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Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St000008
Mp00223: Permutations —runsort⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00130: Permutations —descent tops⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => => [1] => 0
[1,2] => [1,2] => 0 => [2] => 0
[2,1] => [1,2] => 0 => [2] => 0
[1,2,3] => [1,2,3] => 00 => [3] => 0
[1,3,2] => [1,3,2] => 01 => [2,1] => 2
[2,1,3] => [1,3,2] => 01 => [2,1] => 2
[2,3,1] => [1,2,3] => 00 => [3] => 0
[3,1,2] => [1,2,3] => 00 => [3] => 0
[3,2,1] => [1,2,3] => 00 => [3] => 0
[1,2,3,4] => [1,2,3,4] => 000 => [4] => 0
[1,2,4,3] => [1,2,4,3] => 001 => [3,1] => 3
[1,3,2,4] => [1,3,2,4] => 010 => [2,2] => 2
[1,3,4,2] => [1,3,4,2] => 001 => [3,1] => 3
[1,4,2,3] => [1,4,2,3] => 001 => [3,1] => 3
[1,4,3,2] => [1,4,2,3] => 001 => [3,1] => 3
[2,1,3,4] => [1,3,4,2] => 001 => [3,1] => 3
[2,1,4,3] => [1,4,2,3] => 001 => [3,1] => 3
[2,3,1,4] => [1,4,2,3] => 001 => [3,1] => 3
[2,3,4,1] => [1,2,3,4] => 000 => [4] => 0
[2,4,1,3] => [1,3,2,4] => 010 => [2,2] => 2
[2,4,3,1] => [1,2,4,3] => 001 => [3,1] => 3
[3,1,2,4] => [1,2,4,3] => 001 => [3,1] => 3
[3,1,4,2] => [1,4,2,3] => 001 => [3,1] => 3
[3,2,1,4] => [1,4,2,3] => 001 => [3,1] => 3
[3,2,4,1] => [1,2,4,3] => 001 => [3,1] => 3
[3,4,1,2] => [1,2,3,4] => 000 => [4] => 0
[3,4,2,1] => [1,2,3,4] => 000 => [4] => 0
[4,1,2,3] => [1,2,3,4] => 000 => [4] => 0
[4,1,3,2] => [1,3,2,4] => 010 => [2,2] => 2
[4,2,1,3] => [1,3,2,4] => 010 => [2,2] => 2
[4,2,3,1] => [1,2,3,4] => 000 => [4] => 0
[4,3,1,2] => [1,2,3,4] => 000 => [4] => 0
[4,3,2,1] => [1,2,3,4] => 000 => [4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => [5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 0001 => [4,1] => 4
[1,2,4,3,5] => [1,2,4,3,5] => 0010 => [3,2] => 3
[1,2,4,5,3] => [1,2,4,5,3] => 0001 => [4,1] => 4
[1,2,5,3,4] => [1,2,5,3,4] => 0001 => [4,1] => 4
[1,2,5,4,3] => [1,2,5,3,4] => 0001 => [4,1] => 4
[1,3,2,4,5] => [1,3,2,4,5] => 0100 => [2,3] => 2
[1,3,2,5,4] => [1,3,2,5,4] => 0101 => [2,2,1] => 6
[1,3,4,2,5] => [1,3,4,2,5] => 0010 => [3,2] => 3
[1,3,4,5,2] => [1,3,4,5,2] => 0001 => [4,1] => 4
[1,3,5,2,4] => [1,3,5,2,4] => 0001 => [4,1] => 4
[1,3,5,4,2] => [1,3,5,2,4] => 0001 => [4,1] => 4
[1,4,2,3,5] => [1,4,2,3,5] => 0010 => [3,2] => 3
[1,4,2,5,3] => [1,4,2,5,3] => 0011 => [3,1,1] => 7
[1,4,3,2,5] => [1,4,2,5,3] => 0011 => [3,1,1] => 7
[1,4,3,5,2] => [1,4,2,3,5] => 0010 => [3,2] => 3
[1,4,5,2,3] => [1,4,5,2,3] => 0001 => [4,1] => 4
Description
The major index of the composition.
The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see [[Permutations/Descents-Major]].
