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Your data matches 117 different statistics following compositions of up to 3 maps.
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Matching statistic: St000245
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 1
[2,1] => [2,1] => [2,1] => 0
[1,2,3] => [1,3,2] => [1,3,2] => 1
[1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [2,3,1] => [1,3,2] => 1
[3,1,2] => [3,1,2] => [3,1,2] => 1
[3,2,1] => [3,2,1] => [3,2,1] => 0
[1,2,3,4] => [1,4,3,2] => [1,4,3,2] => 1
[1,2,4,3] => [1,4,3,2] => [1,4,3,2] => 1
[1,3,2,4] => [1,4,3,2] => [1,4,3,2] => 1
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 1
[2,1,3,4] => [2,1,4,3] => [2,1,4,3] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1
[2,3,1,4] => [2,4,1,3] => [2,4,1,3] => 2
[2,3,4,1] => [2,4,3,1] => [1,4,3,2] => 1
[2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[2,4,3,1] => [2,4,3,1] => [1,4,3,2] => 1
[3,1,2,4] => [3,1,4,2] => [2,1,4,3] => 1
[3,1,4,2] => [3,1,4,2] => [2,1,4,3] => 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[3,2,4,1] => [3,2,4,1] => [2,1,4,3] => 1
[3,4,1,2] => [3,4,1,2] => [2,4,1,3] => 2
[3,4,2,1] => [3,4,2,1] => [1,4,3,2] => 1
[4,1,2,3] => [4,1,3,2] => [4,1,3,2] => 1
[4,1,3,2] => [4,1,3,2] => [4,1,3,2] => 1
[4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[4,2,3,1] => [4,2,3,1] => [4,1,3,2] => 1
[4,3,1,2] => [4,3,1,2] => [4,3,1,2] => 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,2,3,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,2,4,3,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,2,4,5,3] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,2,5,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,2,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,3,2,4,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,3,2,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,3,4,2,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,3,5,2,4] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,4,2,3,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,4,2,5,3] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,4,3,2,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 1
Description
The number of ascents of a permutation.
Matching statistic: St000672
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 1
[2,1] => [2,1] => [2,1] => 0
[1,2,3] => [1,3,2] => [1,3,2] => 1
[1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [2,3,1] => [1,3,2] => 1
[3,1,2] => [3,1,2] => [3,1,2] => 1
[3,2,1] => [3,2,1] => [3,2,1] => 0
[1,2,3,4] => [1,4,3,2] => [1,4,3,2] => 1
[1,2,4,3] => [1,4,3,2] => [1,4,3,2] => 1
[1,3,2,4] => [1,4,3,2] => [1,4,3,2] => 1
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 1
[2,1,3,4] => [2,1,4,3] => [2,1,4,3] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1
[2,3,1,4] => [2,4,1,3] => [2,4,1,3] => 2
[2,3,4,1] => [2,4,3,1] => [1,4,3,2] => 1
[2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[2,4,3,1] => [2,4,3,1] => [1,4,3,2] => 1
[3,1,2,4] => [3,1,4,2] => [2,1,4,3] => 1
[3,1,4,2] => [3,1,4,2] => [2,1,4,3] => 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[3,2,4,1] => [3,2,4,1] => [2,1,4,3] => 1
[3,4,1,2] => [3,4,1,2] => [2,4,1,3] => 2
[3,4,2,1] => [3,4,2,1] => [1,4,3,2] => 1
[4,1,2,3] => [4,1,3,2] => [4,1,3,2] => 1
[4,1,3,2] => [4,1,3,2] => [4,1,3,2] => 1
[4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[4,2,3,1] => [4,2,3,1] => [4,1,3,2] => 1
[4,3,1,2] => [4,3,1,2] => [4,3,1,2] => 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,2,3,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,2,4,3,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,2,4,5,3] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,2,5,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,2,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,3,2,4,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,3,2,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,3,4,2,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,3,5,2,4] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,4,2,3,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,4,2,5,3] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,4,3,2,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 1
Description
