searching the database
Your data matches 383 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: St001085
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [2,1] => 0
[1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [3,1,2] => 0
[1,1,0,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [2,3,1] => 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
Description
The number of occurrences of the vincular pattern |21-3 in a permutation.
This is the number of occurrences of the pattern $213$, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive.
In other words, this is the number of ascents whose bottom value is strictly smaller and the top value is strictly larger than the first entry of the permutation.
Matching statistic: St000007
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1]]
=> [1] => [1] => 1 = 0 + 1
[1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => [1,2] => 1 = 0 + 1
[1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => [1,2] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => [1,3,2] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => [1,3,2] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => [1,3,2] => 2 = 1 + 1
[1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => [1,2,3] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => [1,3,2,4] => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,4,2,3] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => [1,3,4,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => [1,4,2,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => [1,3,2,4] => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,4,2,3] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [3,2,1,4] => [1,4,2,3] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [2,1,4,3] => [1,4,2,3] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [1,4,2,3] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => [1,2,4,3,5] => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,2,5,3,4] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => [1,3,2,4,5] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => [1,2,4,3,5] => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,2,5,3,4] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => [1,4,2,5,3] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,2,5,3,4] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => [1,5,2,3,4] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [2,1,3,4,5] => [1,3,4,5,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => [1,3,5,2,4] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => [1,4,2,3,5] => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => [1,3,5,2,4] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [2,1,5,4,3] => [1,5,2,3,4] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => [1,3,2,4,5] => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => [1,2,4,3,5] => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,2,5,3,4] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => [1,4,2,5,3] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => [1,2,5,3,4] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => [1,5,2,3,4] => 2 = 1 + 1
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000783
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000783: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000783: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 1 = 0 + 1
[1,0,1,0]
=> [2,1] => [2,1] => [2]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => [2,1]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => [3]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [2,1]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => [2,2]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,1,2] => [4]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,1,3,2] => [3,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,2,1,3] => [3,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => [4]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,3,2] => [2,1,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,4,2,3] => [3,1]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1,4,2] => [4]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => [2,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,1,2,4] => [3,1]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [2,2,1]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,4,3,1,2] => [4,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [5,4,1,3,2] => [5]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,4,2,1,3] => [5]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,1,2,3] => [3,2]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [5,1,4,3,2] => [3,2]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,1,4,2,3] => [5]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,3,1,4,2] => [4,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,3,2,1,4] => [3,2]
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,1,2,4] => [5]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,1,2,4,3] => [4,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,1,3,2,4] => [4,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,2,1,3,4] => [4,1]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [5]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,5,4,3,2] => [2,2,1]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,5,4,2,3] => [4,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,5,2,4,3] => [3,1,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,5,3,2,4] => [3,1,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,5,2,3,4] => [4,1]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,1,5,3,2] => [5]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,1,5,2,3] => [3,2]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,3,1,5,2] => [5]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => [2,2,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,3,1,2,5] => [4,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [4,1,2,5,3] => [5]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,1,3,2,5] => [3,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,2,1,3,5] => [3,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,1]
=> 2 = 1 + 1
Description
The side length of the largest staircase partition fitting into a partition.
For an integer partition $(\lambda_1\geq \lambda_2\geq\dots)$ this is the largest integer $k$ such that $\lambda_i > k-i$ for $i\in\{1,\dots,k\}$.
In other words, this is the length of a longest (strict) north-east chain of cells in the Ferrers diagram of the partition, using the English convention. Equivalently, this is the maximal number of non-attacking rooks that can be placed on the Ferrers diagram.
This is also the maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic records the largest part occurring in any of these partitions.
