searching the database
Your data matches 15 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: St000359
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000359: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000359: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [2,1] => 0
[1,2] => [2] => [1,1,0,0]
=> [2,3,1] => 1
[2,1] => [1,1] => [1,0,1,0]
=> [3,1,2] => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [2,3,4,1] => 2
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => 0
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => 0
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 3
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 4
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 3
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 2
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 3
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 2
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 2
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 2
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 2
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 3
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 2
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 2
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 2
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 2
Description
The number of occurrences of the pattern 23-1.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $23\!\!-\!\!1$.
Matching statistic: St000932
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 95%●distinct values known / distinct values provided: 80%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 95%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [1] => [1,0]
=> ? = 0
[1,2] => [2,1] => [1,1] => [1,0,1,0]
=> 1
[2,1] => [1,2] => [2] => [1,1,0,0]
=> 0
[1,2,3] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,3,2] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 0
[2,3,1] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,1,2] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 0
[3,2,1] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3
[1,2,4,3] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,3,2,4] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,4,2,3] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,4,3,2] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,3,4] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[2,3,1,4] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,3,4,1] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[2,4,1,3] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,4,3,1] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[3,1,2,4] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[3,1,4,2] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,2,1,4] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[3,2,4,1] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,4,1,2] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,4,2,1] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,1,2,3] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[4,1,3,2] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[4,2,1,3] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[4,2,3,1] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[4,3,1,2] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[4,3,2,1] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[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]
=> 4
[1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,2,4,3,5] => [5,4,2,3,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,2,5,3,4] => [5,4,1,3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,3,2,4,5] => [5,3,4,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,3,2,5,4] => [5,3,4,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [5,3,2,4,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,3,5,2,4] => [5,3,1,4,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,3,5,4,2] => [5,3,1,2,4] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,4,2,3,5] => [5,2,4,3,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,4,2,5,3] => [5,2,4,1,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[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] => [5,2,3,1,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,4,5,2,3] => [5,2,1,4,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,4,5,3,2] => [5,2,1,3,4] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[8,7,6,5,4,3,2,1] => [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
[8,7,6,5,4,3,1,2] => [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]
=> ? = 0
[8,7,6,5,4,2,1,3] => [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]
=> ? = 0
[8,7,6,5,3,2,1,4] => [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]
=> ? = 0
[8,7,6,4,3,2,1,5] => [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]
=> ? = 0
[8,7,5,4,3,2,1,6] => [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]
=> ? = 0
[8,6,5,4,3,2,1,7] => [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]
=> ? = 0
[7,6,5,4,3,2,1,8] => [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]
=> ? = 0
[10,9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => [10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 0
[1,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,1] => [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]
=> ? = 8
[1,2,3,4,5,6,7,8,9,10] => [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]
=> ? = 9
[9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[8,9,7,6,5,4,3,2,1] => [2,1,3,4,5,6,7,8,9] => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[8,7,6,5,4,3,2,1,9] => [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]
=> ? = 0
[1,9,8,7,6,5,4,3,2] => [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
[9,8,7,6,5,4,3,1,2] => [1,2,3,4,5,6,7,9,8] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
[7,9,8,6,5,4,3,2,1] => [3,1,2,4,5,6,7,8,9] => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[6,9,8,7,5,4,3,2,1] => [4,1,2,3,5,6,7,8,9] => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[5,9,8,7,6,4,3,2,1] => [5,1,2,3,4,6,7,8,9] => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[4,9,8,7,6,5,3,2,1] => [6,1,2,3,4,5,7,8,9] => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[3,9,8,7,6,5,4,2,1] => [7,1,2,3,4,5,6,8,9] => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[2,9,8,7,6,5,4,3,1] => [8,1,2,3,4,5,6,7,9] => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[9,8,7,6,5,4,2,1,3] => [1,2,3,4,5,6,8,9,7] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
[9,8,7,6,5,3,2,1,4] => [1,2,3,4,5,7,8,9,6] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
[9,8,7,6,4,3,2,1,5] => [1,2,3,4,6,7,8,9,5] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
[9,8,7,5,4,3,2,1,6] => [1,2,3,5,6,7,8,9,4] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
[9,8,6,5,4,3,2,1,7] => [1,2,4,5,6,7,8,9,3] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
[9,7,6,5,4,3,2,1,8] => [1,3,4,5,6,7,8,9,2] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
Description
The number of occurrences of the pattern UDU in a Dyck path.
The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St001067
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 92%●distinct values known / distinct values provided: 70%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 92%●distinct values known / distinct values provided: 70%
Values
[1] => [.,.]