The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Matching statistic: St000834
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 1
[2,1] => [2,1] => [2,1] => 0
[1,2,3] => [1,3,2] => [1,3,2] => 1
[1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [2,3,1] => [1,3,2] => 1
[3,1,2] => [3,1,2] => [3,1,2] => 1
[3,2,1] => [3,2,1] => [3,2,1] => 0
[1,2,3,4] => [1,4,3,2] => [1,4,3,2] => 1
[1,2,4,3] => [1,4,3,2] => [1,4,3,2] => 1
[1,3,2,4] => [1,4,3,2] => [1,4,3,2] => 1
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 1
[2,1,3,4] => [2,1,4,3] => [2,1,4,3] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1
[2,3,1,4] => [2,4,1,3] => [2,4,1,3] => 2
[2,3,4,1] => [2,4,3,1] => [1,4,3,2] => 1
[2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[2,4,3,1] => [2,4,3,1] => [1,4,3,2] => 1
[3,1,2,4] => [3,1,4,2] => [2,1,4,3] => 1
[3,1,4,2] => [3,1,4,2] => [2,1,4,3] => 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[3,2,4,1] => [3,2,4,1] => [2,1,4,3] => 1
[3,4,1,2] => [3,4,1,2] => [2,4,1,3] => 2
[3,4,2,1] => [3,4,2,1] => [1,4,3,2] => 1
[4,1,2,3] => [4,1,3,2] => [4,1,3,2] => 1
[4,1,3,2] => [4,1,3,2] => [4,1,3,2] => 1
[4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[4,2,3,1] => [4,2,3,1] => [4,1,3,2] => 1
[4,3,1,2] => [4,3,1,2] => [4,3,1,2] => 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,2,3,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,2,4,3,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,2,4,5,3] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,2,5,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,2,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,3,2,4,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,3,2,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,3,4,2,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,3,5,2,4] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,4,2,3,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,4,2,5,3] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,4,3,2,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 1
Description
The number of right outer peaks of a permutation.
A right 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 $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St000010
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 1 = 0 + 1
[1,2] => [1,2] => [1,1]
=> 2 = 1 + 1
[2,1] => [2,1] => [2]
=> 1 = 0 + 1
[1,2,3] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[1,3,2] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[2,1,3] => [2,1,3] => [2,1]
=> 2 = 1 + 1
[2,3,1] => [2,3,1] => [2,1]
=> 2 = 1 + 1
[3,1,2] => [3,1,2] => [2,1]
=> 2 = 1 + 1
[3,2,1] => [3,2,1] => [3]
=> 1 = 0 + 1
[1,2,3,4] => [1,4,3,2] => [3,1]
=> 2 = 1 + 1
[1,2,4,3] => [1,4,3,2] => [3,1]
=> 2 = 1 + 1
[1,3,2,4] => [1,4,3,2] => [3,1]
=> 2 = 1 + 1
[1,3,4,2] => [1,4,3,2] => [3,1]
=> 2 = 1 + 1
[1,4,2,3] => [1,4,3,2] => [3,1]
=> 2 = 1 + 1
[1,4,3,2] => [1,4,3,2] => [3,1]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[2,3,1,4] => [2,4,1,3] => [2,1,1]
=> 3 = 2 + 1
[2,3,4,1] => [2,4,3,1] => [3,1]
=> 2 = 1 + 1
[2,4,1,3] => [2,4,1,3] => [2,1,1]
=> 3 = 2 + 1
[2,4,3,1] => [2,4,3,1] => [3,1]
=> 2 = 1 + 1
[3,1,2,4] => [3,1,4,2] => [2,2]
=> 2 = 1 + 1
[3,1,4,2] => [3,1,4,2] => [2,2]
=> 2 = 1 + 1
[3,2,1,4] => [3,2,1,4] => [3,1]
=> 2 = 1 + 1
[3,2,4,1] => [3,2,4,1] => [3,1]
=> 2 = 1 + 1
[3,4,1,2] => [3,4,1,2] => [2,1,1]
=> 3 = 2 + 1
[3,4,2,1] => [3,4,2,1] => [3,1]
=> 2 = 1 + 1
[4,1,2,3] => [4,1,3,2] => [3,1]
=> 2 = 1 + 1
[4,1,3,2] => [4,1,3,2] => [3,1]
=> 2 = 1 + 1
[4,2,1,3] => [4,2,1,3] => [3,1]
=> 2 = 1 + 1
[4,2,3,1] => [4,2,3,1] => [3,1]
=> 2 = 1 + 1
[4,3,1,2] => [4,3,1,2] => [3,1]
=> 2 = 1 + 1
[4,3,2,1] => [4,3,2,1] => [4]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,2,3,5,4] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,2,4,3,5] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,2,4,5,3] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,2,5,3,4] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,3,2,5,4] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,3,4,2,5] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,3,4,5,2] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,3,5,2,4] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
Description
The length of the partition.
