searching the database
Your data matches 27 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000306
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,1,0,0]
=> 0
[1,2] => [2] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 1
Description
The bounce count of a Dyck path.
For a Dyck path $D$ of length $2n$, this is the number of points $(i,i)$ for $1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of $D$.
Matching statistic: St001471
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001471: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 82%●distinct values known / distinct values provided: 67%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001471: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 82%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1,0]
=> 1 = 0 + 1
[1,2] => [2] => [1,1,0,0]
=> 1 = 0 + 1
[2,1] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[7,8,6,5,4,3,2,1] => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 + 1
[8,6,7,5,4,3,2,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[7,6,8,5,4,3,2,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[8,7,6,5,4,2,3,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
[8,7,6,5,3,2,4,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
[8,7,6,4,3,2,5,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
[8,7,5,4,3,2,6,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
[8,6,5,4,3,2,7,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
[7,6,5,4,3,2,8,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
[2,3,4,5,6,7,8,1] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3 + 1
[8,7,6,5,4,2,1,3] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3 + 1
[8,7,6,5,3,2,1,4] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3 + 1
[8,7,6,4,3,2,1,5] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3 + 1
[8,7,5,4,3,2,1,6] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3 + 1
[8,6,5,4,3,2,1,7] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3 + 1
[8,1,2,3,4,5,6,7] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3 + 1
[7,1,2,3,4,5,6,8] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[6,1,2,3,4,5,7,8] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[5,1,2,3,4,6,7,8] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[4,1,2,3,5,6,7,8] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[3,1,2,4,5,6,7,8] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[2,1,3,4,5,6,7,8] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,2,3,4,5,6,7,8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[6,8,7,5,4,3,2,1] => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 + 1
[5,8,7,6,4,3,2,1] => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 + 1
[6,5,8,7,4,3,2,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[4,8,7,6,5,3,2,1] => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 + 1
[5,4,8,7,6,3,2,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[3,8,7,6,5,4,2,1] => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 + 1
[4,3,8,7,6,5,2,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[2,8,7,6,5,4,3,1] => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 + 1
[3,2,8,7,6,5,4,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[1,8,7,6,5,4,3,2] => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,3,4,5,6,7,8,2] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[2,1,8,7,6,5,4,3] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[1,2,4,5,6,7,8,3] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,2,3,5,6,7,8,4] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,2,3,4,6,7,8,5] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,2,3,4,5,7,8,6] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[6,5,4,3,2,1,8,7] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
[1,2,3,4,5,6,8,7] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[8,2,7,6,5,4,3,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[8,3,7,6,5,4,2,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[10,9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5 + 1
[9,1,2,3,4,5,6,7,8] => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[2,3,4,5,6,7,8,9,1] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[5,3,8,7,6,4,2,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
Description
The magnitude of a Dyck path.
The magnitude of a finite dimensional algebra with invertible Cartan matrix C is defined as the sum of all entries of the inverse of C.
We define the magnitude of a Dyck path as the magnitude of the corresponding LNakayama algebra.