=> [1,0]
=> 0
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 0
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 0
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[7,8,6,5,4,3,2,1] => [[[[[[[.,[.,.]],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1
[2,3,4,5,6,7,8,1] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[8,7,6,5,4,3,1,2] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[8,7,6,5,4,2,1,3] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[8,7,6,5,3,2,1,4] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[8,7,6,4,3,2,1,5] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[8,7,5,4,3,2,1,6] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[8,6,5,4,3,2,1,7] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[7,6,5,4,3,2,1,8] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[6,8,7,5,4,3,2,1] => [[[[[[.,[[.,.],.]],.],.],.],.],.]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 1
[5,8,7,6,4,3,2,1] => [[[[[.,[[[.,.],.],.]],.],.],.],.]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[4,8,7,6,5,3,2,1] => [[[[.,[[[[.,.],.],.],.]],.],.],.]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[3,8,7,6,5,4,2,1] => [[[.,[[[[[.,.],.],.],.],.]],.],.]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[2,8,7,6,5,4,3,1] => [[.,[[[[[[.,.],.],.],.],.],.]],.]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,3,4,5,6,7,8,2] => [.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[1,2,4,5,6,7,8,3] => [.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[1,2,3,5,6,7,8,4] => [.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6
[1,2,3,4,6,7,8,5] => [.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[1,2,3,4,5,7,8,6] => [.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6
[1,2,3,4,5,6,8,7] => [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6
[10,9,8,7,6,5,4,3,2,1] => [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 0
[1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
[1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[9,8,7,6,5,4,3,2,1] => [[[[[[[[[.,.],.],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[8,9,7,6,5,4,3,2,1] => [[[[[[[[.,[.,.]],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 1
[8,7,6,5,4,3,2,1,9] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,9,8,7,6,5,4,3,2] => [.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[9,8,7,6,5,4,3,1,2] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
[7,9,8,6,5,4,3,2,1] => ?
=> ?
=> ? = 1
[6,9,8,7,5,4,3,2,1] => ?
=> ?
=> ? = 1
[5,9,8,7,6,4,3,2,1] => ?
=> ?
=> ? = 1
[4,9,8,7,6,5,3,2,1] => ?
=> ?
=> ? = 1
[3,9,8,7,6,5,4,2,1] => [[[.,[[[[[[.,.],.],.],.],.],.]],.],.]
=> ?
=> ? = 1
[2,9,8,7,6,5,4,3,1] => [[.,[[[[[[[.,.],.],.],.],.],.],.]],.]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[9,8,7,6,5,4,2,1,3] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
[9,8,7,6,5,3,2,1,4] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
[9,8,7,6,4,3,2,1,5] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
[9,8,7,5,4,3,2,1,6] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
[9,8,6,5,4,3,2,1,7] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
[9,7,6,5,4,3,2,1,8] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
Description
The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St001189
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 92%●distinct values known / distinct values provided: 70%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 92%●distinct values known / distinct values provided: 70%
Values
[1] => [.,.]
=> [1,0]
=> [1,0]
=> 0
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[7,8,6,5,4,3,2,1] => [[[[[[[.,[.,.]],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[2,3,4,5,6,7,8,1] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6
[8,7,6,5,4,3,1,2] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[8,7,6,5,4,2,1,3] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[8,7,6,5,3,2,1,4] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[8,7,6,4,3,2,1,5] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[8,7,5,4,3,2,1,6] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[8,6,5,4,3,2,1,7] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[7,6,5,4,3,2,1,8] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[6,8,7,5,4,3,2,1] => [[[[[[.,[[.,.],.]],.],.],.],.],.]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[5,8,7,6,4,3,2,1] => [[[[[.,[[[.,.],.],.]],.],.],.],.]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[4,8,7,6,5,3,2,1] => [[[[.,[[[[.,.],.],.],.]],.],.],.]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[3,8,7,6,5,4,2,1] => [[[.,[[[[[.,.],.],.],.],.]],.],.]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
[2,8,7,6,5,4,3,1] => [[.,[[[[[[.,.],.],.],.],.],.]],.]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,3,4,5,6,7,8,2] => [.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[1,2,4,5,6,7,8,3] => [.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 6
[1,2,3,5,6,7,8,4] => [.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 6
[1,2,3,4,6,7,8,5] => [.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6
[1,2,3,4,5,7,8,6] => [.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6
[1,2,3,4,5,6,8,7] => [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 6
[10,9,8,7,6,5,4,3,2,1] => [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 8
[1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 9
[9,8,7,6,5,4,3,2,1] => [[[[[[[[[.,.],.],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[8,9,7,6,5,4,3,2,1] => [[[[[[[[.,[.,.]],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[8,7,6,5,4,3,2,1,9] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[1,9,8,7,6,5,4,3,2] => [.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[9,8,7,6,5,4,3,1,2] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[7,9,8,6,5,4,3,2,1] => ?
=> ?
=> ?
=> ? = 1
[6,9,8,7,5,4,3,2,1] => ?
=> ?
=> ?
=> ? = 1
[5,9,8,7,6,4,3,2,1] => ?
=> ?
=> ?
=> ? = 1
[4,9,8,7,6,5,3,2,1] => ?
=> ?
=> ?
=> ? = 1
[3,9,8,7,6,5,4,2,1] => [[[.,[[[[[[.,.],.],.],.],.],.]],.],.]
=> ?
=> ?