Matching statistic: St000507
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1]]
=> 1 = 0 + 1
[1,2] => [1,2] => [[1,2]]
=> 2 = 1 + 1
[2,1] => [2,1] => [[1],[2]]
=> 1 = 0 + 1
[1,2,3] => [1,3,2] => [[1,2],[3]]
=> 2 = 1 + 1
[1,3,2] => [1,3,2] => [[1,2],[3]]
=> 2 = 1 + 1
[2,1,3] => [2,1,3] => [[1,3],[2]]
=> 2 = 1 + 1
[2,3,1] => [2,3,1] => [[1,2],[3]]
=> 2 = 1 + 1
[3,1,2] => [3,1,2] => [[1,3],[2]]
=> 2 = 1 + 1
[3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 1 = 0 + 1
[1,2,3,4] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,2,4,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,3,2,4] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,3,4,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,4,2,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,4,3] => [[1,3],[2,4]]
=> 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 2 = 1 + 1
[2,3,1,4] => [2,4,1,3] => [[1,2],[3,4]]
=> 3 = 2 + 1
[2,3,4,1] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[2,4,1,3] => [2,4,1,3] => [[1,2],[3,4]]
=> 3 = 2 + 1
[2,4,3,1] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[3,1,2,4] => [3,1,4,2] => [[1,3],[2,4]]
=> 2 = 1 + 1
[3,1,4,2] => [3,1,4,2] => [[1,3],[2,4]]
=> 2 = 1 + 1
[3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2 = 1 + 1
[3,2,4,1] => [3,2,4,1] => [[1,3],[2],[4]]
=> 2 = 1 + 1
[3,4,1,2] => [3,4,1,2] => [[1,2],[3,4]]
=> 3 = 2 + 1
[3,4,2,1] => [3,4,2,1] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[4,1,2,3] => [4,1,3,2] => [[1,3],[2],[4]]
=> 2 = 1 + 1
[4,1,3,2] => [4,1,3,2] => [[1,3],[2],[4]]
=> 2 = 1 + 1
[4,2,1,3] => [4,2,1,3] => [[1,4],[2],[3]]
=> 2 = 1 + 1
[4,2,3,1] => [4,2,3,1] => [[1,3],[2],[4]]
=> 2 = 1 + 1
[4,3,1,2] => [4,3,1,2] => [[1,4],[2],[3]]
=> 2 = 1 + 1
[4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,2,3,5,4] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,2,4,3,5] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,2,4,5,3] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,2,5,3,4] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,3,2,5,4] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,3,4,2,5] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,3,4,5,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,3,5,2,4] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000012
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> 0
[1,2] => [1,2] => [2] => [1,1,0,0]
=> 1
[2,1] => [2,1] => [1,1] => [1,0,1,0]
=> 0
[1,2,3] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,3,2] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,1,3] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,3,1] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[3,1,2] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,2,1] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,2,4,3] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,3,2,4] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,3,4,2] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,4,2,3] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,4,3,2] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,3,4] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,1,4,3] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,3,1,4] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,4,1] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,4,1,3] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,4,3,1] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[3,1,2,4] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,1,4,2] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,2,1,4] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[3,2,4,1] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,4,1,2] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,4,2,1] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[4,1,2,3] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[4,1,3,2] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[4,2,1,3] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[4,2,3,1] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[4,3,1,2] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,4,3,5] => [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,4,5,3] => [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,5,3,4] => [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,5,4,3] => [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,3,2,4,5] => [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,3,4,2,5] => [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,3,5,2,4] => [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,4,2,3,5] => [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,4,2,5,3] => [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,4,3,2,5] => [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
Description
The area of a Dyck path.