Matching statistic: St001801
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001801: Permutations ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 67%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001801: Permutations ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1,0]
=> [1] => 0
[1,2] => [2] => [1,1,0,0]
=> [1,2] => 0
[2,1] => [1,1] => [1,0,1,0]
=> [2,1] => 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 2
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[2,1,3,4,5,6,7] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? = 1
[2,1,7,6,5,4,3] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[3,1,2,4,5,6,7] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? = 1
[3,1,7,6,5,4,2] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[3,2,1,7,6,5,4] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[3,2,7,6,5,4,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[4,1,2,3,5,6,7] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? = 1
[4,1,7,6,5,3,2] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[4,2,1,7,6,5,3] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[4,2,7,6,5,3,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[4,3,1,7,6,5,2] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[4,3,2,1,7,6,5] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => ? = 3
[4,3,2,7,6,5,1] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[4,3,7,6,5,2,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[5,1,2,3,4,6,7] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? = 1
[5,1,7,6,4,3,2] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[5,2,1,7,6,4,3] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[5,2,7,6,4,3,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[5,3,1,7,6,4,2] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[5,3,2,1,7,6,4] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => ? = 3
[5,3,2,7,6,4,1] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[5,3,7,6,4,2,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[5,4,1,7,6,3,2] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[5,4,2,1,7,6,3] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => ? = 3
[5,4,2,7,6,3,1] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[5,4,3,1,7,6,2] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => ? = 3
[5,4,3,2,1,6,7] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,5,1,6,7] => ? = 2
[5,4,3,2,1,7,6] => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,5,1,7,6] => ? = 3
[5,4,3,2,7,6,1] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => ? = 3
[5,4,3,7,6,2,1] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[5,4,7,6,3,2,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[6,1,2,3,4,5,7] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? = 1
[6,1,7,5,4,3,2] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[6,2,1,7,5,4,3] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[6,2,7,5,4,3,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[6,3,1,7,5,4,2] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[6,3,2,1,7,5,4] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => ? = 3
[6,3,2,7,5,4,1] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[6,3,7,5,4,2,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[6,4,1,7,5,3,2] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[6,4,2,1,7,5,3] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => ? = 3
[6,4,2,7,5,3,1] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[6,4,3,1,7,5,2] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => ? = 3
[6,4,3,2,1,5,7] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,5,1,6,7] => ? = 2
[6,4,3,2,1,7,5] => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,5,1,7,6] => ? = 3
[6,4,3,2,7,5,1] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => ? = 3
[6,4,3,7,5,2,1] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[6,4,7,5,3,2,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[6,5,1,7,4,3,2] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[6,5,2,1,7,4,3] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => ? = 3
Description
Half the number of preimage-image pairs of different parity in a permutation.
Matching statistic: St000539
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000539: Permutations ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 67%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000539: Permutations ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1,0]
=> [1] => ? = 0
[1,2] => [2] => [1,1,0,0]
=> [1,2] => 0
[2,1] => [1,1] => [1,0,1,0]
=> [2,1] => 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 2
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1
[2,1,3,4,5,6,7] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? = 1
[2,1,7,6,5,4,3] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[3,1,2,4,5,6,7] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? = 1
[3,1,7,6,5,4,2] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[3,2,1,7,6,5,4] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[3,2,7,6,5,4,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[4,1,2,3,5,6,7] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? = 1
[4,1,7,6,5,3,2] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[4,2,1,7,6,5,3] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[4,2,7,6,5,3,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[4,3,1,7,6,5,2] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[4,3,2,1,7,6,5] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => ? = 3
[4,3,2,7,6,5,1] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[4,3,7,6,5,2,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[5,1,2,3,4,6,7] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? = 1
[5,1,7,6,4,3,2] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[5,2,1,7,6,4,3] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[5,2,7,6,4,3,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[5,3,1,7,6,4,2] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[5,3,2,1,7,6,4] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => ? = 3
[5,3,2,7,6,4,1] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[5,3,7,6,4,2,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[5,4,1,7,6,3,2] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[5,4,2,1,7,6,3] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => ? = 3
[5,4,2,7,6,3,1] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[5,4,3,1,7,6,2] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => ? = 3
[5,4,3,2,1,6,7] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,5,1,6,7] => ? = 2
[5,4,3,2,1,7,6] => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,5,1,7,6] => ? = 3
[5,4,3,2,7,6,1] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => ? = 3
[5,4,3,7,6,2,1] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[5,4,7,6,3,2,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[6,1,2,3,4,5,7] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? = 1
[6,1,7,5,4,3,2] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[6,2,1,7,5,4,3] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[6,2,7,5,4,3,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[6,3,1,7,5,4,2] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[6,3,2,1,7,5,4] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => ? = 3
[6,3,2,7,5,4,1] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[6,3,7,5,4,2,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[6,4,1,7,5,3,2] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[6,4,2,1,7,5,3] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => ? = 3
[6,4,2,7,5,3,1] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[6,4,3,1,7,5,2] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => ? = 3
[6,4,3,2,1,5,7] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,5,1,6,7] => ? = 2
[6,4,3,2,1,7,5] => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,5,1,7,6] => ? = 3
[6,4,3,2,7,5,1] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => ? = 3
[6,4,3,7,5,2,1] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
[6,4,7,5,3,2,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 3
[6,5,1,7,4,3,2] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 3
Description
The number of odd inversions of a permutation.