Matching statistic: St000391
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00130: Permutations —descent tops⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => => ? = 0
[1,2] => [1,2] => 0 => 0
[2,1] => [1,2] => 0 => 0
[1,2,3] => [1,2,3] => 00 => 0
[1,3,2] => [1,3,2] => 01 => 2
[2,1,3] => [1,3,2] => 01 => 2
[2,3,1] => [1,2,3] => 00 => 0
[3,1,2] => [1,2,3] => 00 => 0
[3,2,1] => [1,2,3] => 00 => 0
[1,2,3,4] => [1,2,3,4] => 000 => 0
[1,2,4,3] => [1,2,4,3] => 001 => 3
[1,3,2,4] => [1,3,2,4] => 010 => 2
[1,3,4,2] => [1,3,4,2] => 001 => 3
[1,4,2,3] => [1,4,2,3] => 001 => 3
[1,4,3,2] => [1,4,2,3] => 001 => 3
[2,1,3,4] => [1,3,4,2] => 001 => 3
[2,1,4,3] => [1,4,2,3] => 001 => 3
[2,3,1,4] => [1,4,2,3] => 001 => 3
[2,3,4,1] => [1,2,3,4] => 000 => 0
[2,4,1,3] => [1,3,2,4] => 010 => 2
[2,4,3,1] => [1,2,4,3] => 001 => 3
[3,1,2,4] => [1,2,4,3] => 001 => 3
[3,1,4,2] => [1,4,2,3] => 001 => 3
[3,2,1,4] => [1,4,2,3] => 001 => 3
[3,2,4,1] => [1,2,4,3] => 001 => 3
[3,4,1,2] => [1,2,3,4] => 000 => 0
[3,4,2,1] => [1,2,3,4] => 000 => 0
[4,1,2,3] => [1,2,3,4] => 000 => 0
[4,1,3,2] => [1,3,2,4] => 010 => 2
[4,2,1,3] => [1,3,2,4] => 010 => 2
[4,2,3,1] => [1,2,3,4] => 000 => 0
[4,3,1,2] => [1,2,3,4] => 000 => 0
[4,3,2,1] => [1,2,3,4] => 000 => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => [1,2,3,5,4] => 0001 => 4
[1,2,4,3,5] => [1,2,4,3,5] => 0010 => 3
[1,2,4,5,3] => [1,2,4,5,3] => 0001 => 4
[1,2,5,3,4] => [1,2,5,3,4] => 0001 => 4
[1,2,5,4,3] => [1,2,5,3,4] => 0001 => 4
[1,3,2,4,5] => [1,3,2,4,5] => 0100 => 2
[1,3,2,5,4] => [1,3,2,5,4] => 0101 => 6
[1,3,4,2,5] => [1,3,4,2,5] => 0010 => 3
[1,3,4,5,2] => [1,3,4,5,2] => 0001 => 4
[1,3,5,2,4] => [1,3,5,2,4] => 0001 => 4
[1,3,5,4,2] => [1,3,5,2,4] => 0001 => 4
[1,4,2,3,5] => [1,4,2,3,5] => 0010 => 3
[1,4,2,5,3] => [1,4,2,5,3] => 0011 => 7
[1,4,3,2,5] => [1,4,2,5,3] => 0011 => 7
[1,4,3,5,2] => [1,4,2,3,5] => 0010 => 3
[1,4,5,2,3] => [1,4,5,2,3] => 0001 => 4
[1,4,5,3,2] => [1,4,5,2,3] => 0001 => 4
Description
The sum of the positions of the ones in a binary word.