=> ? = 1
[2,9,8,7,6,5,4,3,1] => [[.,[[[[[[[.,.],.],.],.],.],.],.]],.]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[9,8,7,6,5,4,2,1,3] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[9,8,7,6,5,3,2,1,4] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[9,8,7,6,4,3,2,1,5] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[9,8,7,5,4,3,2,1,6] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[9,8,6,5,4,3,2,1,7] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[9,7,6,5,4,3,2,1,8] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
Description
The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000864
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000864: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 90%●distinct values known / distinct values provided: 60%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000864: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 90%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [1,0]
=> [1] => 0
[1,2] => [2] => [1,1,0,0]
=> [2,1] => 1
[2,1] => [1,1] => [1,0,1,0]
=> [1,2] => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [3,2,1] => 2
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 4
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 3
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 3
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 2
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 2
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 3
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 2
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 2
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2
[1,2,3,4,5,6,7] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => ? = 6
[1,2,3,4,5,7,6] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 5
[1,2,3,4,6,7,5] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 5
[1,2,3,5,6,7,4] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 5
[1,2,4,5,6,7,3] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 5
[1,3,4,5,6,7,2] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 5
[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]
=> [2,1,3,4,5,6,7] => ? = 1
[2,3,4,5,6,7,1] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 5
[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]
=> [2,1,3,4,5,6,7] => ? = 1
[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]
=> [2,1,3,4,5,6,7] => ? = 1
[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]
=> [2,1,3,4,5,6,7] => ? = 1
[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]
=> [2,1,3,4,5,6,7] => ? = 1
[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]
=> [2,1,3,4,5,6,7] => ? = 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]
=> [1,2,3,4,5,6,7,8] => ? = 0
[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]
=> [2,1,3,4,5,6,7,8] => ? = 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]
=> [7,6,5,4,3,2,1,8] => ? = 6
[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]
=> [1,2,3,4,5,6,8,7] => ? = 0
[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]
=> [1,2,3,4,5,6,8,7] => ? = 0
[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]
=> [1,2,3,4,5,6,8,7] => ? = 0
[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]
=> [1,2,3,4,5,6,8,7] => ? = 0
[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]
=> [1,2,3,4,5,6,8,7] => ? = 0
[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]
=> [1,2,3,4,5,6,8,7] => ? = 0
[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]
=> [1,2,3,4,5,6,8,7] => ? = 0
[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]
=> [8,7,6,5,4,3,2,1] => ? = 7
[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]
=> [2,1,3,4,5,6,7,8] => ? = 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]
=> [2,1,3,4,5,6,7,8] => ? = 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]
=> [2,1,3,4,5,6,7,8] => ? = 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]
=> [2,1,3,4,5,6,7,8] => ? = 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]
=> [2,1,3,4,5,6,7,8] => ? = 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]
=> [2,1,3,4,5,6,7,8] => ? = 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]
=> [7,6,5,4,3,2,1,8] => ? = 6
[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]
=> [7,6,5,4,3,2,1,8] => ? = 6
[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]
=> [7,6,5,4,3,2,1,8] => ? = 6
[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]
=> [7,6,5,4,3,2,1,8] => ? = 6
[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]
=> [7,6,5,4,3,2,1,8] => ? = 6
[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]
=> [7,6,5,4,3,2,1,8] => ? = 6
[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]
=> [1,2,3,4,5,6,7,8,9,10] => ? = 0
[1,2,3,4,5,6,7,8,9] => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [9,8,7,6,5,4,3,2,1] => ? = 8
[1,2,3,4,5,6,7,8,9,10] => [10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [10,9,8,7,6,5,4,3,2,1] => ? = 9
[9,8,7,6,5,4,3,2,1] => [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]
=> [1,2,3,4,5,6,7,8,9] => ? = 0
[8,9,7,6,5,4,3,2,1] => [2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9] => ? = 1
[8,7,6,5,4,3,2,1,9] => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => ? = 0
[1,9,8,7,6,5,4,3,2] => [2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9] => ? = 1
[9,8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => ? = 0
[7,9,8,6,5,4,3,2,1] => [2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9] => ? = 1
[6,9,8,7,5,4,3,2,1] => [2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9] => ? = 1
[5,9,8,7,6,4,3,2,1] => [2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9] => ? = 1
[4,9,8,7,6,5,3,2,1] => [2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9] => ? = 1
[3,9,8,7,6,5,4,2,1] => [2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9] => ? = 1
[2,9,8,7,6,5,4,3,1] => [2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9] => ? = 1
Description
The number of circled entries of the shifted recording tableau of a permutation.
The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing.
The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled.
This statistic records the number of circled entries in $Q$.