An inversion $i < j$ of a permutation is odd if $i \not\equiv j\ (\operatorname{mod} 2)$. See [[St000538]] for even inversions.
Matching statistic: St001928
(load all 29 compositions to match this statistic)
(load all 29 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 67%
St001928: Permutations ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => 0
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [1,3,2] => 1
[3,1,2] => [3,1,2] => 1
[3,2,1] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,2,4,3] => 1
[1,4,2,3] => [1,4,2,3] => 1
[1,4,3,2] => [1,4,3,2] => 1
[2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [1,3,2,4] => 1
[2,3,4,1] => [1,2,4,3] => 1
[2,4,1,3] => [2,4,1,3] => 1
[2,4,3,1] => [1,4,3,2] => 1
[3,1,2,4] => [3,1,2,4] => 1
[3,1,4,2] => [2,1,4,3] => 2
[3,2,1,4] => [3,2,1,4] => 1
[3,2,4,1] => [2,1,4,3] => 2
[3,4,1,2] => [2,4,1,3] => 1
[3,4,2,1] => [1,4,3,2] => 1
[4,1,2,3] => [4,1,2,3] => 1
[4,1,3,2] => [4,1,3,2] => 2
[4,2,1,3] => [4,2,1,3] => 1
[4,2,3,1] => [4,1,3,2] => 2
[4,3,1,2] => [4,3,1,2] => 1
[4,3,2,1] => [4,3,2,1] => 2
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,3,5,4] => 1
[1,2,5,3,4] => [1,2,5,3,4] => 1
[1,2,5,4,3] => [1,2,5,4,3] => 1
[1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,2,4,3,5] => 1
[1,3,4,5,2] => [1,2,3,5,4] => 1
[1,3,5,2,4] => [1,3,5,2,4] => 1
[1,3,5,4,2] => [1,2,5,4,3] => 1
[1,4,2,3,5] => [1,4,2,3,5] => 1
[1,4,2,5,3] => [1,3,2,5,4] => 2
[1,4,3,2,5] => [1,4,3,2,5] => 1
[1,4,3,5,2] => [1,3,2,5,4] => 2
[1,4,5,2,3] => [1,3,5,2,4] => 1
[1,7,2,6,5,4,3] => [1,7,2,6,5,4,3] => ? = 3
[1,7,3,6,5,4,2] => [1,7,2,6,5,4,3] => ? = 3
[1,7,4,6,5,3,2] => [1,7,2,6,5,4,3] => ? = 3
[1,7,5,6,4,3,2] => [1,7,2,6,5,4,3] => ? = 3
[1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 3
[2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 1
[2,1,7,6,5,4,3] => [2,1,7,6,5,4,3] => ? = 3
[2,7,1,6,5,4,3] => [2,7,1,6,5,4,3] => ? = 3
[2,7,3,6,5,4,1] => [1,7,2,6,5,4,3] => ? = 3
[2,7,4,6,5,3,1] => [1,7,2,6,5,4,3] => ? = 3