Matching statistic: St000726
Mp00223: Permutations —runsort⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000726: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 61%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000726: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 61%
Values
[1] => [1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [2,1] => [1,2] => 0
[2,1] => [1,2] => [2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => [2,3,1] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,2,1] => [2,1,3] => 2
[2,1,3] => [1,3,2] => [3,2,1] => [2,1,3] => 2
[2,3,1] => [1,2,3] => [2,3,1] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [2,3,1] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [2,3,1] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [2,4,3,1] => [1,3,2,4] => 3
[1,3,2,4] => [1,3,2,4] => [3,2,4,1] => [2,1,3,4] => 2
[1,3,4,2] => [1,3,4,2] => [4,2,3,1] => [2,3,1,4] => 3
[1,4,2,3] => [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 3
[1,4,3,2] => [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 3
[2,1,3,4] => [1,3,4,2] => [4,2,3,1] => [2,3,1,4] => 3
[2,1,4,3] => [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 3
[2,3,1,4] => [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 3
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [3,2,4,1] => [2,1,3,4] => 2
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => [1,3,2,4] => 3
[3,1,2,4] => [1,2,4,3] => [2,4,3,1] => [1,3,2,4] => 3
[3,1,4,2] => [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 3
[3,2,1,4] => [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 3
[3,2,4,1] => [1,2,4,3] => [2,4,3,1] => [1,3,2,4] => 3
[3,4,1,2] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [3,2,4,1] => [2,1,3,4] => 2
[4,2,1,3] => [1,3,2,4] => [3,2,4,1] => [2,1,3,4] => 2
[4,2,3,1] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => [1,2,4,3,5] => 4
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => [1,3,2,4,5] => 3
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,1] => [1,3,4,2,5] => 4
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,1] => [1,4,2,3,5] => 4
[1,2,5,4,3] => [1,2,5,3,4] => [2,4,5,3,1] => [1,4,2,3,5] => 4
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,1] => [2,1,3,4,5] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,1] => [2,1,4,3,5] => 6
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,1] => [2,3,1,4,5] => 3
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,1] => [2,3,4,1,5] => 4
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => [2,4,1,3,5] => 4
[1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => [2,4,1,3,5] => 4
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,1] => [3,1,2,4,5] => 3
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,1] => [3,1,4,2,5] => 7
[1,4,3,2,5] => [1,4,2,5,3] => [3,5,2,4,1] => [3,1,4,2,5] => 7
[1,4,3,5,2] => [1,4,2,3,5] => [3,4,2,5,1] => [3,1,2,4,5] => 3
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,1] => [3,4,1,2,5] => 4
[1,4,5,3,2] => [1,4,5,2,3] => [4,5,2,3,1] => [3,4,1,2,5] => 4
[1,2,6,3,4,5,7] => [1,2,6,3,4,5,7] => [2,4,5,6,3,7,1] => [1,5,2,3,4,6,7] => ? = 5
[1,2,6,3,4,7,5] => [1,2,6,3,4,7,5] => [2,4,5,7,3,6,1] => [1,5,2,3,6,4,7] => ? = 11
[1,2,6,3,5,4,7] => [1,2,6,3,5,4,7] => [2,4,6,5,3,7,1] => [1,5,2,4,3,6,7] => ? = 9
