searching the database
Your data matches 4 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: St000001
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000001: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000001: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 6
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 5
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 4
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => 5
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 9
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 10
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 5
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 16
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 10
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 5
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 9
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 5
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => 6
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => 14
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => 15
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 14
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 35
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 20
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 21
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 21
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 35
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => 15
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 14
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => 14
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => 6
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => 20
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => 21
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 28
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => 64
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => 35
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => 14
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 70
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 56
Description
The number of reduced words for a permutation.
This is the number of ways to write a permutation as a minimal length product of simple transpositions. E.g., there are two reduced words for the permutation $[3,2,1]$, which are $(1,2)(2,3)(1,2) = (2,3)(1,2)(2,3)$.
Matching statistic: St000003
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St000003: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 99%
Values
[1]
=> 1
[2]
=> 1
[1,1]
=> 1
[3]
=> 1
[2,1]
=> 2
[1,1,1]
=> 1
[4]
=> 1
[3,1]
=> 3
[2,2]
=> 2
[2,1,1]
=> 3
[1,1,1,1]
=> 1
[5]
=> 1
[4,1]
=> 4
[3,2]
=> 5
[3,1,1]
=> 6
[2,2,1]
=> 5
[2,1,1,1]
=> 4
[1,1,1,1,1]
=> 1
[6]
=> 1
[5,1]
=> 5
[4,2]
=> 9
[4,1,1]
=> 10
[3,3]
=> 5
[3,2,1]
=> 16
[3,1,1,1]
=> 10
[2,2,2]
=> 5
[2,2,1,1]
=> 9
[2,1,1,1,1]
=> 5
[1,1,1,1,1,1]
=> 1
[6,1]
=> 6
[5,2]
=> 14
[5,1,1]
=> 15
[4,3]
=> 14
[4,2,1]
=> 35
[4,1,1,1]
=> 20
[3,3,1]
=> 21
[3,2,2]
=> 21
[3,2,1,1]
=> 35
[3,1,1,1,1]
=> 15
[2,2,2,1]
=> 14
[2,2,1,1,1]
=> 14
[2,1,1,1,1,1]
=> 6
[6,2]
=> 20
[6,1,1]
=> 21
[5,3]
=> 28
[5,2,1]
=> 64
[5,1,1,1]
=> 35
[4,4]
=> 14
[4,3,1]
=> 70
[4,2,2]
=> 56
[5,5,2,2]
=> ? = 21450
[5,3,3,3]
=> ? = 15015
[5,4,4,2,2]
=> ? = 1361360
Description
The number of [[/StandardTableaux|standard Young tableaux]] of the partition.
Matching statistic: St000100
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 55% ●values known / values provided: 65%●distinct values known / distinct values provided: 55%
Mp00185: Skew partitions —cell poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 55% ●values known / values provided: 65%●distinct values known / distinct values provided: 55%
Values
[1]
=> [[1],[]]
=> ([],1)
=> ? = 1
[2]