[2,7,5,6,4,3,1] => [1,7,2,6,5,4,3] => ? = 3
[2,7,6,5,4,3,1] => [1,7,6,5,4,3,2] => ? = 3
[3,1,2,4,5,6,7] => [3,1,2,4,5,6,7] => ? = 1
[3,1,7,6,5,4,2] => [2,1,7,6,5,4,3] => ? = 3
[3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ? = 3
[3,2,7,6,5,4,1] => [2,1,7,6,5,4,3] => ? = 3
[3,7,1,6,5,4,2] => [2,7,1,6,5,4,3] => ? = 3
[3,7,2,6,5,4,1] => [2,7,1,6,5,4,3] => ? = 3
[3,7,4,6,5,2,1] => [1,7,2,6,5,4,3] => ? = 3
[3,7,5,6,4,2,1] => [1,7,2,6,5,4,3] => ? = 3
[3,7,6,5,4,2,1] => [1,7,6,5,4,3,2] => ? = 3
[4,1,2,3,5,6,7] => [4,1,2,3,5,6,7] => ? = 1
[4,1,7,6,5,3,2] => [2,1,7,6,5,4,3] => ? = 3
[4,2,1,7,6,5,3] => [3,2,1,7,6,5,4] => ? = 3
[4,2,7,6,5,3,1] => [2,1,7,6,5,4,3] => ? = 3
[4,3,1,7,6,5,2] => [3,2,1,7,6,5,4] => ? = 3
[4,3,2,1,7,6,5] => [4,3,2,1,7,6,5] => ? = 3
[4,3,2,7,6,5,1] => [3,2,1,7,6,5,4] => ? = 3
[4,3,7,6,5,2,1] => [2,1,7,6,5,4,3] => ? = 3
[4,7,1,6,5,3,2] => [2,7,1,6,5,4,3] => ? = 3
[4,7,2,6,5,3,1] => [2,7,1,6,5,4,3] => ? = 3
[4,7,3,6,5,2,1] => [2,7,1,6,5,4,3] => ? = 3
[4,7,5,6,3,2,1] => [1,7,2,6,5,4,3] => ? = 3
[4,7,6,5,3,2,1] => [1,7,6,5,4,3,2] => ? = 3
[5,1,2,3,4,6,7] => [5,1,2,3,4,6,7] => ? = 1
[5,1,7,6,4,3,2] => [2,1,7,6,5,4,3] => ? = 3
[5,2,1,7,6,4,3] => [3,2,1,7,6,5,4] => ? = 3
[5,2,7,6,4,3,1] => [2,1,7,6,5,4,3] => ? = 3
[5,3,1,7,6,4,2] => [3,2,1,7,6,5,4] => ? = 3
[5,3,2,1,7,6,4] => [4,3,2,1,7,6,5] => ? = 3
[5,3,2,7,6,4,1] => [3,2,1,7,6,5,4] => ? = 3
[5,3,7,6,4,2,1] => [2,1,7,6,5,4,3] => ? = 3
[5,4,1,7,6,3,2] => [3,2,1,7,6,5,4] => ? = 3
[5,4,2,1,7,6,3] => [4,3,2,1,7,6,5] => ? = 3
[5,4,2,7,6,3,1] => [3,2,1,7,6,5,4] => ? = 3
[5,4,3,1,7,6,2] => [4,3,2,1,7,6,5] => ? = 3
[5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => ? = 2
[5,4,3,2,1,7,6] => [5,4,3,2,1,7,6] => ? = 3
[5,4,3,2,7,6,1] => [4,3,2,1,7,6,5] => ? = 3
[5,4,3,7,6,2,1] => [3,2,1,7,6,5,4] => ? = 3
Description
The number of non-overlapping descents in a permutation.
In other words, any maximal descending subsequence $\pi_i,\pi_{i+1},\dots,\pi_k$ contributes $\lfloor\frac{k-i+1}{2}\rfloor$ to the total count.