This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic.
1. Dyck paths are bijection with '''area sequences''' $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$.
2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$
3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.
Matching statistic: St000142
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1]
=> 0
[1,2] => [1,2] => [2,1] => [2]
=> 1
[2,1] => [2,1] => [1,2] => [1,1]
=> 0
[1,2,3] => [1,3,2] => [2,3,1] => [2,1]
=> 1
[1,3,2] => [1,3,2] => [2,3,1] => [2,1]
=> 1
[2,1,3] => [2,1,3] => [3,1,2] => [2,1]
=> 1
[2,3,1] => [2,3,1] => [1,3,2] => [2,1]
=> 1
[3,1,2] => [3,1,2] => [2,1,3] => [2,1]
=> 1
[3,2,1] => [3,2,1] => [1,2,3] => [1,1,1]
=> 0
[1,2,3,4] => [1,4,3,2] => [2,3,4,1] => [2,1,1]
=> 1
[1,2,4,3] => [1,4,3,2] => [2,3,4,1] => [2,1,1]
=> 1
[1,3,2,4] => [1,4,3,2] => [2,3,4,1] => [2,1,1]
=> 1
[1,3,4,2] => [1,4,3,2] => [2,3,4,1] => [2,1,1]
=> 1
[1,4,2,3] => [1,4,3,2] => [2,3,4,1] => [2,1,1]
=> 1
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => [2,1,1]
=> 1
[2,1,3,4] => [2,1,4,3] => [3,4,1,2] => [2,1,1]
=> 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [2,1,1]
=> 1
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => [2,2]
=> 2
[2,3,4,1] => [2,4,3,1] => [1,3,4,2] => [2,1,1]
=> 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [2,2]
=> 2
[2,4,3,1] => [2,4,3,1] => [1,3,4,2] => [2,1,1]
=> 1
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => [2,1,1]
=> 1
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => [2,1,1]
=> 1
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [2,1,1]
=> 1
[3,2,4,1] => [3,2,4,1] => [1,4,2,3] => [2,1,1]
=> 1
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => [2,2]
=> 2
[3,4,2,1] => [3,4,2,1] => [1,2,4,3] => [2,1,1]
=> 1
[4,1,2,3] => [4,1,3,2] => [2,3,1,4] => [2,1,1]
=> 1
[4,1,3,2] => [4,1,3,2] => [2,3,1,4] => [2,1,1]
=> 1
[4,2,1,3] => [4,2,1,3] => [3,1,2,4] => [2,1,1]
=> 1
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => [2,1,1]
=> 1
[4,3,1,2] => [4,3,1,2] => [2,1,3,4] => [2,1,1]
=> 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,2,3,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,2,3,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,2,4,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,2,4,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,2,5,3,4] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,2,5,4,3] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,3,2,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,3,2,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,3,4,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,3,5,2,4] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,4,2,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,4,2,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,4,3,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 1
Description
The number of even parts of a partition.