=> [[2],[]]
=> ([(0,1)],2)
=> 1
[1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 1
[3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 2
[1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1
[4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 3
[2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 3
[1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[5]
=> [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 4
[3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 5
[3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 6
[2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 5
[2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 4
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 5
[4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 9
[4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 10
[3,3]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5
[3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> 16
[3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 10
[2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5
[2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 9
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 5
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[6,1]
=> [[6,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> 6
[5,2]
=> [[5,2],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> 14
[5,1,1]
=> [[5,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> 15
[4,3]
=> [[4,3],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> 14
[4,2,1]
=> [[4,2,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> 35
[4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> 20
[3,3,1]
=> [[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> 21
[3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> 21
[3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> 35
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> 15
[2,2,2,1]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> 14
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> 14
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> 6
[6,2]
=> [[6,2],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> 20
[6,1,1]
=> [[6,1,1],[]]
=> ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
=> 21
[5,3]
=> [[5,3],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> 28
[5,2,1]
=> [[5,2,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
=> 64
[5,1,1,1]
=> [[5,1,1,1],[]]
=> ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
=> 35
[4,4]
=> [[4,4],[]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 14
[4,3,1]
=> [[4,3,1],[]]
=> ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> 70
[4,2,2]
=> [[4,2,2],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> 56
[4,2,1,1]
=> [[4,2,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(5,7),(6,4),(6,7)],8)
=> 90
[6,5]
=> [[6,5],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(3,9),(4,3),(4,10),(5,1),(5,8),(6,4),(6,7),(7,10),(9,8),(10,9)],11)
=> ? = 132
[5,4,2]
=> [[5,4,2],[]]
=> ([(0,5),(0,6),(2,9),(3,4),(3,10),(4,1),(4,8),(5,3),(5,7),(6,2),(6,7),(7,9),(7,10),(10,8)],11)
=> ? = 990
[5,4,1,1]
=> [[5,4,1,1],[]]
=> ([(0,6),(0,7),(3,2),(4,5),(4,10),(5,1),(5,9),(6,4),(6,8),(7,3),(7,8),(8,10),(10,9)],11)
=> ? = 1155
[5,3,3]
=> [[5,3,3],[]]
=> ([(0,5),(0,6),(2,9),(3,1),(4,3),(4,8),(5,4),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 660
[5,3,2,1]
=> [[5,3,2,1],[]]
=> ([(0,6),(0,7),(3,2),(4,3),(4,9),(5,1),(5,10),(6,4),(6,8),(7,5),(7,8),(8,9),(8,10)],11)
=> ? = 2310
[5,3,1,1,1]
=> [[5,3,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,6),(4,10),(5,1),(6,2),(7,3),(7,9),(8,4),(8,9),(9,10)],11)
=> ? = 1540
[5,2,2,2]
=> [[5,2,2,2],[]]
=> ([(0,6),(0,7),(2,9),(3,4),(4,1),(5,2),(5,10),(6,3),(6,8),(7,5),(7,8),(8,10),(10,9)],11)