Matching statistic: St001165
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001165: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 67%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001165: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1,0]
=> 1 = 0 + 1
[1,2] => [2] => [1,1,0,0]
=> 1 = 0 + 1
[2,1] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,3,4,5,6,7] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,2,3,4,5,7,6] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,2,3,4,6,7,5] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,2,3,5,6,7,4] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,2,4,5,6,7,3] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,3,2,7,6,5,4] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,3,4,5,6,7,2] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,4,2,7,6,5,3] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,4,3,7,6,5,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,5,2,7,6,4,3] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,5,3,7,6,4,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,5,4,7,6,3,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,6,2,7,5,4,3] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,6,3,7,5,4,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,6,4,7,5,3,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,6,5,7,4,3,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,7,2,6,5,4,3] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,7,3,6,5,4,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,7,4,6,5,3,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,7,5,6,4,3,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,7,6,5,4,3,2] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 + 1
[2,1,3,4,5,6,7] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[2,1,7,6,5,4,3] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3 + 1
[2,3,1,7,6,5,4] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[2,3,4,5,6,7,1] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[2,4,1,7,6,5,3] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[2,4,3,7,6,5,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[2,5,1,7,6,4,3] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[2,5,3,7,6,4,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[2,5,4,7,6,3,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[2,6,1,7,5,4,3] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[2,6,3,7,5,4,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[2,6,4,7,5,3,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[2,6,5,7,4,3,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[2,7,1,6,5,4,3] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[2,7,3,6,5,4,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[2,7,4,6,5,3,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[2,7,5,6,4,3,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[2,7,6,5,4,3,1] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 + 1
[3,1,2,4,5,6,7] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[3,1,7,6,5,4,2] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3 + 1
[3,2,1,7,6,5,4] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[3,2,7,6,5,4,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3 + 1
[3,4,1,7,6,5,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[3,4,2,7,6,5,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[3,5,1,7,6,4,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[3,5,2,7,6,4,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[3,5,4,7,6,2,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[3,6,1,7,5,4,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[3,6,2,7,5,4,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
Description
Number of simple modules with even projective dimension in the corresponding Nakayama algebra.
Matching statistic: St000329
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 67%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1,0]
=> 0
[1,2] => [2,1] => [1,1] => [1,0,1,0]
=> 0
[2,1] => [1,2] => [2] => [1,1,0,0]
=> 1
[1,2,3] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,1,3] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,3,1] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[3,1,2] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,2,1] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,3,2,4] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,4,2,3] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,4,3,2] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,1,3,4] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[2,1,4,3] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,3,4,1] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,4,3,1] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,1,2,4] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[3,1,4,2] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[3,2,4,1] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,4,2,1] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,1,2,3] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[4,1,3,2] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,2,1,3] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,2,3,1] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,3,1,2] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,3,2,1] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 2
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,2,5,4] => [4,5,2,3,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,4,5,2] => [2,5,4,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,5,4,2] => [2,4,5,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,3,5,2] => [2,5,3,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,3,2,7,6,5,4] => [4,5,6,7,2,3,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,3,4,5,6,7,2] => [2,7,6,5,4,3,1] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,4,2,7,6,5,3] => [3,5,6,7,2,4,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,4,3,7,6,5,2] => [2,5,6,7,3,4,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,5,2,7,6,4,3] => [3,4,6,7,2,5,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,5,3,7,6,4,2] => [2,4,6,7,3,5,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,5,4,7,6,3,2] => [2,3,6,7,4,5,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,6,2,7,5,4,3] => [3,4,5,7,2,6,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,6,3,7,5,4,2] => [2,4,5,7,3,6,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,6,4,7,5,3,2] => [2,3,5,7,4,6,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,6,5,7,4,3,2] => [2,3,4,7,5,6,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,7,2,6,5,4,3] => [3,4,5,6,2,7,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,7,3,6,5,4,2] => [2,4,5,6,3,7,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,7,4,6,5,3,2] => [2,3,5,6,4,7,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,7,5,6,4,3,2] => [2,3,4,6,5,7,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 3
[2,1,3,4,5,6,7] => [7,6,5,4,3,1,2] => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[2,1,7,6,5,4,3] => [3,4,5,6,7,1,2] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 3
[2,3,1,7,6,5,4] => [4,5,6,7,1,3,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,3,4,5,6,7,1] => [1,7,6,5,4,3,2] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[2,4,1,7,6,5,3] => [3,5,6,7,1,4,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,4,3,7,6,5,1] => [1,5,6,7,3,4,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,5,1,7,6,4,3] => [3,4,6,7,1,5,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,5,3,7,6,4,1] => [1,4,6,7,3,5,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,5,4,7,6,3,1] => [1,3,6,7,4,5,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,6,1,7,5,4,3] => [3,4,5,7,1,6,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,6,3,7,5,4,1] => [1,4,5,7,3,6,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,6,4,7,5,3,1] => [1,3,5,7,4,6,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,6,5,7,4,3,1] => [1,3,4,7,5,6,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,7,3,6,5,4,1] => [1,4,5,6,3,7,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,7,4,6,5,3,1] => [1,3,5,6,4,7,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,7,5,6,4,3,1] => [1,3,4,6,5,7,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 3
[3,1,2,4,5,6,7] => [7,6,5,4,2,1,3] => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[3,1,7,6,5,4,2] => [2,4,5,6,7,1,3] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 3
[3,2,1,7,6,5,4] => [4,5,6,7,1,2,3] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 3
[3,2,7,6,5,4,1] => [1,4,5,6,7,2,3] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 3
[3,4,1,7,6,5,2] => [2,5,6,7,1,4,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[3,4,2,7,6,5,1] => [1,5,6,7,2,4,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[3,5,1,7,6,4,2] => [2,4,6,7,1,5,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[3,5,2,7,6,4,1] => [1,4,6,7,2,5,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[3,5,4,7,6,2,1] => [1,2,6,7,4,5,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[3,6,1,7,5,4,2] => [2,4,5,7,1,6,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[3,6,2,7,5,4,1] => [1,4,5,7,2,6,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
Description
The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.