Matching statistic: St001629
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 10% ●values known / values provided: 52%●distinct values known / distinct values provided: 10%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 10% ●values known / values provided: 52%●distinct values known / distinct values provided: 10%
Values
[1] => [[1]]
=> [1] => [1] => ? = 0
[1,2] => [[1,2]]
=> [2] => [1] => ? = 1
[2,1] => [[1],[2]]
=> [2] => [1] => ? = 0
[1,2,3] => [[1,2,3]]
=> [3] => [1] => ? = 2
[1,3,2] => [[1,2],[3]]
=> [3] => [1] => ? = 1
[2,1,3] => [[1,3],[2]]
=> [2,1] => [1,1] => ? = 0
[2,3,1] => [[1,2],[3]]
=> [3] => [1] => ? = 1
[3,1,2] => [[1,3],[2]]
=> [2,1] => [1,1] => ? = 0
[3,2,1] => [[1],[2],[3]]
=> [3] => [1] => ? = 0
[1,2,3,4] => [[1,2,3,4]]
=> [4] => [1] => ? = 3
[1,2,4,3] => [[1,2,3],[4]]
=> [4] => [1] => ? = 2
[1,3,2,4] => [[1,2,4],[3]]
=> [3,1] => [1,1] => ? = 1
[1,3,4,2] => [[1,2,3],[4]]
=> [4] => [1] => ? = 2
[1,4,2,3] => [[1,2,4],[3]]
=> [3,1] => [1,1] => ? = 1
[1,4,3,2] => [[1,2],[3],[4]]
=> [4] => [1] => ? = 1
[2,1,3,4] => [[1,3,4],[2]]
=> [2,2] => [2] => ? = 1
[2,1,4,3] => [[1,3],[2,4]]
=> [2,2] => [2] => ? = 0
[2,3,1,4] => [[1,2,4],[3]]
=> [3,1] => [1,1] => ? = 1
[2,3,4,1] => [[1,2,3],[4]]
=> [4] => [1] => ? = 2
[2,4,1,3] => [[1,2],[3,4]]
=> [3,1] => [1,1] => ? = 1
[2,4,3,1] => [[1,2],[3],[4]]
=> [4] => [1] => ? = 1
[3,1,2,4] => [[1,3,4],[2]]
=> [2,2] => [2] => ? = 1
[3,1,4,2] => [[1,3],[2,4]]
=> [2,2] => [2] => ? = 0
[3,2,1,4] => [[1,4],[2],[3]]
=> [3,1] => [1,1] => ? = 0
[3,2,4,1] => [[1,3],[2],[4]]
=> [2,2] => [2] => ? = 0
[3,4,1,2] => [[1,2],[3,4]]
=> [3,1] => [1,1] => ? = 1
[3,4,2,1] => [[1,2],[3],[4]]
=> [4] => [1] => ? = 1
[4,1,2,3] => [[1,3,4],[2]]
=> [2,2] => [2] => ? = 1
[4,1,3,2] => [[1,3],[2],[4]]
=> [2,2] => [2] => ? = 0
[4,2,1,3] => [[1,4],[2],[3]]
=> [3,1] => [1,1] => ? = 0
[4,2,3,1] => [[1,3],[2],[4]]
=> [2,2] => [2] => ? = 0
[4,3,1,2] => [[1,4],[2],[3]]
=> [3,1] => [1,1] => ? = 0
[4,3,2,1] => [[1],[2],[3],[4]]
=> [4] => [1] => ? = 0
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [5] => [1] => ? = 4
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [5] => [1] => ? = 3
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [4,1] => [1,1] => ? = 2
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> [5] => [1] => ? = 3
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> [4,1] => [1,1] => ? = 2
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [5] => [1] => ? = 2
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [3,2] => [1,1] => ? = 2
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> [3,2] => [1,1] => ? = 1
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> [4,1] => [1,1] => ? = 2
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> [5] => [1] => ? = 3
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> [4,1] => [1,1] => ? = 2
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> [5] => [1] => ? = 2
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> [3,2] => [1,1] => ? = 2
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> [3,2] => [1,1] => ? = 1
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [4,1] => [1,1] => ? = 1
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> [3,2] => [1,1] => ? = 1
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,1] => [1,1] => ? = 2
[2,1,4,3,5] => [[1,3,5],[2,4]]
=> [2,2,1] => [2,1] => 0
[2,1,5,3,4] => [[1,3,5],[2,4]]
=> [2,2,1] => [2,1] => 0
[3,1,4,2,5] => [[1,3,5],[2,4]]
=> [2,2,1] => [2,1] => 0
[3,1,5,2,4] => [[1,3,5],[2,4]]
=> [2,2,1] => [2,1] => 0
[3,2,4,1,5] => [[1,3,5],[2],[4]]
=> [2,2,1] => [2,1] => 0
[3,2,5,1,4] => [[1,3],[2,5],[4]]
=> [2,2,1] => [2,1] => 0
[4,1,3,2,5] => [[1,3,5],[2],[4]]
=> [2,2,1] => [2,1] => 0
[4,1,5,2,3] => [[1,3,5],[2,4]]
=> [2,2,1] => [2,1] => 0
[4,2,3,1,5] => [[1,3,5],[2],[4]]
=> [2,2,1] => [2,1] => 0
[4,2,5,1,3] => [[1,3],[2,5],[4]]
=> [2,2,1] => [2,1] => 0
[4,3,5,1,2] => [[1,3],[2,5],[4]]
=> [2,2,1] => [2,1] => 0
[5,1,3,2,4] => [[1,3,5],[2],[4]]
=> [2,2,1] => [2,1] => 0
[5,1,4,2,3] => [[1,3,5],[2],[4]]
=> [2,2,1] => [2,1] => 0
[5,2,3,1,4] => [[1,3,5],[2],[4]]
=> [2,2,1] => [2,1] => 0
[5,2,4,1,3] => [[1,3],[2,5],[4]]
=> [2,2,1] => [2,1] => 0
[5,3,4,1,2] => [[1,3],[2,5],[4]]
=> [2,2,1] => [2,1] => 0
[2,1,4,3,6,5,7] => [[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => [3,1] => 0
[2,1,4,3,7,5,6] => [[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => [3,1] => 0
[2,1,5,3,6,4,7] => [[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => [3,1] => 0
[2,1,5,3,7,4,6] => [[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => [3,1] => 0
[2,1,5,4,6,3,7] => [[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => [3,1] => 0
[2,1,5,4,7,3,6] => [[1,3,5],[2,4,7],[6]]
=> [2,2,2,1] => [3,1] => 0
[2,1,6,3,5,4,7] => [[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => [3,1] => 0
[2,1,6,3,7,4,5] => [[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => [3,1] => 0
[2,1,6,4,5,3,7] => [[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => [3,1] => 0
[2,1,6,4,7,3,5] => [[1,3,5],[2,4,7],[6]]
=> [2,2,2,1] => [3,1] => 0
[2,1,6,5,7,3,4] => [[1,3,5],[2,4,7],[6]]
=> [2,2,2,1] => [3,1] => 0
[2,1,7,3,5,4,6] => [[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => [3,1] => 0
[2,1,7,3,6,4,5] => [[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => [3,1] => 0
[2,1,7,4,5,3,6] => [[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => [3,1] => 0
[2,1,7,4,6,3,5] => [[1,3,5],[2,4,7],[6]]
=> [2,2,2,1] => [3,1] => 0
[2,1,7,5,6,3,4] => [[1,3,5],[2,4,7],[6]]
=> [2,2,2,1] => [3,1] => 0
[3,1,4,2,6,5,7] => [[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => [3,1] => 0
