[1,1,3,2] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[1,2,1,3] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[1,2,3,1] => [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 0
[1,3,1,2] => [1,3,2,4] => [1,4,3,2] => [1,3,4,2] => 0
[1,3,2,1] => [1,3,2,4] => [1,4,3,2] => [1,3,4,2] => 0
[2,1,1,3] => [2,1,3,4] => [2,1,4,3] => [2,1,4,3] => 0
[2,1,3,1] => [2,1,3,4] => [2,1,4,3] => [2,1,4,3] => 0
[2,3,1,1] => [3,1,2,4] => [3,1,4,2] => [4,3,1,2] => 2
[3,1,1,2] => [2,3,1,4] => [2,4,1,3] => [4,2,1,3] => 1
[3,1,2,1] => [2,3,1,4] => [2,4,1,3] => [4,2,1,3] => 1
Description
The number of occurrences of the pattern 32-1.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $32\!\!-\!\!1$.
Matching statistic: St001021
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001021: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001021: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> 0
[1,1] => [1,2] => [1,2] => [1,0,1,0]
=> 0
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> 0
[2,1] => [2,1] => [2,1] => [1,1,0,0]
=> 0
[1,1,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,1,2] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,2,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[2,1,1] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 0
[1,1,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,3,1] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 0
[3,1,1] => [2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[1,2,2] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[2,1,2] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 0
[2,2,1] => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 0
[2,1,3] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 0
[2,3,1] => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> 0
[3,1,2] => [2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[3,2,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[1,1,1,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,1,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[2,1,1,1] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 0
[1,1,1,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,3,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,3,1,1] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 0
[3,1,1,1] => [2,3,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,4,1] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0
[1,4,1,1] => [1,3,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[4,1,1,1] => [2,3,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[1,1,2,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[2,1,1,2] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 0
[2,1,2,1] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 0
[2,2,1,1] => [3,1,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 0
[1,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,3,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,1,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,3,1,2] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 0
[1,3,2,1] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 0
[2,1,1,3] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 0
[2,1,3,1] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 0
[2,3,1,1] => [3,1,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 0
[3,1,1,2] => [2,3,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[3,1,2,1] => [2,3,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
Description
Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001229
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001229: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001229: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> 0
[1,1] => [1,2] => [1,2] => [1,0,1,0]
=> 0
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> 0
[2,1] => [2,1] => [2,1] => [1,1,0,0]
=> 0
[1,1,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,1,2] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,2,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[2,1,1] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 0
[1,1,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,3,1] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[3,1,1] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 0
[1,2,2] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[2,1,2] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 0
[2,2,1] => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 0
[2,3,1] => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> 0
[3,1,2] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 0
[3,2,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[1,1,1,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,1,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[2,1,1,1] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 0
[1,1,1,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,3,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,3,1,1] => [1,3,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[3,1,1,1] => [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 0
[1,1,1,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,4,1] => [1,2,4,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[1,4,1,1] => [1,3,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1
[4,1,1,1] => [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,2,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[2,1,1,2] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 0
[2,1,2,1] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 0
[2,2,1,1] => [3,1,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 0
[1,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,3,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,1,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,3,1,2] => [1,3,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[1,3,2,1] => [1,3,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[2,1,1,3] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 0
[2,1,3,1] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 0
[2,3,1,1] => [3,1,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 0
[3,1,1,2] => [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 0
[3,1,2,1] => [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 0
Description
The vector space dimension of the first extension group between the Jacobson radical J and J^2.
The vector space dimension of $Ext_A^1(J,J^2)$.
Matching statistic: St001584
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001584: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001584: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> 0
[1,1] => [1,2] => [1,2] => [1,0,1,0]
=> 0
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> 0
[2,1] => [2,1] => [2,1] => [1,1,0,0]
=> 0
[1,1,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,1,2] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,2,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[2,1,1] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 0
[1,1,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,3,1] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[3,1,1] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 0
[1,2,2] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[2,1,2] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 0
[2,2,1] => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 0
[2,3,1] => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> 0
[3,1,2] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 0
[3,2,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[1,1,1,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,1,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[2,1,1,1] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 0
[1,1,1,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,3,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,3,1,1] => [1,3,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[3,1,1,1] => [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 0
[1,1,1,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,4,1] => [1,2,4,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[1,4,1,1] => [1,3,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1
[4,1,1,1] => [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,2,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[2,1,1,2] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 0
[2,1,2,1] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 0
[2,2,1,1] => [3,1,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 0
[1,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,3,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,1,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,3,1,2] => [1,3,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[1,3,2,1] => [1,3,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[2,1,1,3] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 0
[2,1,3,1] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 0
[2,3,1,1] => [3,1,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 0
[3,1,1,2] => [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 0
[3,1,2,1] => [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 0
Description
The area statistic between a Dyck path and its bounce path.