Matching statistic: St001359
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001359: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001359: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [1] => [1] => 1 = 0 + 1
[1,0,1,0]
=> [3,1,2] => [1,2] => [1,2] => 1 = 0 + 1
[1,1,0,0]
=> [2,3,1] => [2,1] => [2,1] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,2,3] => [1,3,2] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [2,1,3] => [2,1,3] => 1 = 0 + 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,1] => [2,3,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,2,3,4] => [1,4,3,2] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,2,3] => [4,1,3,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,1,2,4] => [3,1,4,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,4,2] => [3,1,4,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,1,3,4] => [2,1,4,3] => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,2,4] => [3,1,4,2] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,1,2,3] => [4,1,3,2] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [4,3,1,2] => [4,3,1,2] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [2,3,1,4] => [2,4,1,3] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => [2,4,3,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,2,3,4,5] => [1,5,4,3,2] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,2,3,4] => [5,1,4,3,2] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,2,3,5] => [4,1,5,3,2] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,2,5,3,4] => [1,5,4,3,2] => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,2,5,3] => [4,1,5,3,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,2,4,5] => [3,1,5,4,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,5,2,4] => [3,1,5,4,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,4,2,3,5] => [1,5,4,3,2] => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,5,2,3,4] => [1,5,4,3,2] => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [5,1,4,2,3] => [5,1,4,3,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,4,2,5] => [3,1,5,4,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [3,1,5,2,4] => [3,1,5,4,2] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,4,5,2,3] => [1,5,4,3,2] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,4,5,2] => [3,1,5,4,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,1,3,4,5] => [2,1,5,4,3] => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,5,1,3,4] => [2,5,1,4,3] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,4,1,3,5] => [2,5,1,4,3] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [2,1,5,3,4] => [2,1,5,4,3] => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,4,1,5,3] => [2,5,1,4,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [3,1,2,4,5] => [3,1,5,4,2] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [5,3,1,2,4] => [5,3,1,4,2] => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,1,2,3,5] => [4,1,5,3,2] => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [5,1,2,3,4] => [5,1,4,3,2] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [5,4,1,2,3] => [5,4,1,3,2] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [4,3,1,2,5] => [4,3,1,5,2] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [3,1,5,2,4] => [3,1,5,4,2] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [4,1,5,2,3] => [4,1,5,3,2] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [4,3,1,5,2] => [4,3,1,5,2] => 2 = 1 + 1
Description
The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles.
In other words, this is $2^k$ where $k$ is the number of cycles of length at least three ([[St000486]]) in its cycle decomposition.
The generating function for the number of equivalence classes, $f(n)$, is
$$\sum_{n\geq 0} f(n)\frac{x^n}{n!} = \frac{e(\frac{x}{2} + \frac{x^2}{4})}{\sqrt{1-x}}.$$
Matching statistic: St001432
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 1 = 0 + 1
[1,0,1,0]
=> [2,1] => [2,1] => [2]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => [2,1]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => [3]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [2,1]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => [2,2]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,1,2] => [4]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,1,3,2] => [3,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,2,1,3] => [3,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => [4]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,3,2] => [2,1,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,4,2,3] => [3,1]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1,4,2] => [4]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => [2,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,1,2,4] => [3,1]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [2,2,1]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,4,3,1,2] => [4,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [5,4,1,3,2] => [5]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,4,2,1,3] => [5]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,1,2,3] => [3,2]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [5,1,4,3,2] => [3,2]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,1,4,2,3] => [5]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,3,1,4,2] => [4,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,3,2,1,4] => [3,2]
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,1,2,4] => [5]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,1,2,4,3] => [4,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,1,3,2,4] => [4,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,2,1,3,4] => [4,1]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [5]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,5,4,3,2] => [2,2,1]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,5,4,2,3] => [4,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,5,2,4,3] => [3,1,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,5,3,2,4] => [3,1,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,5,2,3,4] => [4,1]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,1,5,3,2] => [5]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,1,5,2,3] => [3,2]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,3,1,5,2] => [5]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => [2,2,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,3,1,2,5] => [4,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [4,1,2,5,3] => [5]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,1,3,2,5] => [3,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,2,1,3,5] => [3,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,1]
=> 2 = 1 + 1
Description
The order dimension of the partition.
Given a partition $\lambda$, let $I(\lambda)$ be the principal order ideal in the Young lattice generated by $\lambda$. The order dimension of a partition is defined as the order dimension of the poset $I(\lambda)$.
Matching statistic: St001913
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001913: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001913: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1],[]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[1,0,1,0]
=> [[1,1],[]]
=> ([(0,1)],2)
=> [2]
=> 1 = 0 + 1
[1,1,0,0]
=> [[2],[]]
=> ([(0,1)],2)
=> [2]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> [2,1]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> [4,2]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ([(0,2),(0,4),(1,3),(1,4),(3,5),(4,5)],6)
=> [3,2,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> [3,3]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> [3,2,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> [4,3]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> [4,2,1]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> [3,2,1]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> [3,3]
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> ([(0,2),(0,3),(1,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
=> [4,2,1]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> ([(0,6),(1,3),(1,6),(2,4),(3,2),(3,5),(5,4),(6,5)],7)
=> [4,3]
=> 2 = 1 + 1
Description
The number of preimages of an integer partition in Bulgarian solitaire.
A move in Bulgarian solitaire consists of removing the first column of the Ferrers diagram and inserting it as a new row.
Partitions without preimages are called garden of eden partitions [1].