Matching statistic: St000157
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [[1]]
=> 0
[1,2] => [1,2] => [2,1] => [[1],[2]]
=> 1
[2,1] => [2,1] => [1,2] => [[1,2]]
=> 0
[1,2,3] => [1,3,2] => [2,3,1] => [[1,2],[3]]
=> 1
[1,3,2] => [1,3,2] => [2,3,1] => [[1,2],[3]]
=> 1
[2,1,3] => [2,1,3] => [3,1,2] => [[1,3],[2]]
=> 1
[2,3,1] => [2,3,1] => [1,3,2] => [[1,2],[3]]
=> 1
[3,1,2] => [3,1,2] => [2,1,3] => [[1,3],[2]]
=> 1
[3,2,1] => [3,2,1] => [1,2,3] => [[1,2,3]]
=> 0
[1,2,3,4] => [1,4,3,2] => [2,3,4,1] => [[1,2,3],[4]]
=> 1
[1,2,4,3] => [1,4,3,2] => [2,3,4,1] => [[1,2,3],[4]]
=> 1
[1,3,2,4] => [1,4,3,2] => [2,3,4,1] => [[1,2,3],[4]]
=> 1
[1,3,4,2] => [1,4,3,2] => [2,3,4,1] => [[1,2,3],[4]]
=> 1
[1,4,2,3] => [1,4,3,2] => [2,3,4,1] => [[1,2,3],[4]]
=> 1
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => [[1,2,3],[4]]
=> 1
[2,1,3,4] => [2,1,4,3] => [3,4,1,2] => [[1,2],[3,4]]
=> 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [[1,2],[3,4]]
=> 1
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => [[1,3],[2,4]]
=> 2
[2,3,4,1] => [2,4,3,1] => [1,3,4,2] => [[1,2,3],[4]]
=> 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [[1,3],[2,4]]
=> 2
[2,4,3,1] => [2,4,3,1] => [1,3,4,2] => [[1,2,3],[4]]
=> 1
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => [[1,2],[3,4]]
=> 1
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => [[1,2],[3,4]]
=> 1
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [[1,3,4],[2]]
=> 1
[3,2,4,1] => [3,2,4,1] => [1,4,2,3] => [[1,2,4],[3]]
=> 1
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[3,4,2,1] => [3,4,2,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[4,1,2,3] => [4,1,3,2] => [2,3,1,4] => [[1,2,4],[3]]
=> 1
[4,1,3,2] => [4,1,3,2] => [2,3,1,4] => [[1,2,4],[3]]
=> 1
[4,2,1,3] => [4,2,1,3] => [3,1,2,4] => [[1,3,4],[2]]
=> 1
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[4,3,1,2] => [4,3,1,2] => [2,1,3,4] => [[1,3,4],[2]]
=> 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,2,3,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,2,3,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,2,4,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,2,4,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,2,5,3,4] => [1,5,4,3,2] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,2,5,4,3] => [1,5,4,3,2] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,3,2,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,3,2,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,3,4,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,3,5,2,4] => [1,5,4,3,2] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,4,2,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,4,2,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,4,3,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000291
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1 => 0
[1,2] => [1,2] => [2] => 10 => 1
[2,1] => [2,1] => [1,1] => 11 => 0
[1,2,3] => [1,3,2] => [2,1] => 101 => 1
[1,3,2] => [1,3,2] => [2,1] => 101 => 1
[2,1,3] => [2,1,3] => [1,2] => 110 => 1
[2,3,1] => [2,3,1] => [2,1] => 101 => 1
[3,1,2] => [3,1,2] => [1,2] => 110 => 1
[3,2,1] => [3,2,1] => [1,1,1] => 111 => 0
[1,2,3,4] => [1,4,3,2] => [2,1,1] => 1011 => 1
[1,2,4,3] => [1,4,3,2] => [2,1,1] => 1011 => 1
[1,3,2,4] => [1,4,3,2] => [2,1,1] => 1011 => 1
[1,3,4,2] => [1,4,3,2] => [2,1,1] => 1011 => 1
[1,4,2,3] => [1,4,3,2] => [2,1,1] => 1011 => 1
[1,4,3,2] => [1,4,3,2] => [2,1,1] => 1011 => 1
[2,1,3,4] => [2,1,4,3] => [1,2,1] => 1101 => 1
[2,1,4,3] => [2,1,4,3] => [1,2,1] => 1101 => 1
[2,3,1,4] => [2,4,1,3] => [2,2] => 1010 => 2
[2,3,4,1] => [2,4,3,1] => [2,1,1] => 1011 => 1
[2,4,1,3] => [2,4,1,3] => [2,2] => 1010 => 2
[2,4,3,1] => [2,4,3,1] => [2,1,1] => 1011 => 1
[3,1,2,4] => [3,1,4,2] => [1,2,1] => 1101 => 1
[3,1,4,2] => [3,1,4,2] => [1,2,1] => 1101 => 1
[3,2,1,4] => [3,2,1,4] => [1,1,2] => 1110 => 1