The bounce path [[Mp00099]] is weakly below a given Dyck path and this statistic records the number of boxes between the two paths.
Matching statistic: St001745
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
St001745: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
St001745: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,1] => [1,2] => [1,2] => [1,2] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 0
[1,1,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,1] => [1,3,2] => [3,1,2] => [2,3,1] => 0
[2,1,1] => [2,3,1] => [2,3,1] => [3,1,2] => 0
[1,1,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,1] => [1,3,2] => [3,1,2] => [2,3,1] => 0
[3,1,1] => [2,3,1] => [2,3,1] => [3,1,2] => 0
[1,2,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,1,2] => [2,1,3] => [2,1,3] => [1,3,2] => 0
[2,2,1] => [3,1,2] => [1,3,2] => [2,1,3] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,1,2] => [2,3,1] => 0
[2,1,3] => [2,1,3] => [2,1,3] => [1,3,2] => 0
[2,3,1] => [3,1,2] => [1,3,2] => [2,1,3] => 1
[3,1,2] => [2,3,1] => [2,3,1] => [3,1,2] => 0
[3,2,1] => [3,2,1] => [3,2,1] => [3,2,1] => 0
[1,1,1,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,2,1] => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 0
[1,2,1,1] => [1,3,4,2] => [3,1,4,2] => [4,2,1,3] => 0
[2,1,1,1] => [2,3,4,1] => [2,3,4,1] => [3,4,1,2] => 0
[1,1,1,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,3,1] => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 0
[1,3,1,1] => [1,3,4,2] => [3,1,4,2] => [4,2,1,3] => 0
[3,1,1,1] => [2,3,4,1] => [2,3,4,1] => [3,4,1,2] => 0
[1,1,1,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,4,1] => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 0
[1,4,1,1] => [1,3,4,2] => [3,1,4,2] => [4,2,1,3] => 0
[4,1,1,1] => [2,3,4,1] => [2,3,4,1] => [3,4,1,2] => 0
[1,1,2,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,1,2] => [1,3,2,4] => [3,1,2,4] => [1,4,2,3] => 0
[1,2,2,1] => [1,4,2,3] => [1,4,2,3] => [2,1,3,4] => 2
[2,1,1,2] => [2,3,1,4] => [2,3,1,4] => [1,3,4,2] => 0
[2,1,2,1] => [2,4,1,3] => [2,1,4,3] => [3,2,1,4] => 1
[2,2,1,1] => [3,4,1,2] => [1,3,4,2] => [2,4,1,3] => 0
[1,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,3,2] => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 0
[1,2,1,3] => [1,3,2,4] => [3,1,2,4] => [1,4,2,3] => 0
[1,2,3,1] => [1,4,2,3] => [1,4,2,3] => [2,1,3,4] => 2
[1,3,1,2] => [1,3,4,2] => [3,1,4,2] => [4,2,1,3] => 0
[1,3,2,1] => [1,4,3,2] => [4,3,1,2] => [4,2,3,1] => 0
[2,1,1,3] => [2,3,1,4] => [2,3,1,4] => [1,3,4,2] => 0
[2,1,3,1] => [2,4,1,3] => [2,1,4,3] => [3,2,1,4] => 1
[2,3,1,1] => [3,4,1,2] => [1,3,4,2] => [2,4,1,3] => 0
[3,1,1,2] => [2,3,4,1] => [2,3,4,1] => [3,4,1,2] => 0
[3,1,2,1] => [2,4,3,1] => [4,2,3,1] => [3,4,2,1] => 0
Description
The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation.
Let $\nu$ be a (partial) permutation of $[k]$ with $m$ letters together with dashes between some of its letters. An occurrence of $\nu$ in a permutation $\tau$ is a subsequence $\tau_{a_1},\dots,\tau_{a_m}$
such that $a_i + 1 = a_{i+1}$ whenever there is a dash between the $i$-th and the $(i+1)$-st letter of $\nu$, which is order isomorphic to $\nu$.
Thus, $\nu$ is a vincular pattern, except that it is not required to be a permutation.
An arrow pattern of size $k$ consists of such a generalized vincular pattern $\nu$ and arrows $b_1\to c_1, b_2\to c_2,\dots$, such that precisely the numbers $1,\dots,k$ appear in the vincular pattern and the arrows.