Matching statistic: St000486
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000486: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000486: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [1] => [1] => ? = 0
[1,0,1,0]
=> [3,1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,3,1] => [2,1] => [2,1] => 0
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,2,3] => [1,3,2] => 0
[1,0,1,1,0,0]
=> [3,1,4,2] => [3,1,2] => [3,1,2] => 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [2,1,3] => [2,1,3] => 0
[1,1,0,1,0,0]
=> [4,3,1,2] => [3,1,2] => [3,1,2] => 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,1] => [2,3,1] => 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,2,3,4] => [1,4,3,2] => 0
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,2,3] => [4,1,3,2] => 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,1,2,4] => [3,1,4,2] => 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 0
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,4,2] => [3,1,4,2] => 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,1,3,4] => [2,1,4,3] => 0
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [2,4,1,3] => [2,4,1,3] => 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,2,4] => [3,1,4,2] => 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,1,2,3] => [4,1,3,2] => 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [4,3,1,2] => [4,3,1,2] => 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [2,3,1,4] => [2,4,1,3] => 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [2,4,1,3] => [2,4,1,3] => 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => [2,4,3,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,2,3,4,5] => [1,5,4,3,2] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,2,3,4] => [5,1,4,3,2] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,2,3,5] => [4,1,5,3,2] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,2,5,3,4] => [1,5,4,3,2] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,2,5,3] => [4,1,5,3,2] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,2,4,5] => [3,1,5,4,2] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,5,2,4] => [3,1,5,4,2] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,4,2,3,5] => [1,5,4,3,2] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,5,2,3,4] => [1,5,4,3,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [5,1,4,2,3] => [5,1,4,3,2] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,4,2,5] => [3,1,5,4,2] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [3,1,5,2,4] => [3,1,5,4,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,4,5,2,3] => [1,5,4,3,2] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,4,5,2] => [3,1,5,4,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,1,3,4,5] => [2,1,5,4,3] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,5,1,3,4] => [2,5,1,4,3] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,4,1,3,5] => [2,5,1,4,3] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [2,1,5,3,4] => [2,1,5,4,3] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,4,1,5,3] => [2,5,1,4,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [3,1,2,4,5] => [3,1,5,4,2] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [5,3,1,2,4] => [5,3,1,4,2] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,1,2,3,5] => [4,1,5,3,2] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [5,1,2,3,4] => [5,1,4,3,2] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [5,4,1,2,3] => [5,4,1,3,2] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [4,3,1,2,5] => [4,3,1,5,2] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [3,1,5,2,4] => [3,1,5,4,2] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [4,1,5,2,3] => [4,1,5,3,2] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [4,3,1,5,2] => [4,3,1,5,2] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,3,1,4,5] => [2,5,1,4,3] => 1
Description
The number of cycles of length at least 3 of a permutation.
Matching statistic: St001174
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001174: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001174: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1] => ? = 0
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 0
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => 0
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => 1
Description
The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St001440
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001440: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001440: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [] => []
=> ? = 0
[1,0,1,0]
=> [1,2] => [1] => [1]
=> 0
[1,1,0,0]
=> [2,1] => [1] => [1]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => [1,1]
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => [1,1]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => [2]
=> 0
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => [2]
=> 0
[1,1,1,0,0,0]
=> [3,1,2] => [1,2] => [1,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => [1,1,1]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => [1,1,1]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => [2,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => [2,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,2,3] => [1,1,1]
=> 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => [2,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => [2,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => [2,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => [2,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,3] => [2,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1,2] => [2,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [3,1,2] => [2,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,1,2] => [2,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [1,2,3] => [1,1,1]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => [2,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => [2,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => [2,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => [2,1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => [2,1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => [2,1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,3,2,4] => [2,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,4,2,3] => [2,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,4,2,3] => [2,1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,4,2,3] => [2,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => [2,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => [2,2]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => [2,2]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,3,4] => [2,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => [2,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => [2,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => [2,1,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => [2,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,3,1,4] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,4,1,3] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,4,1,3] => [2,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,4,1,3] => [2,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,3,4] => [2,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4] => [2,1,1]
=> 1
Description
The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition.