[1,2,6,3,5,7,4] => [1,2,6,3,5,7,4] => [2,4,7,5,3,6,1] => [1,5,2,4,6,3,7] => ? = 11
[1,2,6,3,7,4,5] => [1,2,6,3,7,4,5] => [2,4,6,7,3,5,1] => [1,5,2,6,3,4,7] => ? = 11
[1,2,6,3,7,5,4] => [1,2,6,3,7,4,5] => [2,4,6,7,3,5,1] => [1,5,2,6,3,4,7] => ? = 11
[1,2,6,4,3,5,7] => [1,2,6,3,5,7,4] => [2,4,7,5,3,6,1] => [1,5,2,4,6,3,7] => ? = 11
[1,2,6,4,3,7,5] => [1,2,6,3,7,4,5] => [2,4,6,7,3,5,1] => [1,5,2,6,3,4,7] => ? = 11
[1,2,6,4,5,3,7] => [1,2,6,3,7,4,5] => [2,4,6,7,3,5,1] => [1,5,2,6,3,4,7] => ? = 11
[1,2,6,4,5,7,3] => [1,2,6,3,4,5,7] => [2,4,5,6,3,7,1] => [1,5,2,3,4,6,7] => ? = 5
[1,2,6,4,7,3,5] => [1,2,6,3,5,4,7] => [2,4,6,5,3,7,1] => [1,5,2,4,3,6,7] => ? = 9
[1,2,6,4,7,5,3] => [1,2,6,3,4,7,5] => [2,4,5,7,3,6,1] => [1,5,2,3,6,4,7] => ? = 11
[1,2,6,5,3,4,7] => [1,2,6,3,4,7,5] => [2,4,5,7,3,6,1] => [1,5,2,3,6,4,7] => ? = 11
[1,2,6,5,3,7,4] => [1,2,6,3,7,4,5] => [2,4,6,7,3,5,1] => [1,5,2,6,3,4,7] => ? = 11
[1,2,6,5,4,3,7] => [1,2,6,3,7,4,5] => [2,4,6,7,3,5,1] => [1,5,2,6,3,4,7] => ? = 11
[1,2,6,5,4,7,3] => [1,2,6,3,4,7,5] => [2,4,5,7,3,6,1] => [1,5,2,3,6,4,7] => ? = 11
[1,2,6,5,7,3,4] => [1,2,6,3,4,5,7] => [2,4,5,6,3,7,1] => [1,5,2,3,4,6,7] => ? = 5
[1,2,6,5,7,4,3] => [1,2,6,3,4,5,7] => [2,4,5,6,3,7,1] => [1,5,2,3,4,6,7] => ? = 5
[1,2,6,7,3,4,5] => [1,2,6,7,3,4,5] => [2,5,6,7,3,4,1] => [1,5,6,2,3,4,7] => ? = 6
[1,2,6,7,3,5,4] => [1,2,6,7,3,5,4] => [2,5,7,6,3,4,1] => [1,5,6,2,4,3,7] => ? = 10
[1,2,6,7,4,3,5] => [1,2,6,7,3,5,4] => [2,5,7,6,3,4,1] => [1,5,6,2,4,3,7] => ? = 10
[1,2,6,7,4,5,3] => [1,2,6,7,3,4,5] => [2,5,6,7,3,4,1] => [1,5,6,2,3,4,7] => ? = 6
[1,2,6,7,5,3,4] => [1,2,6,7,3,4,5] => [2,5,6,7,3,4,1] => [1,5,6,2,3,4,7] => ? = 6
[1,2,6,7,5,4,3] => [1,2,6,7,3,4,5] => [2,5,6,7,3,4,1] => [1,5,6,2,3,4,7] => ? = 6
[1,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => [2,4,5,6,7,3,1] => [1,6,2,3,4,5,7] => ? = 6
[1,2,7,3,4,6,5] => [1,2,7,3,4,6,5] => [2,4,5,7,6,3,1] => [1,6,2,3,5,4,7] => ? = 11
[1,2,7,3,5,4,6] => [1,2,7,3,5,4,6] => [2,4,6,5,7,3,1] => [1,6,2,4,3,5,7] => ? = 10
[1,2,7,3,5,6,4] => [1,2,7,3,5,6,4] => [2,4,7,5,6,3,1] => [1,6,2,4,5,3,7] => ? = 11
[1,2,7,3,6,4,5] => [1,2,7,3,6,4,5] => [2,4,6,7,5,3,1] => [1,6,2,5,3,4,7] => ? = 11
[1,2,7,3,6,5,4] => [1,2,7,3,6,4,5] => [2,4,6,7,5,3,1] => [1,6,2,5,3,4,7] => ? = 11
[1,2,7,4,3,5,6] => [1,2,7,3,5,6,4] => [2,4,7,5,6,3,1] => [1,6,2,4,5,3,7] => ? = 11
[1,2,7,4,3,6,5] => [1,2,7,3,6,4,5] => [2,4,6,7,5,3,1] => [1,6,2,5,3,4,7] => ? = 11
[1,2,7,4,5,3,6] => [1,2,7,3,6,4,5] => [2,4,6,7,5,3,1] => [1,6,2,5,3,4,7] => ? = 11
[1,2,7,4,5,6,3] => [1,2,7,3,4,5,6] => [2,4,5,6,7,3,1] => [1,6,2,3,4,5,7] => ? = 6
[1,2,7,4,6,3,5] => [1,2,7,3,5,4,6] => [2,4,6,5,7,3,1] => [1,6,2,4,3,5,7] => ? = 10
[1,2,7,4,6,5,3] => [1,2,7,3,4,6,5] => [2,4,5,7,6,3,1] => [1,6,2,3,5,4,7] => ? = 11
[1,2,7,5,3,4,6] => [1,2,7,3,4,6,5] => [2,4,5,7,6,3,1] => [1,6,2,3,5,4,7] => ? = 11
[1,2,7,5,3,6,4] => [1,2,7,3,6,4,5] => [2,4,6,7,5,3,1] => [1,6,2,5,3,4,7] => ? = 11
[1,2,7,5,4,3,6] => [1,2,7,3,6,4,5] => [2,4,6,7,5,3,1] => [1,6,2,5,3,4,7] => ? = 11
[1,2,7,5,4,6,3] => [1,2,7,3,4,6,5] => [2,4,5,7,6,3,1] => [1,6,2,3,5,4,7] => ? = 11
[1,2,7,5,6,3,4] => [1,2,7,3,4,5,6] => [2,4,5,6,7,3,1] => [1,6,2,3,4,5,7] => ? = 6
[1,2,7,5,6,4,3] => [1,2,7,3,4,5,6] => [2,4,5,6,7,3,1] => [1,6,2,3,4,5,7] => ? = 6