Let $\Phi$ be the map [[Mp00087]]. Let $\tau$ be a permutation and $\sigma = \Phi(\tau)$. Then a subsequence $w = (x_{a_1},\dots,x_{a_m})$ of $\tau$ is an occurrence of the arrow pattern if $w$ is an occurrence of $\nu$, for each arrow $b\to c$ we have $\sigma(x_b) = x_c$ and $x_1 < x_2 < \dots < x_k$.
Matching statistic: St001811
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St001811: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St001811: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ? = 0
[1,1] => [1,2] => [1,2] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 0
[1,1,1] => [1,2,3] => [1,2,3] => 0
[1,1,2] => [1,2,3] => [1,2,3] => 0
[1,2,1] => [1,2,3] => [1,2,3] => 0
[2,1,1] => [2,1,3] => [2,1,3] => 0
[1,1,3] => [1,2,3] => [1,2,3] => 0
[1,3,1] => [1,3,2] => [3,1,2] => 0
[3,1,1] => [2,3,1] => [2,3,1] => 0
[1,2,2] => [1,2,3] => [1,2,3] => 0
[2,1,2] => [2,1,3] => [2,1,3] => 0
[2,2,1] => [3,1,2] => [1,3,2] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,1,2] => 0
[2,1,3] => [2,1,3] => [2,1,3] => 0
[2,3,1] => [3,1,2] => [1,3,2] => 1
[3,1,2] => [2,3,1] => [2,3,1] => 0
[3,2,1] => [3,2,1] => [3,2,1] => 0
[1,1,1,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,1,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,1,1] => [2,1,3,4] => [2,1,3,4] => 0
[1,1,1,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,1,1] => [1,3,2,4] => [3,1,2,4] => 0
[3,1,1,1] => [2,3,1,4] => [2,3,1,4] => 0
[1,1,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,4,1] => [1,2,4,3] => [4,1,2,3] => 0
[1,4,1,1] => [1,3,4,2] => [3,4,1,2] => 0
[4,1,1,1] => [2,3,4,1] => [2,3,4,1] => 0
[1,1,2,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,1,2] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,2,1] => [2,1,3,4] => [2,1,3,4] => 0
[2,2,1,1] => [3,1,2,4] => [1,3,2,4] => 1
[1,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,3,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,1,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,1,2] => [1,3,2,4] => [3,1,2,4] => 0
[1,3,2,1] => [1,3,2,4] => [3,1,2,4] => 0
[2,1,1,3] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,3,1] => [2,1,3,4] => [2,1,3,4] => 0
[2,3,1,1] => [3,1,2,4] => [1,3,2,4] => 1
[3,1,1,2] => [2,3,1,4] => [2,3,1,4] => 0
[3,1,2,1] => [2,3,1,4] => [2,3,1,4] => 0
[3,2,1,1] => [3,2,1,4] => [3,2,1,4] => 0
Description
The Castelnuovo-Mumford regularity of a permutation.
The ''Castelnuovo-Mumford regularity'' of a permutation $\sigma$ is the ''Castelnuovo-Mumford regularity'' of the ''matrix Schubert variety'' $X_\sigma$.
Equivalently, it is the difference between the degrees of the ''Grothendieck polynomial'' and the ''Schubert polynomial'' for $\sigma$. It can be computed by subtracting the ''Coxeter length'' [[St000018]] from the ''Rajchgot index'' [[St001759]].
The following 105 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000572The dimension exponent of a set partition. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001082The number of boxed occurrences of 123 in a permutation. St000988The orbit size of a permutation under Foata's bijection. St001964The interval resolution global dimension of a poset. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000379The number of Hamiltonian cycles in a graph. St000506The number of standard desarrangement tableaux of shape equal to the given partition. 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. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000938The number of zeros of the symmetric group character corresponding to the 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. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000454The largest eigenvalue of a graph if it is integral. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001330The hat guessing number of a graph. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000455The second largest eigenvalue of a graph if it is integral. St000944The 3-degree of an integer partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000699The toughness times the least common multiple of 1,. St000807The sum of the heights of the valleys of the associated bargraph. St000177The number of free tiles in the pattern. St000178Number of free entries. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001881The number of factors of a lattice as a Cartesian product of lattices. St000936The number of even values of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001498The normalised height of a Nakayama algebra with magnitude 1. St001624The breadth of a lattice. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. 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. St001875The number of simple modules with projective dimension at most 1. St001651The Frankl number of a lattice. St001490The number of connected components of a skew partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000929The constant term of the character polynomial of an integer partition. 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. St000567The sum of the products of all pairs of parts. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. 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. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St001857The number of edges in the reduced word graph of a signed permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000181The number of connected components of the Hasse diagram for the poset. St001487The number of inner corners of a skew partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001890The maximum magnitude of the Möbius function of a poset.
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