Matching statistic: St001508
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001508: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 67%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001508: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1,0]
=> 0
[1,2] => [2,1] => [1,1] => [1,0,1,0]
=> 0
[2,1] => [1,2] => [2] => [1,1,0,0]
=> 1
[1,2,3] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,1,3] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,3,1] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[3,1,2] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,2,1] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,3,2,4] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,4,2,3] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,4,3,2] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,1,3,4] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[2,1,4,3] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,3,4,1] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,4,3,1] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,1,2,4] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[3,1,4,2] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[3,2,4,1] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,4,2,1] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,1,2,3] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[4,1,3,2] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,2,1,3] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,2,3,1] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,3,1,2] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,3,2,1] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 2
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,2,5,4] => [4,5,2,3,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,4,5,2] => [2,5,4,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,5,4,2] => [2,4,5,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,3,5,2] => [2,5,3,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,3,2,7,6,5,4] => [4,5,6,7,2,3,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,3,4,5,6,7,2] => [2,7,6,5,4,3,1] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,4,2,7,6,5,3] => [3,5,6,7,2,4,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,4,3,7,6,5,2] => [2,5,6,7,3,4,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,5,2,7,6,4,3] => [3,4,6,7,2,5,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,5,3,7,6,4,2] => [2,4,6,7,3,5,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,5,4,7,6,3,2] => [2,3,6,7,4,5,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,6,2,7,5,4,3] => [3,4,5,7,2,6,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,6,3,7,5,4,2] => [2,4,5,7,3,6,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,6,4,7,5,3,2] => [2,3,5,7,4,6,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,6,5,7,4,3,2] => [2,3,4,7,5,6,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,7,2,6,5,4,3] => [3,4,5,6,2,7,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,7,3,6,5,4,2] => [2,4,5,6,3,7,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,7,4,6,5,3,2] => [2,3,5,6,4,7,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,7,5,6,4,3,2] => [2,3,4,6,5,7,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 3
[2,1,3,4,5,6,7] => [7,6,5,4,3,1,2] => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[2,1,7,6,5,4,3] => [3,4,5,6,7,1,2] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 3
[2,3,1,7,6,5,4] => [4,5,6,7,1,3,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,3,4,5,6,7,1] => [1,7,6,5,4,3,2] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[2,4,1,7,6,5,3] => [3,5,6,7,1,4,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,4,3,7,6,5,1] => [1,5,6,7,3,4,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,5,1,7,6,4,3] => [3,4,6,7,1,5,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,5,3,7,6,4,1] => [1,4,6,7,3,5,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,5,4,7,6,3,1] => [1,3,6,7,4,5,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,6,1,7,5,4,3] => [3,4,5,7,1,6,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,6,3,7,5,4,1] => [1,4,5,7,3,6,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,6,4,7,5,3,1] => [1,3,5,7,4,6,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,6,5,7,4,3,1] => [1,3,4,7,5,6,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,7,3,6,5,4,1] => [1,4,5,6,3,7,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,7,4,6,5,3,1] => [1,3,5,6,4,7,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,7,5,6,4,3,1] => [1,3,4,6,5,7,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 3
[3,1,2,4,5,6,7] => [7,6,5,4,2,1,3] => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[3,1,7,6,5,4,2] => [2,4,5,6,7,1,3] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 3
[3,2,1,7,6,5,4] => [4,5,6,7,1,2,3] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 3
[3,2,7,6,5,4,1] => [1,4,5,6,7,2,3] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 3
[3,4,1,7,6,5,2] => [2,5,6,7,1,4,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[3,4,2,7,6,5,1] => [1,5,6,7,2,4,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[3,5,1,7,6,4,2] => [2,4,6,7,1,5,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[3,5,2,7,6,4,1] => [1,4,6,7,2,5,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[3,5,4,7,6,2,1] => [1,2,6,7,4,5,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[3,6,1,7,5,4,2] => [2,4,5,7,1,6,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[3,6,2,7,5,4,1] => [1,4,5,7,2,6,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
Description
The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary.