[1,2,7,6,3,4,5] => [1,2,7,3,4,5,6] => [2,4,5,6,7,3,1] => [1,6,2,3,4,5,7] => ? = 6
[1,2,7,6,3,5,4] => [1,2,7,3,5,4,6] => [2,4,6,5,7,3,1] => [1,6,2,4,3,5,7] => ? = 10
[1,2,7,6,4,3,5] => [1,2,7,3,5,4,6] => [2,4,6,5,7,3,1] => [1,6,2,4,3,5,7] => ? = 10
[1,2,7,6,4,5,3] => [1,2,7,3,4,5,6] => [2,4,5,6,7,3,1] => [1,6,2,3,4,5,7] => ? = 6
[1,2,7,6,5,3,4] => [1,2,7,3,4,5,6] => [2,4,5,6,7,3,1] => [1,6,2,3,4,5,7] => ? = 6
[1,2,7,6,5,4,3] => [1,2,7,3,4,5,6] => [2,4,5,6,7,3,1] => [1,6,2,3,4,5,7] => ? = 6
[1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [3,2,4,5,6,7,1] => [2,1,3,4,5,6,7] => ? = 2
Description
The normalized sum of the leaf labels of the increasing binary tree associated to a permutation.
The sum of the leaf labels is at least the size of the permutation, equality is attained for the binary trees that have only one leaf. This statistic is the sum of the leaf labels minus the size of the permutation.
Matching statistic: St000111
Mp00223: Permutations —runsort⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000111: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 50%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000111: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [1,2] => 0
[2,1] => [1,2] => [2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => [2,3,1] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,2,1] => [2,1,3] => 2
[2,1,3] => [1,3,2] => [3,2,1] => [2,1,3] => 2
[2,3,1] => [1,2,3] => [2,3,1] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [2,3,1] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [2,3,1] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [2,4,3,1] => [1,3,2,4] => 3
[1,3,2,4] => [1,3,2,4] => [3,2,4,1] => [2,1,3,4] => 2
[1,3,4,2] => [1,3,4,2] => [4,2,3,1] => [2,3,1,4] => 3
[1,4,2,3] => [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 3
[1,4,3,2] => [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 3
[2,1,3,4] => [1,3,4,2] => [4,2,3,1] => [2,3,1,4] => 3
[2,1,4,3] => [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 3
[2,3,1,4] => [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 3
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [3,2,4,1] => [2,1,3,4] => 2
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => [1,3,2,4] => 3
[3,1,2,4] => [1,2,4,3] => [2,4,3,1] => [1,3,2,4] => 3
[3,1,4,2] => [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 3
[3,2,1,4] => [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 3
[3,2,4,1] => [1,2,4,3] => [2,4,3,1] => [1,3,2,4] => 3
[3,4,1,2] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [3,2,4,1] => [2,1,3,4] => 2
[4,2,1,3] => [1,3,2,4] => [3,2,4,1] => [2,1,3,4] => 2
[4,2,3,1] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => [1,2,4,3,5] => 4
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => [1,3,2,4,5] => 3
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,1] => [1,3,4,2,5] => 4
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,1] => [1,4,2,3,5] => 4
[1,2,5,4,3] => [1,2,5,3,4] => [2,4,5,3,1] => [1,4,2,3,5] => 4
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,1] => [2,1,3,4,5] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,1] => [2,1,4,3,5] => 6
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,1] => [2,3,1,4,5] => 3