Matching statistic: St001195
(load all 56 compositions to match this statistic)
(load all 56 compositions to match this statistic)
St001195: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> ? = 0
[1,0,1,0]
=> ? ∊ {0,0}
[1,1,0,0]
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> 0
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
The following 373 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000402Half the size of the symmetry class of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000864The number of circled entries of the shifted recording tableau of a permutation. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St000897The number of different multiplicities of parts of an integer partition. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001204Call 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 special CNakayama algebra. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001498The normalised height of a Nakayama algebra with magnitude 1. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000730The maximal arc length of a set partition. St000929The constant term of the character polynomial of an integer partition. St001737The number of descents of type 2 in a permutation. St000260The radius of a connected graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000284The Plancherel distribution on integer partitions. St000618The number of self-evacuating tableaux of given shape. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000781The number of proper colouring schemes of a Ferrers diagram. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000891The number of distinct diagonal sums of a permutation matrix. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St000003The number of standard Young tableaux of the partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000182The number of permutations whose cycle type is the given integer partition. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000326The position of the first one in a binary word after appending a 1 at the end. St000517The Kreweras number of an integer partition. St000913The number of ways to refine the partition into singletons. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001520The number of strict 3-descents. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001638The book thickness of a graph. St001820The size of the image of the pop stack sorting operator. St000893The number of distinct diagonal sums of an alternating sign matrix. St000628The balance of a binary word. St000920The logarithmic height of a Dyck path. St001569The maximal modular displacement of a permutation. St000903The number of different parts of an integer composition. St000298The order dimension or Dushnik-Miller dimension of a poset. St000905The number of different multiplicities of parts of an integer composition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000630The length of the shortest palindromic decomposition of a binary word. St001884The number of borders of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000640The rank of the largest boolean interval in a poset. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001964The interval resolution global dimension of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000455The second largest eigenvalue of a graph if it is integral. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000307The number of rowmotion orbits of a poset. St001812The biclique partition number of a graph. St000100The number of linear extensions of a poset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000635The number of strictly order preserving maps of a poset into itself. St000658The number of rises of length 2 of a Dyck path. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001139The number of occurrences of hills of size 2 in a Dyck path. St001890The maximum magnitude of the Möbius function of a poset. St001624The breadth of a lattice. St000647The number of big descents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000862The number of parts of the shifted shape of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001335The cardinality of a minimal cycle-isolating set of a graph. St001651The Frankl number of a lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001330The hat guessing number of a graph. St001875The number of simple modules with projective dimension at most 1. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000035The number of left outer peaks of a permutation. St000665The number of rafts of a permutation. St000871The number of very big ascents of a permutation. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001864The number of excedances of a signed permutation. St001881The number of factors of a lattice as a Cartesian product of lattices. St001896The number of right descents of a signed permutations. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000137The Grundy value of an integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000562The number of internal points of a set partition. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001280The number of parts of an integer partition that are at least two. St001383The BG-rank of an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001396Number of triples of incomparable elements in a finite poset. St001525The number of symmetric hooks on the diagonal of a partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001532The leading coefficient of the Poincare polynomial of the poset cone. St000356The number of occurrences of the pattern 13-2. St000567The sum of the products of all pairs of parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000782The indicator function of whether a given perfect matching is an L & P matching. St000928The sum of the coefficients of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001083The number of boxed occurrences of 132 in a permutation. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001394The genus of a permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001933The largest multiplicity of a part in an integer partition. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000422The energy of a graph, if it is integral. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001863The number of weak excedances of a signed permutation. St001822The number of alignments of a signed permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000023The number of inner peaks of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000089The absolute variation of a composition. St000091The descent variation of a composition. St000233The number of nestings of a set partition. St000252The number of nodes of degree 3 of a binary tree. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. St000357The number of occurrences of the pattern 12-3. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000367The number of simsun double descents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000407The number of occurrences of the pattern 2143 in a permutation. St000516The number of stretching pairs of a permutation. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000648The number of 2-excedences of a permutation. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000664The number of right ropes of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000732The number of double deficiencies of a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000779The tier of a permutation. St000872The number of very big descents of a permutation. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001403The number of vertical separators in a permutation. St001470The cyclic holeyness of a permutation. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001556The number of inversions of the third entry of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001578The minimal number of edges to add or remove to make a graph a line graph. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001811The Castelnuovo-Mumford regularity of a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001866The nesting alignments of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000099The number of valleys of a permutation, including the boundary. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000354The number of recoils of a permutation. St000388The number of orbits of vertices of a graph under automorphisms. St000570The Edelman-Greene number of a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000829The Ulam distance of a permutation to the identity permutation. St000886The number of permutations with the same antidiagonal sums. St000899The maximal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001220The width of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001405The number of bonds in a permutation. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001423The number of distinct cubes in a binary word. St001489The maximum of the number of descents and the number of inverse descents. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001768The number of reduced words of a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001823The Stasinski-Voll length of a signed permutation. St001874Lusztig's a-function for the symmetric group. St001935The number of ascents in a parking function. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000390The number of runs of ones in a binary word. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001488The number of corners of a skew partition. St001566The length of the longest arithmetic progression in a permutation. St001642The Prague dimension of a graph. St001672The restrained domination number of a graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000735The last entry on the main diagonal of a standard tableau. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St001301The first Betti number of the order complex associated with the poset. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001948The number of augmented double ascents of a permutation. St000654The first descent of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000908The length of the shortest maximal antichain in a poset. St000911The number of maximal antichains of maximal size in a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000824The sum of the number of descents and the number of recoils of a permutation. St000907The number of maximal antichains of minimal length in a poset. St001516The number of cyclic bonds of a permutation. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset.
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!