Given two lattice paths $U,L$ from $(0,0)$ to $(d,n-d)$, [1] describes a bijection between lattice paths weakly between $U$ and $L$ and subsets of $\{1,\dots,n\}$ such that the set of all such subsets gives the standard complex of the lattice path matroid $M[U,L]$.
This statistic gives the cardinality of the image of this bijection when a Dyck path is considered as a path weakly above the diagonal and relative to the diagonal boundary.
Matching statistic: St001873
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001873: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 67%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001873: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1,0]
=> 0
[1,2] => [2,1] => [1,1] => [1,0,1,0]
=> 0
[2,1] => [1,2] => [2] => [1,1,0,0]
=> 1
[1,2,3] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,1,3] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,3,1] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[3,1,2] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,2,1] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,3,2,4] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,4,2,3] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,4,3,2] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,1,3,4] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[2,1,4,3] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,3,4,1] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,4,3,1] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,1,2,4] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[3,1,4,2] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[3,2,4,1] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,4,2,1] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,1,2,3] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[4,1,3,2] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,2,1,3] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,2,3,1] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,3,1,2] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,3,2,1] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 2
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,2,5,4] => [4,5,2,3,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,4,5,2] => [2,5,4,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,5,4,2] => [2,4,5,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,3,5,2] => [2,5,3,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,3,2,7,6,5,4] => [4,5,6,7,2,3,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,3,4,5,6,7,2] => [2,7,6,5,4,3,1] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,4,2,7,6,5,3] => [3,5,6,7,2,4,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,4,3,7,6,5,2] => [2,5,6,7,3,4,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,5,2,7,6,4,3] => [3,4,6,7,2,5,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,5,3,7,6,4,2] => [2,4,6,7,3,5,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,5,4,7,6,3,2] => [2,3,6,7,4,5,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,6,2,7,5,4,3] => [3,4,5,7,2,6,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,6,3,7,5,4,2] => [2,4,5,7,3,6,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,6,4,7,5,3,2] => [2,3,5,7,4,6,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,6,5,7,4,3,2] => [2,3,4,7,5,6,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,7,2,6,5,4,3] => [3,4,5,6,2,7,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,7,3,6,5,4,2] => [2,4,5,6,3,7,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,7,4,6,5,3,2] => [2,3,5,6,4,7,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,7,5,6,4,3,2] => [2,3,4,6,5,7,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 3
[2,1,3,4,5,6,7] => [7,6,5,4,3,1,2] => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[2,1,7,6,5,4,3] => [3,4,5,6,7,1,2] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 3
[2,3,1,7,6,5,4] => [4,5,6,7,1,3,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,3,4,5,6,7,1] => [1,7,6,5,4,3,2] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[2,4,1,7,6,5,3] => [3,5,6,7,1,4,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,4,3,7,6,5,1] => [1,5,6,7,3,4,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,5,1,7,6,4,3] => [3,4,6,7,1,5,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,5,3,7,6,4,1] => [1,4,6,7,3,5,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,5,4,7,6,3,1] => [1,3,6,7,4,5,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,6,1,7,5,4,3] => [3,4,5,7,1,6,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,6,3,7,5,4,1] => [1,4,5,7,3,6,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,6,4,7,5,3,1] => [1,3,5,7,4,6,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,6,5,7,4,3,1] => [1,3,4,7,5,6,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,7,3,6,5,4,1] => [1,4,5,6,3,7,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,7,4,6,5,3,1] => [1,3,5,6,4,7,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,7,5,6,4,3,1] => [1,3,4,6,5,7,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 3
[3,1,2,4,5,6,7] => [7,6,5,4,2,1,3] => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[3,1,7,6,5,4,2] => [2,4,5,6,7,1,3] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 3
[3,2,1,7,6,5,4] => [4,5,6,7,1,2,3] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 3
[3,2,7,6,5,4,1] => [1,4,5,6,7,2,3] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 3