[3,2,4,1] => [3,2,4,1] => [1,2,1] => 1101 => 1
[3,4,1,2] => [3,4,1,2] => [2,2] => 1010 => 2
[3,4,2,1] => [3,4,2,1] => [2,1,1] => 1011 => 1
[4,1,2,3] => [4,1,3,2] => [1,2,1] => 1101 => 1
[4,1,3,2] => [4,1,3,2] => [1,2,1] => 1101 => 1
[4,2,1,3] => [4,2,1,3] => [1,1,2] => 1110 => 1
[4,2,3,1] => [4,2,3,1] => [1,2,1] => 1101 => 1
[4,3,1,2] => [4,3,1,2] => [1,1,2] => 1110 => 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1] => 1111 => 0
[1,2,3,4,5] => [1,5,4,3,2] => [2,1,1,1] => 10111 => 1
[1,2,3,5,4] => [1,5,4,3,2] => [2,1,1,1] => 10111 => 1
[1,2,4,3,5] => [1,5,4,3,2] => [2,1,1,1] => 10111 => 1
[1,2,4,5,3] => [1,5,4,3,2] => [2,1,1,1] => 10111 => 1
[1,2,5,3,4] => [1,5,4,3,2] => [2,1,1,1] => 10111 => 1
[1,2,5,4,3] => [1,5,4,3,2] => [2,1,1,1] => 10111 => 1
[1,3,2,4,5] => [1,5,4,3,2] => [2,1,1,1] => 10111 => 1
[1,3,2,5,4] => [1,5,4,3,2] => [2,1,1,1] => 10111 => 1
[1,3,4,2,5] => [1,5,4,3,2] => [2,1,1,1] => 10111 => 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,1,1,1] => 10111 => 1
[1,3,5,2,4] => [1,5,4,3,2] => [2,1,1,1] => 10111 => 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,1,1,1] => 10111 => 1
[1,4,2,3,5] => [1,5,4,3,2] => [2,1,1,1] => 10111 => 1
[1,4,2,5,3] => [1,5,4,3,2] => [2,1,1,1] => 10111 => 1
[1,4,3,2,5] => [1,5,4,3,2] => [2,1,1,1] => 10111 => 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,1,1,1] => 10111 => 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,1,1,1] => 10111 => 1
Description
The number of descents of a binary word.
Matching statistic: St000319
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> [1]
=> 0
[1,2] => [1,2] => [1,1]
=> [2]
=> 1
[2,1] => [2,1] => [2]
=> [1,1]
=> 0
[1,2,3] => [1,3,2] => [2,1]
=> [2,1]
=> 1
[1,3,2] => [1,3,2] => [2,1]
=> [2,1]
=> 1
[2,1,3] => [2,1,3] => [2,1]
=> [2,1]
=> 1
[2,3,1] => [2,3,1] => [2,1]
=> [2,1]
=> 1
[3,1,2] => [3,1,2] => [2,1]
=> [2,1]
=> 1
[3,2,1] => [3,2,1] => [3]
=> [1,1,1]
=> 0
[1,2,3,4] => [1,4,3,2] => [3,1]
=> [2,1,1]
=> 1
[1,2,4,3] => [1,4,3,2] => [3,1]
=> [2,1,1]
=> 1
[1,3,2,4] => [1,4,3,2] => [3,1]
=> [2,1,1]
=> 1
[1,3,4,2] => [1,4,3,2] => [3,1]
=> [2,1,1]
=> 1
[1,4,2,3] => [1,4,3,2] => [3,1]
=> [2,1,1]
=> 1
[1,4,3,2] => [1,4,3,2] => [3,1]
=> [2,1,1]
=> 1
[2,1,3,4] => [2,1,4,3] => [2,2]
=> [2,2]
=> 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2,2]
=> 1
[2,3,1,4] => [2,4,1,3] => [2,1,1]
=> [3,1]
=> 2
[2,3,4,1] => [2,4,3,1] => [3,1]
=> [2,1,1]
=> 1
[2,4,1,3] => [2,4,1,3] => [2,1,1]
=> [3,1]
=> 2
[2,4,3,1] => [2,4,3,1] => [3,1]
=> [2,1,1]
=> 1
[3,1,2,4] => [3,1,4,2] => [2,2]
=> [2,2]
=> 1
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2,2]
=> 1
[3,2,1,4] => [3,2,1,4] => [3,1]
=> [2,1,1]
=> 1
[3,2,4,1] => [3,2,4,1] => [3,1]
=> [2,1,1]
=> 1
[3,4,1,2] => [3,4,1,2] => [2,1,1]
=> [3,1]
=> 2
[3,4,2,1] => [3,4,2,1] => [3,1]
=> [2,1,1]
=> 1
[4,1,2,3] => [4,1,3,2] => [3,1]
=> [2,1,1]
=> 1
[4,1,3,2] => [4,1,3,2] => [3,1]
=> [2,1,1]
=> 1
[4,2,1,3] => [4,2,1,3] => [3,1]
=> [2,1,1]
=> 1
[4,2,3,1] => [4,2,3,1] => [3,1]
=> [2,1,1]
=> 1
[4,3,1,2] => [4,3,1,2] => [3,1]
=> [2,1,1]
=> 1
[4,3,2,1] => [4,3,2,1] => [4]
=> [1,1,1,1]
=> 0
[1,2,3,4,5] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 1
[1,2,3,5,4] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 1
[1,2,4,3,5] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 1
[1,2,4,5,3] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 1
[1,2,5,3,4] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 1
[1,2,5,4,3] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 1
[1,3,2,4,5] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 1
[1,3,2,5,4] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 1
[1,3,4,2,5] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 1
[1,3,4,5,2] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 1
[1,3,5,2,4] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 1