[3,1,4,2,7,5,6] => [[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => [3,1] => 0
[3,1,5,2,6,4,7] => [[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => [3,1] => 0
[3,1,5,2,7,4,6] => [[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => [3,1] => 0
[3,1,5,4,6,2,7] => [[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => [3,1] => 0
[3,1,5,4,7,2,6] => [[1,3,5],[2,4,7],[6]]
=> [2,2,2,1] => [3,1] => 0
[3,1,6,2,5,4,7] => [[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => [3,1] => 0
[3,1,6,2,7,4,5] => [[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => [3,1] => 0
[3,1,6,4,5,2,7] => [[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => [3,1] => 0
[3,1,6,4,7,2,5] => [[1,3,5],[2,4,7],[6]]
=> [2,2,2,1] => [3,1] => 0
[3,1,6,5,7,2,4] => [[1,3,5],[2,4,7],[6]]
=> [2,2,2,1] => [3,1] => 0
[3,1,7,2,5,4,6] => [[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => [3,1] => 0
[3,1,7,2,6,4,5] => [[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => [3,1] => 0
[3,1,7,4,5,2,6] => [[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => [3,1] => 0
[3,1,7,4,6,2,5] => [[1,3,5],[2,4,7],[6]]
=> [2,2,2,1] => [3,1] => 0
[3,1,7,5,6,2,4] => [[1,3,5],[2,4,7],[6]]
=> [2,2,2,1] => [3,1] => 0
[3,2,4,1,6,5,7] => [[1,3,5,7],[2,6],[4]]
=> [2,2,2,1] => [3,1] => 0
[3,2,4,1,7,5,6] => [[1,3,5,7],[2,6],[4]]
=> [2,2,2,1] => [3,1] => 0
Description
The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
Matching statistic: St000264
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 52%●distinct values known / distinct values provided: 10%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 52%●distinct values known / distinct values provided: 10%
Values
[1] => [[1]]
=> [1] => ([],1)
=> ? = 0 + 3
[1,2] => [[1,2]]
=> [2] => ([],2)
=> ? = 1 + 3
[2,1] => [[1],[2]]
=> [2] => ([],2)
=> ? = 0 + 3
[1,2,3] => [[1,2,3]]
=> [3] => ([],3)
=> ? = 2 + 3
[1,3,2] => [[1,2],[3]]
=> [3] => ([],3)
=> ? = 1 + 3
[2,1,3] => [[1,3],[2]]
=> [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 3
[2,3,1] => [[1,2],[3]]
=> [3] => ([],3)
=> ? = 1 + 3
[3,1,2] => [[1,3],[2]]
=> [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 3
[3,2,1] => [[1],[2],[3]]
=> [3] => ([],3)
=> ? = 0 + 3
[1,2,3,4] => [[1,2,3,4]]
=> [4] => ([],4)
=> ? = 3 + 3
[1,2,4,3] => [[1,2,3],[4]]
=> [4] => ([],4)
=> ? = 2 + 3
[1,3,2,4] => [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 3
[1,3,4,2] => [[1,2,3],[4]]
=> [4] => ([],4)
=> ? = 2 + 3
[1,4,2,3] => [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 3
[1,4,3,2] => [[1,2],[3],[4]]
=> [4] => ([],4)
=> ? = 1 + 3
[2,1,3,4] => [[1,3,4],[2]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 3
[2,1,4,3] => [[1,3],[2,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 3
[2,3,1,4] => [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 3
[2,3,4,1] => [[1,2,3],[4]]
=> [4] => ([],4)
=> ? = 2 + 3
[2,4,1,3] => [[1,2],[3,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 3
[2,4,3,1] => [[1,2],[3],[4]]
=> [4] => ([],4)
=> ? = 1 + 3
[3,1,2,4] => [[1,3,4],[2]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 3
[3,1,4,2] => [[1,3],[2,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 3
[3,2,1,4] => [[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 3
[3,2,4,1] => [[1,3],[2],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 3
[3,4,1,2] => [[1,2],[3,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 3
[3,4,2,1] => [[1,2],[3],[4]]
=> [4] => ([],4)
=> ? = 1 + 3
[4,1,2,3] => [[1,3,4],[2]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 3
[4,1,3,2] => [[1,3],[2],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 3
[4,2,1,3] => [[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 3
[4,2,3,1] => [[1,3],[2],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 3
[4,3,1,2] => [[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 3
[4,3,2,1] => [[1],[2],[3],[4]]
=> [4] => ([],4)
=> ? = 0 + 3
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? = 4 + 3
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [5] => ([],5)
=> ? = 3 + 3
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 3
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> [5] => ([],5)
=> ? = 3 + 3
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 3
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [5] => ([],5)
=> ? = 2 + 3
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 3
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 3
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 3
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> [5] => ([],5)
=> ? = 3 + 3
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 3
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> [5] => ([],5)
=> ? = 2 + 3
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 3
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 3
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 3
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 3
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 3