[3,4,1,7,6,5,2] => [2,5,6,7,1,4,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[3,4,2,7,6,5,1] => [1,5,6,7,2,4,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[3,5,1,7,6,4,2] => [2,4,6,7,1,5,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[3,5,2,7,6,4,1] => [1,4,6,7,2,5,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[3,5,4,7,6,2,1] => [1,2,6,7,4,5,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[3,6,1,7,5,4,2] => [2,4,5,7,1,6,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[3,6,2,7,5,4,1] => [1,4,5,7,2,6,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
Description
For 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).
The statistic gives half of the rank of the matrix C^t-C.
Matching statistic: St001203
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001203: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 67%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001203: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,2] => [2] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,3,4,5,6,7] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,2,3,4,5,7,6] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[1,2,3,4,6,7,5] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[1,2,3,5,6,7,4] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[1,2,4,5,6,7,3] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[1,3,2,7,6,5,4] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[1,3,4,5,6,7,2] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[1,4,2,7,6,5,3] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[1,4,3,7,6,5,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[1,5,2,7,6,4,3] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[1,5,3,7,6,4,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[1,5,4,7,6,3,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[1,6,2,7,5,4,3] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[1,6,3,7,5,4,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[1,6,4,7,5,3,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[1,6,5,7,4,3,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[1,7,2,6,5,4,3] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[1,7,3,6,5,4,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[1,7,4,6,5,3,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[1,7,5,6,4,3,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[1,7,6,5,4,3,2] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[2,1,3,4,5,6,7] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[2,1,7,6,5,4,3] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[2,3,1,7,6,5,4] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[2,3,4,5,6,7,1] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[2,4,1,7,6,5,3] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[2,4,3,7,6,5,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[2,5,1,7,6,4,3] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[2,5,3,7,6,4,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[2,5,4,7,6,3,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[2,6,1,7,5,4,3] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[2,6,3,7,5,4,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[2,6,4,7,5,3,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[2,6,5,7,4,3,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[2,7,1,6,5,4,3] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[2,7,3,6,5,4,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[2,7,4,6,5,3,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[2,7,5,6,4,3,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[2,7,6,5,4,3,1] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[3,1,2,4,5,6,7] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[3,1,7,6,5,4,2] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[3,2,1,7,6,5,4] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[3,2,7,6,5,4,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[3,4,1,7,6,5,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[3,4,2,7,6,5,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[3,5,1,7,6,4,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[3,5,2,7,6,4,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[3,5,4,7,6,2,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[3,6,1,7,5,4,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[3,6,2,7,5,4,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
Description
We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
In the list $L$ delete the first entry $c_0$ and substract from all other entries $n-1$ and then append the last element 1 (this was suggested by Christian Stump). The result is a Kupisch series of an LNakayama algebra.
Example:
[5,6,6,6,6] goes into [2,2,2,2,1].
Now associate to the CNakayama algebra with the above properties the Dyck path corresponding to the Kupisch series of the LNakayama algebra.
The statistic return the global dimension of the CNakayama algebra divided by 2.
The following 17 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001394The genus of a permutation. St001720The minimal length of a chain of small intervals in a lattice. St000264The girth of a graph, which is not a tree. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001597The Frobenius rank of a skew partition. 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. St001624The breadth of a lattice.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!