[1,4,2,3,5] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 1
[1,4,2,5,3] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 1
[1,4,3,2,5] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 1
[1,4,3,5,2] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 1
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
The following 107 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000320The dinv adjustment of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000519The largest length of a factor maximising the subword complexity. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001280The number of parts of an integer partition that are at least two. St001657The number of twos in an integer partition. St000147The largest part of an integer partition. St000288The number of ones in a binary word. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000378The diagonal inversion number of an integer partition. St000733The row containing the largest entry of a standard tableau. St000935The number of ordered refinements of an integer partition. St001389The number of partitions of the same length below the given integer partition. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000984The number of boxes below precisely one peak. St001139The number of occurrences of hills of size 2 in a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000035The number of left outer peaks of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000884The number of isolated descents of a permutation. St001176The size of a partition minus its first part. St000919The number of maximal left branches of a binary tree. St000662The staircase size of the code of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000251The number of nonsingleton blocks of a set partition. St000167The number of leaves of an ordered tree. St000676The number of odd rises of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000996The number of exclusive left-to-right maxima of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. 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. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St000105The number of blocks in the set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St001333The cardinality of a minimal edge-isolating set of a graph. St001427The number of descents of a signed permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000702The number of weak deficiencies of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001665The number of pure excedances of a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000260The radius of a connected graph. St000542The number of left-to-right-minima of a permutation. St000021The number of descents of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000325The width of the tree associated to a permutation. St000155The number of exceedances (also excedences) of a permutation. St000168The number of internal nodes of an ordered tree. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001298The number of repeated entries in the Lehmer code of a permutation. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001874Lusztig's a-function for the symmetric group. St000015The number of peaks of a Dyck path. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000083The number of left oriented leafs of a binary tree except the first one. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001480The number of simple summands of the module J^2/J^3. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000455The second largest eigenvalue of a graph if it is integral. St001863The number of weak excedances of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000264The girth of a graph, which is not a tree. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001624The breadth of a lattice.
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