[2,1,4,3,5] => [[1,3,5],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,1,5,3,4] => [[1,3,5],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[3,1,4,2,5] => [[1,3,5],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[3,1,5,2,4] => [[1,3,5],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[3,2,4,1,5] => [[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[3,2,5,1,4] => [[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[4,1,3,2,5] => [[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[4,1,5,2,3] => [[1,3,5],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[4,2,3,1,5] => [[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[4,2,5,1,3] => [[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[4,3,5,1,2] => [[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[5,1,3,2,4] => [[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[5,1,4,2,3] => [[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[5,2,3,1,4] => [[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[5,2,4,1,3] => [[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[5,3,4,1,2] => [[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,1,4,3,6,5,7] => [[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[2,1,4,3,7,5,6] => [[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[2,1,5,3,6,4,7] => [[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[2,1,5,3,7,4,6] => [[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[2,1,5,4,6,3,7] => [[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[2,1,5,4,7,3,6] => [[1,3,5],[2,4,7],[6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[2,1,6,3,5,4,7] => [[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[2,1,6,3,7,4,5] => [[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[2,1,6,4,5,3,7] => [[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[2,1,6,4,7,3,5] => [[1,3,5],[2,4,7],[6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[2,1,6,5,7,3,4] => [[1,3,5],[2,4,7],[6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[2,1,7,3,5,4,6] => [[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[2,1,7,3,6,4,5] => [[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[2,1,7,4,5,3,6] => [[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[2,1,7,4,6,3,5] => [[1,3,5],[2,4,7],[6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[2,1,7,5,6,3,4] => [[1,3,5],[2,4,7],[6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[3,1,4,2,6,5,7] => [[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[3,1,4,2,7,5,6] => [[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[3,1,5,2,6,4,7] => [[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[3,1,5,2,7,4,6] => [[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[3,1,5,4,6,2,7] => [[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[3,1,5,4,7,2,6] => [[1,3,5],[2,4,7],[6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[3,1,6,2,5,4,7] => [[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[3,1,6,2,7,4,5] => [[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[3,1,6,4,5,2,7] => [[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[3,1,6,4,7,2,5] => [[1,3,5],[2,4,7],[6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[3,1,6,5,7,2,4] => [[1,3,5],[2,4,7],[6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[3,1,7,2,5,4,6] => [[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[3,1,7,2,6,4,5] => [[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[3,1,7,4,5,2,6] => [[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[3,1,7,4,6,2,5] => [[1,3,5],[2,4,7],[6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[3,1,7,5,6,2,4] => [[1,3,5],[2,4,7],[6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[3,2,4,1,6,5,7] => [[1,3,5,7],[2,6],[4]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[3,2,4,1,7,5,6] => [[1,3,5,7],[2,6],[4]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St000214
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => [1] => 0
[1,2] => [.,[.,.]]
=> [2,1] => [2,1] => 1
[2,1] => [[.,.],.]
=> [1,2] => [1,2] => 0
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 2
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => 1
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => [2,3,1] => 0
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 1
[3,1,2] => [[.,.],[.,.]]
=> [3,1,2] => [2,3,1] => 0
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,4,3,2] => 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,4,3,1] => 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,4,3] => 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,4,3,1] => 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,4,3] => 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,4,2,1] => 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,4,2] => 0
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,2,4,1] => 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,2,4,1] => 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,3,2,4] => 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,4,2,1] => 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,4,2] => 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,3,4,1] => 0
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,3,1,4] => 0
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,2,4,1] => 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,4,2,1] => 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,4,2] => 0
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,3,4,1] => 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,3,1,4] => 0
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,3,4,1] => 0
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,5,4,3,2] => 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [2,5,4,3,1] => 2
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,1,5,4,3] => 3
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [2,5,4,3,1] => 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [3,5,4,2,1] => 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [3,1,5,4,2] => 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,5,4,1] => 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,1,5,4] => 3