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,1] => [2,3,4,1,5] => 4
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => [2,4,1,3,5] => 4
[1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => [2,4,1,3,5] => 4
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,1] => [3,1,2,4,5] => 3
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,1] => [3,1,4,2,5] => 7
[1,4,3,2,5] => [1,4,2,5,3] => [3,5,2,4,1] => [3,1,4,2,5] => 7
[1,4,3,5,2] => [1,4,2,3,5] => [3,4,2,5,1] => [3,1,2,4,5] => 3
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,1] => [3,4,1,2,5] => 4
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ? = 0
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => [1,2,3,4,6,5,7] => ? = 6
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => [1,2,3,5,4,6,7] => ? = 5
[1,2,3,4,6,7,5] => [1,2,3,4,6,7,5] => [2,3,4,7,5,6,1] => [1,2,3,5,6,4,7] => ? = 6
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => [1,2,3,6,4,5,7] => ? = 6
[1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => [1,2,3,6,4,5,7] => ? = 6
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [2,3,5,4,6,7,1] => [1,2,4,3,5,6,7] => ? = 4
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => [1,2,4,3,6,5,7] => ? = 10
[1,2,3,5,6,4,7] => [1,2,3,5,6,4,7] => [2,3,6,4,5,7,1] => [1,2,4,5,3,6,7] => ? = 5
[1,2,3,5,6,7,4] => [1,2,3,5,6,7,4] => [2,3,7,4,5,6,1] => [1,2,4,5,6,3,7] => ? = 6
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [2,3,6,4,7,5,1] => [1,2,4,6,3,5,7] => ? = 6
[1,2,3,5,7,6,4] => [1,2,3,5,7,4,6] => [2,3,6,4,7,5,1] => [1,2,4,6,3,5,7] => ? = 6
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [2,3,5,6,4,7,1] => [1,2,5,3,4,6,7] => ? = 5
[1,2,3,6,4,7,5] => [1,2,3,6,4,7,5] => [2,3,5,7,4,6,1] => [1,2,5,3,6,4,7] => ? = 11
[1,2,3,6,5,4,7] => [1,2,3,6,4,7,5] => [2,3,5,7,4,6,1] => [1,2,5,3,6,4,7] => ? = 11
[1,2,3,6,5,7,4] => [1,2,3,6,4,5,7] => [2,3,5,6,4,7,1] => [1,2,5,3,4,6,7] => ? = 5
[1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => [2,3,6,7,4,5,1] => [1,2,5,6,3,4,7] => ? = 6
[1,2,3,6,7,5,4] => [1,2,3,6,7,4,5] => [2,3,6,7,4,5,1] => [1,2,5,6,3,4,7] => ? = 6
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [2,3,5,6,7,4,1] => [1,2,6,3,4,5,7] => ? = 6
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [2,3,5,7,6,4,1] => [1,2,6,3,5,4,7] => ? = 11
[1,2,3,7,5,4,6] => [1,2,3,7,4,6,5] => [2,3,5,7,6,4,1] => [1,2,6,3,5,4,7] => ? = 11
[1,2,3,7,5,6,4] => [1,2,3,7,4,5,6] => [2,3,5,6,7,4,1] => [1,2,6,3,4,5,7] => ? = 6
[1,2,3,7,6,4,5] => [1,2,3,7,4,5,6] => [2,3,5,6,7,4,1] => [1,2,6,3,4,5,7] => ? = 6
[1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [2,3,5,6,7,4,1] => [1,2,6,3,4,5,7] => ? = 6
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [2,4,3,5,6,7,1] => [1,3,2,4,5,6,7] => ? = 3
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [2,4,3,5,7,6,1] => [1,3,2,4,6,5,7] => ? = 9
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [2,4,3,6,5,7,1] => [1,3,2,5,4,6,7] => ? = 8
[1,2,4,3,6,7,5] => [1,2,4,3,6,7,5] => [2,4,3,7,5,6,1] => [1,3,2,5,6,4,7] => ? = 9
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [2,4,3,6,7,5,1] => [1,3,2,6,4,5,7] => ? = 9
[1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [2,4,3,6,7,5,1] => [1,3,2,6,4,5,7] => ? = 9
[1,2,4,5,3,6,7] => [1,2,4,5,3,6,7] => [2,5,3,4,6,7,1] => [1,3,4,2,5,6,7] => ? = 4
[1,2,4,5,3,7,6] => [1,2,4,5,3,7,6] => [2,5,3,4,7,6,1] => [1,3,4,2,6,5,7] => ? = 10
[1,2,4,5,6,3,7] => [1,2,4,5,6,3,7] => [2,6,3,4,5,7,1] => [1,3,4,5,2,6,7] => ? = 5
[1,2,4,5,6,7,3] => [1,2,4,5,6,7,3] => [2,7,3,4,5,6,1] => [1,3,4,5,6,2,7] => ? = 6
[1,2,4,5,7,3,6] => [1,2,4,5,7,3,6] => [2,6,3,4,7,5,1] => [1,3,4,6,2,5,7] => ? = 6
[1,2,4,5,7,6,3] => [1,2,4,5,7,3,6] => [2,6,3,4,7,5,1] => [1,3,4,6,2,5,7] => ? = 6
[1,2,4,6,3,5,7] => [1,2,4,6,3,5,7] => [2,5,3,6,4,7,1] => [1,3,5,2,4,6,7] => ? = 5
[1,2,4,6,3,7,5] => [1,2,4,6,3,7,5] => [2,5,3,7,4,6,1] => [1,3,5,2,6,4,7] => ? = 11
[1,2,4,6,5,3,7] => [1,2,4,6,3,7,5] => [2,5,3,7,4,6,1] => [1,3,5,2,6,4,7] => ? = 11
[1,2,4,6,5,7,3] => [1,2,4,6,3,5,7] => [2,5,3,6,4,7,1] => [1,3,5,2,4,6,7] => ? = 5
[1,2,4,6,7,3,5] => [1,2,4,6,7,3,5] => [2,6,3,7,4,5,1] => [1,3,5,6,2,4,7] => ? = 6
[1,2,4,6,7,5,3] => [1,2,4,6,7,3,5] => [2,6,3,7,4,5,1] => [1,3,5,6,2,4,7] => ? = 6
[1,2,4,7,3,5,6] => [1,2,4,7,3,5,6] => [2,5,3,6,7,4,1] => [1,3,6,2,4,5,7] => ? = 6
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [2,5,3,7,6,4,1] => [1,3,6,2,5,4,7] => ? = 11
[1,2,4,7,5,3,6] => [1,2,4,7,3,6,5] => [2,5,3,7,6,4,1] => [1,3,6,2,5,4,7] => ? = 11
[1,2,4,7,5,6,3] => [1,2,4,7,3,5,6] => [2,5,3,6,7,4,1] => [1,3,6,2,4,5,7] => ? = 6
[1,2,4,7,6,3,5] => [1,2,4,7,3,5,6] => [2,5,3,6,7,4,1] => [1,3,6,2,4,5,7] => ? = 6
[1,2,4,7,6,5,3] => [1,2,4,7,3,5,6] => [2,5,3,6,7,4,1] => [1,3,6,2,4,5,7] => ? = 6
[1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [2,4,5,3,6,7,1] => [1,4,2,3,5,6,7] => ? = 4
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [2,4,5,3,7,6,1] => [1,4,2,3,6,5,7] => ? = 10
Description
The sum of the descent tops (or Genocchi descents) of a permutation.
This statistic is given by
$$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} \pi_i.$$
Matching statistic: St001207
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 17%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 17%
Values
[1] => [1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [2,3,1] => 2
[2,1,3] => [1,3,2] => [1,3,2] => [2,3,1] => 2
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [2,3,4,1] => 3
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [2,3,1,4] => 2
[1,3,4,2] => [1,3,4,2] => [1,4,3,2] => [3,4,2,1] => 3
[1,4,2,3] => [1,4,2,3] => [1,4,3,2] => [3,4,2,1] => 3
[1,4,3,2] => [1,4,2,3] => [1,4,3,2] => [3,4,2,1] => 3
[2,1,3,4] => [1,3,4,2] => [1,4,3,2] => [3,4,2,1] => 3
[2,1,4,3] => [1,4,2,3] => [1,4,3,2] => [3,4,2,1] => 3
[2,3,1,4] => [1,4,2,3] => [1,4,3,2] => [3,4,2,1] => 3
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => [2,3,1,4] => 2
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [2,3,4,1] => 3
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => [2,3,4,1] => 3
[3,1,4,2] => [1,4,2,3] => [1,4,3,2] => [3,4,2,1] => 3
[3,2,1,4] => [1,4,2,3] => [1,4,3,2] => [3,4,2,1] => 3
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => [2,3,4,1] => 3
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => [2,3,1,4] => 2
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => [2,3,1,4] => 2
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => ? = 4
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => ? = 3