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,5,4,1] => 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,3,2,5,4] => 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [3,5,4,2,1] => 2
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [3,1,5,4,2] => 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,5,4,1] => 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [2,3,1,5,4] => 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,5,4,1] => 2
[1,2,3,5,6,7,4] => [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [6,5,4,7,3,2,1] => [3,2,1,7,6,5,4] => ? = 5
[1,2,4,5,6,7,3] => [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [6,5,4,3,7,2,1] => [4,3,2,1,7,6,5] => ? = 5
[2,1,4,3,6,5,7] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => ? = 0
[2,1,4,3,7,5,6] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => ? = 0
[2,1,5,3,6,4,7] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => ? = 0
[2,1,5,3,7,4,6] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => ? = 0
[2,1,5,4,6,3,7] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [7,5,3,4,6,1,2] => [3,6,4,2,7,5,1] => ? = 0
[2,1,5,4,7,3,6] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [7,5,3,4,6,1,2] => [3,6,4,2,7,5,1] => ? = 0
[2,1,6,3,5,4,7] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => ? = 0
[2,1,6,3,7,4,5] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => ? = 0
[2,1,6,4,5,3,7] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [7,5,3,4,6,1,2] => [3,6,4,2,7,5,1] => ? = 0
[2,1,6,4,7,3,5] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [7,5,3,4,6,1,2] => [3,6,4,2,7,5,1] => ? = 0
[2,1,6,5,7,3,4] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [7,5,3,4,6,1,2] => [3,6,4,2,7,5,1] => ? = 0
[2,1,7,3,5,4,6] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => ? = 0
[2,1,7,3,6,4,5] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => ? = 0
[2,1,7,4,5,3,6] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [7,5,3,4,6,1,2] => [3,6,4,2,7,5,1] => ? = 0
[2,1,7,4,6,3,5] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [7,5,3,4,6,1,2] => [3,6,4,2,7,5,1] => ? = 0
[2,1,7,5,6,3,4] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [7,5,3,4,6,1,2] => [3,6,4,2,7,5,1] => ? = 0
[3,1,4,2,6,5,7] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => ? = 0
[3,1,4,2,7,5,6] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => ? = 0
[3,1,5,2,6,4,7] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => ? = 0
[3,1,5,2,7,4,6] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => ? = 0
[3,1,5,4,6,2,7] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [7,5,3,4,6,1,2] => [3,6,4,2,7,5,1] => ? = 0
[3,1,5,4,7,2,6] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [7,5,3,4,6,1,2] => [3,6,4,2,7,5,1] => ? = 0
[3,1,6,2,5,4,7] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => ? = 0
[3,1,6,2,7,4,5] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => ? = 0
[3,1,6,4,5,2,7] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [7,5,3,4,6,1,2] => [3,6,4,2,7,5,1] => ? = 0
[3,1,6,4,7,2,5] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [7,5,3,4,6,1,2] => [3,6,4,2,7,5,1] => ? = 0
[3,1,6,5,7,2,4] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [7,5,3,4,6,1,2] => [3,6,4,2,7,5,1] => ? = 0
[3,1,7,2,5,4,6] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => ? = 0
[3,1,7,2,6,4,5] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => ? = 0
[3,1,7,4,5,2,6] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [7,5,3,4,6,1,2] => [3,6,4,2,7,5,1] => ? = 0
[3,1,7,4,6,2,5] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [7,5,3,4,6,1,2] => [3,6,4,2,7,5,1] => ? = 0
[3,1,7,5,6,2,4] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [7,5,3,4,6,1,2] => [3,6,4,2,7,5,1] => ? = 0
[3,2,4,1,6,5,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [7,5,6,3,1,2,4] => [5,6,4,2,7,3,1] => ? = 0
[3,2,4,1,7,5,6] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [7,5,6,3,1,2,4] => [5,6,4,2,7,3,1] => ? = 0
[3,2,5,1,6,4,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [7,5,6,3,1,2,4] => [5,6,4,2,7,3,1] => ? = 0
[3,2,5,1,7,4,6] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [7,5,6,3,1,2,4] => [5,6,4,2,7,3,1] => ? = 0
[3,2,5,4,6,1,7] => [[[.,.],[[.,.],[.,.]]],[.,.]]
=> [7,5,3,4,1,2,6] => [5,3,6,4,2,7,1] => ? = 0
[3,2,5,4,7,1,6] => [[[.,.],[[.,.],[.,.]]],[.,.]]
=> [7,5,3,4,1,2,6] => [5,3,6,4,2,7,1] => ? = 0
[3,2,6,1,5,4,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [7,5,6,3,1,2,4] => [5,6,4,2,7,3,1] => ? = 0
[3,2,6,1,7,4,5] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [7,5,6,3,1,2,4] => [5,6,4,2,7,3,1] => ? = 0
[3,2,6,4,5,1,7] => [[[.,.],[[.,.],[.,.]]],[.,.]]
=> [7,5,3,4,1,2,6] => [5,3,6,4,2,7,1] => ? = 0
[3,2,6,4,7,1,5] => [[[.,.],[[.,.],[.,.]]],[.,.]]
=> [7,5,3,4,1,2,6] => [5,3,6,4,2,7,1] => ? = 0
[3,2,6,5,7,1,4] => [[[.,.],[[.,.],[.,.]]],[.,.]]
=> [7,5,3,4,1,2,6] => [5,3,6,4,2,7,1] => ? = 0
[3,2,7,1,5,4,6] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [7,5,6,3,1,2,4] => [5,6,4,2,7,3,1] => ? = 0
[3,2,7,1,6,4,5] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [7,5,6,3,1,2,4] => [5,6,4,2,7,3,1] => ? = 0
[3,2,7,4,5,1,6] => [[[.,.],[[.,.],[.,.]]],[.,.]]
=> [7,5,3,4,1,2,6] => [5,3,6,4,2,7,1] => ? = 0
[3,2,7,4,6,1,5] => [[[.,.],[[.,.],[.,.]]],[.,.]]
=> [7,5,3,4,1,2,6] => [5,3,6,4,2,7,1] => ? = 0
[3,2,7,5,6,1,4] => [[[.,.],[[.,.],[.,.]]],[.,.]]
=> [7,5,3,4,1,2,6] => [5,3,6,4,2,7,1] => ? = 0
Description
The number of adjacencies of a permutation.
An adjacency of a permutation $\pi$ is an index $i$ such that $\pi(i)-1 = \pi(i+1)$. Adjacencies are also known as ''small descents''.