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [3,4,5,2,1] => ? = 4
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => ? = 4
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => ? = 4
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [2,3,1,4,5] => ? = 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [3,4,2,5,1] => ? = 6
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => [3,4,2,1,5] => ? = 3
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,3,4,2] => [3,5,1,4,2] => ? = 4
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,5,2,3] => [2,1,5,3,4] => ? = 4
[1,3,5,4,2] => [1,3,5,2,4] => [1,4,5,2,3] => [2,1,5,3,4] => ? = 4
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,3,2,5] => [3,4,2,1,5] => ? = 3
[1,4,2,5,3] => [1,4,2,5,3] => [1,5,3,4,2] => [3,5,1,4,2] => ? = 7
[1,4,3,2,5] => [1,4,2,5,3] => [1,5,3,4,2] => [3,5,1,4,2] => ? = 7
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,3,2,5] => [3,4,2,1,5] => ? = 3
[1,4,5,2,3] => [1,4,5,2,3] => [1,5,4,3,2] => [4,5,3,2,1] => ? = 4
[1,4,5,3,2] => [1,4,5,2,3] => [1,5,4,3,2] => [4,5,3,2,1] => ? = 4
[1,5,2,3,4] => [1,5,2,3,4] => [1,5,3,4,2] => [3,5,1,4,2] => ? = 4
[1,5,2,4,3] => [1,5,2,4,3] => [1,5,3,4,2] => [3,5,1,4,2] => ? = 7
[1,5,3,2,4] => [1,5,2,4,3] => [1,5,3,4,2] => [3,5,1,4,2] => ? = 7
[1,5,3,4,2] => [1,5,2,3,4] => [1,5,3,4,2] => [3,5,1,4,2] => ? = 4
[1,5,4,2,3] => [1,5,2,3,4] => [1,5,3,4,2] => [3,5,1,4,2] => ? = 4
[1,5,4,3,2] => [1,5,2,3,4] => [1,5,3,4,2] => [3,5,1,4,2] => ? = 4
[2,1,3,4,5] => [1,3,4,5,2] => [1,5,3,4,2] => [3,5,1,4,2] => ? = 4
[2,1,3,5,4] => [1,3,5,2,4] => [1,4,5,2,3] => [2,1,5,3,4] => ? = 4
[2,1,4,3,5] => [1,4,2,3,5] => [1,4,3,2,5] => [3,4,2,1,5] => ? = 3
[2,1,4,5,3] => [1,4,5,2,3] => [1,5,4,3,2] => [4,5,3,2,1] => ? = 4
[2,1,5,3,4] => [1,5,2,3,4] => [1,5,3,4,2] => [3,5,1,4,2] => ? = 4
[2,1,5,4,3] => [1,5,2,3,4] => [1,5,3,4,2] => [3,5,1,4,2] => ? = 4
[2,3,1,4,5] => [1,4,5,2,3] => [1,5,4,3,2] => [4,5,3,2,1] => ? = 4
[2,3,1,5,4] => [1,5,2,3,4] => [1,5,3,4,2] => [3,5,1,4,2] => ? = 4
[2,3,4,1,5] => [1,5,2,3,4] => [1,5,3,4,2] => [3,5,1,4,2] => ? = 4
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[2,3,5,1,4] => [1,4,2,3,5] => [1,4,3,2,5] => [3,4,2,1,5] => ? = 3
[2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => ? = 4
[2,4,1,3,5] => [1,3,5,2,4] => [1,4,5,2,3] => [2,1,5,3,4] => ? = 4
[2,4,1,5,3] => [1,5,2,4,3] => [1,5,3,4,2] => [3,5,1,4,2] => ? = 7
[2,4,3,1,5] => [1,5,2,4,3] => [1,5,3,4,2] => [3,5,1,4,2] => ? = 7
[2,4,3,5,1] => [1,2,4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => ? = 3
[2,4,5,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => [2,3,1,4,5] => ? = 2
[2,4,5,3,1] => [1,2,4,5,3] => [1,2,5,4,3] => [3,4,5,2,1] => ? = 4
[2,5,1,3,4] => [1,3,4,2,5] => [1,4,3,2,5] => [3,4,2,1,5] => ? = 3
[2,5,1,4,3] => [1,4,2,5,3] => [1,5,3,4,2] => [3,5,1,4,2] => ? = 7
[2,5,3,1,4] => [1,4,2,5,3] => [1,5,3,4,2] => [3,5,1,4,2] => ? = 7
[2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => ? = 4
[2,5,4,1,3] => [1,3,2,5,4] => [1,3,2,5,4] => [3,4,2,5,1] => ? = 6
[2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => ? = 4
[3,1,2,4,5] => [1,2,4,5,3] => [1,2,5,4,3] => [3,4,5,2,1] => ? = 4
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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