This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].
Matching statistic: St001223
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001223: Dyck paths ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 60%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001223: Dyck paths ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 60%
Values
[1] => [.,.]
=> [1,0]
=> 0
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 0
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 0
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 5
[1,2,3,4,6,7,5] => [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 5
[1,2,3,5,6,7,4] => [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 5
[1,2,4,5,6,7,3] => [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[1,3,4,5,6,7,2] => [.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
[1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[2,1,4,3,6,5,7] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[2,1,4,3,7,5,6] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[2,1,5,3,6,4,7] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[2,1,5,3,7,4,6] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[2,1,5,4,6,3,7] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 0
[2,1,5,4,7,3,6] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 0
[2,1,6,3,5,4,7] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[2,1,6,3,7,4,5] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[2,1,6,4,5,3,7] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 0
[2,1,6,4,7,3,5] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 0
[2,1,6,5,7,3,4] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 0
[2,1,7,3,5,4,6] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[2,1,7,3,6,4,5] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[2,1,7,4,5,3,6] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 0
[2,1,7,4,6,3,5] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 0
[2,1,7,5,6,3,4] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 0
[2,3,4,5,6,7,1] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
[2,7,6,5,4,3,1] => [[.,[[[[[.,.],.],.],.],.]],.]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[3,1,4,2,6,5,7] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[3,1,4,2,7,5,6] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[3,1,5,2,6,4,7] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[3,1,5,2,7,4,6] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[3,1,5,4,6,2,7] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 0
[3,1,5,4,7,2,6] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 0
[3,1,6,2,5,4,7] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[3,1,6,2,7,4,5] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[3,1,6,4,5,2,7] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 0
[3,1,6,4,7,2,5] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 0
[3,1,6,5,7,2,4] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 0
[3,1,7,2,5,4,6] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[3,1,7,2,6,4,5] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[3,1,7,4,5,2,6] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 0
[3,1,7,4,6,2,5] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 0
[3,1,7,5,6,2,4] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 0
[3,2,4,1,6,5,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 0
[3,2,4,1,7,5,6] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 0
[3,2,5,1,6,4,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 0
[3,2,5,1,7,4,6] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 0
[3,2,5,4,6,1,7] => [[[.,.],[[.,.],[.,.]]],[.,.]]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 0
[3,2,5,4,7,1,6] => [[[.,.],[[.,.],[.,.]]],[.,.]]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 0
[3,2,6,1,5,4,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 0
[3,2,6,1,7,4,5] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 0
[3,2,6,4,5,1,7] => [[[.,.],[[.,.],[.,.]]],[.,.]]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 0
Description
Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless.
Matching statistic: St001233
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001233: Dyck paths ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 60%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001233: Dyck paths ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 60%
Values
[1] => [.,.]
=> [1,0]
=> 0
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 2
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 0
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 0
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 6
[1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 5
[1,2,3,4,6,7,5] => [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 5
[1,2,3,5,6,7,4] => [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 5
[1,2,4,5,6,7,3] => [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 5
[1,3,4,5,6,7,2] => [.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 5
[1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[2,1,4,3,6,5,7] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[2,1,4,3,7,5,6] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[2,1,5,3,6,4,7] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[2,1,5,3,7,4,6] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[2,1,5,4,6,3,7] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[2,1,5,4,7,3,6] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[2,1,6,3,5,4,7] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[2,1,6,3,7,4,5] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[2,1,6,4,5,3,7] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[2,1,6,4,7,3,5] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[2,1,6,5,7,3,4] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[2,1,7,3,5,4,6] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[2,1,7,3,6,4,5] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[2,1,7,4,5,3,6] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[2,1,7,4,6,3,5] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[2,1,7,5,6,3,4] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[2,3,4,5,6,7,1] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 5
[2,7,6,5,4,3,1] => [[.,[[[[[.,.],.],.],.],.]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1
[3,1,4,2,6,5,7] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[3,1,4,2,7,5,6] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[3,1,5,2,6,4,7] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[3,1,5,2,7,4,6] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[3,1,5,4,6,2,7] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[3,1,5,4,7,2,6] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[3,1,6,2,5,4,7] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[3,1,6,2,7,4,5] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[3,1,6,4,5,2,7] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[3,1,6,4,7,2,5] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[3,1,6,5,7,2,4] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[3,1,7,2,5,4,6] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[3,1,7,2,6,4,5] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[3,1,7,4,5,2,6] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[3,1,7,4,6,2,5] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[3,1,7,5,6,2,4] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[3,2,4,1,6,5,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 0
[3,2,4,1,7,5,6] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 0
[3,2,5,1,6,4,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 0
[3,2,5,1,7,4,6] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 0
[3,2,5,4,6,1,7] => [[[.,.],[[.,.],[.,.]]],[.,.]]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> ? = 0
[3,2,5,4,7,1,6] => [[[.,.],[[.,.],[.,.]]],[.,.]]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> ? = 0
[3,2,6,1,5,4,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 0
[3,2,6,1,7,4,5] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 0
[3,2,6,4,5,1,7] => [[[.,.],[[.,.],[.,.]]],[.,.]]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> ? = 0
Description
The number of indecomposable 2-dimensional modules with projective dimension one.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000931The number of occurrences of the pattern UUU in a Dyck path. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001948The number of augmented double ascents of a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition.
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!