=> ? = 825
[5,2,2,1,1]
=> [[5,2,2,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,6),(4,10),(5,1),(6,2),(7,3),(7,9),(8,4),(8,9),(9,10)],11)
=> ? = 1540
[4,4,3]
=> [[4,4,3],[]]
=> ([(0,4),(0,5),(1,8),(2,7),(3,2),(3,9),(4,3),(4,6),(5,1),(5,6),(6,8),(6,9),(8,10),(9,7),(9,10)],11)
=> ? = 462
[4,4,2,1]
=> [[4,4,2,1],[]]
=> ([(0,5),(0,6),(2,8),(3,2),(3,10),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(10,8)],11)
=> ? = 1320
[4,4,1,1,1]
=> [[4,4,1,1,1],[]]
=> ([(0,6),(0,7),(2,9),(3,4),(4,1),(5,2),(5,10),(6,3),(6,8),(7,5),(7,8),(8,10),(10,9)],11)
=> ? = 825
[4,3,3,1]
=> [[4,3,3,1],[]]
=> ([(0,5),(0,6),(3,2),(3,8),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 1188
[4,3,2,2]
=> [[4,3,2,2],[]]
=> ([(0,5),(0,6),(2,8),(3,2),(3,10),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(10,8)],11)
=> ? = 1320
[4,3,2,1,1]
=> [[4,3,2,1,1],[]]
=> ([(0,6),(0,7),(3,2),(4,3),(4,9),(5,1),(5,10),(6,4),(6,8),(7,5),(7,8),(8,9),(8,10)],11)
=> ? = 2310
[4,2,2,2,1]
=> [[4,2,2,2,1],[]]
=> ([(0,6),(0,7),(3,2),(4,5),(4,10),(5,1),(5,9),(6,4),(6,8),(7,3),(7,8),(8,10),(10,9)],11)
=> ? = 1155
[3,3,3,2]
=> [[3,3,3,2],[]]
=> ([(0,4),(0,5),(1,8),(2,7),(3,2),(3,9),(4,3),(4,6),(5,1),(5,6),(6,8),(6,9),(8,10),(9,7),(9,10)],11)
=> ? = 462
[3,3,3,1,1]
=> [[3,3,3,1,1],[]]
=> ([(0,5),(0,6),(2,9),(3,1),(4,3),(4,8),(5,4),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 660
[3,3,2,2,1]
=> [[3,3,2,2,1],[]]
=> ([(0,5),(0,6),(2,9),(3,4),(3,10),(4,1),(4,8),(5,3),(5,7),(6,2),(6,7),(7,9),(7,10),(10,8)],11)
=> ? = 990
[5,4,3]
=> [[5,4,3],[]]
=> ([(0,5),(0,6),(2,9),(3,4),(3,10),(4,1),(4,8),(5,3),(5,7),(6,2),(6,7),(7,9),(7,10),(9,11),(10,8),(10,11)],12)
=> ? = 2112
[5,4,2,1]
=> [[5,4,2,1],[]]
=> ([(0,6),(0,7),(3,4),(3,11),(4,1),(4,9),(5,2),(5,10),(6,5),(6,8),(7,3),(7,8),(8,10),(8,11),(11,9)],12)
=> ? = 5775
[5,4,1,1,1]
=> [[5,4,1,1,1],[]]
=> ([(0,7),(0,8),(3,4),(4,2),(5,6),(5,11),(6,1),(6,10),(7,3),(7,9),(8,5),(8,9),(9,11),(11,10)],12)
=> ? = 3520
[5,3,3,1]
=> [[5,3,3,1],[]]
=> ([(0,6),(0,7),(3,2),(4,3),(4,9),(5,1),(5,10),(6,4),(6,8),(7,5),(7,8),(8,9),(8,10),(9,11),(10,11)],12)
=> ? = 4158
[5,3,2,2]
=> [[5,3,2,2],[]]
=> ([(0,6),(0,7),(2,9),(3,1),(4,3),(4,10),(5,2),(5,11),(6,4),(6,8),(7,5),(7,8),(8,10),(8,11),(11,9)],12)
=> ? = 4455
[5,3,2,1,1]
=> [[5,3,2,1,1],[]]
=> ([(0,7),(0,8),(3,2),(4,1),(5,3),(5,10),(6,4),(6,11),(7,5),(7,9),(8,6),(8,9),(9,10),(9,11)],12)
=> ? = 7700
[5,2,2,2,1]
=> [[5,2,2,2,1],[]]
=> ([(0,7),(0,8),(3,4),(4,2),(5,6),(5,11),(6,1),(6,10),(7,3),(7,9),(8,5),(8,9),(9,11),(11,10)],12)
=> ? = 3520
[4,4,4]
=> [[4,4,4],[]]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 462
[4,4,3,1]
=> [[4,4,3,1],[]]
=> ([(0,5),(0,6),(2,8),(3,2),(3,10),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(9,11),(10,8),(10,11)],12)
=> ? = 2970
[4,4,2,2]
=> [[4,4,2,2],[]]
=> ([(0,5),(0,6),(1,9),(2,8),(3,2),(3,10),(4,1),(4,11),(5,3),(5,7),(6,4),(6,7),(7,10),(7,11),(10,8),(11,9)],12)
=> ? = 2640
[4,4,2,1,1]
=> [[4,4,2,1,1],[]]
=> ([(0,6),(0,7),(2,9),(3,1),(4,3),(4,10),(5,2),(5,11),(6,4),(6,8),(7,5),(7,8),(8,10),(8,11),(11,9)],12)
=> ? = 4455
[4,3,3,2]
=> [[4,3,3,2],[]]
=> ([(0,5),(0,6),(2,8),(3,2),(3,10),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(9,11),(10,8),(10,11)],12)
=> ? = 2970
[4,3,3,1,1]
=> [[4,3,3,1,1],[]]
=> ([(0,6),(0,7),(3,2),(4,3),(4,9),(5,1),(5,10),(6,4),(6,8),(7,5),(7,8),(8,9),(8,10),(9,11),(10,11)],12)
=> ? = 4158
[4,3,2,2,1]
=> [[4,3,2,2,1],[]]
=> ([(0,6),(0,7),(3,4),(3,11),(4,1),(4,9),(5,2),(5,10),(6,5),(6,8),(7,3),(7,8),(8,10),(8,11),(11,9)],12)
=> ? = 5775
[3,3,3,2,1]
=> [[3,3,3,2,1],[]]
=> ([(0,5),(0,6),(2,9),(3,4),(3,10),(4,1),(4,8),(5,3),(5,7),(6,2),(6,7),(7,9),(7,10),(9,11),(10,8),(10,11)],12)
=> ? = 2112
[5,4,3,1]
=> [[5,4,3,1],[]]
=> ([(0,6),(0,7),(3,4),(3,11),(4,1),(4,9),(5,2),(5,10),(6,5),(6,8),(7,3),(7,8),(8,10),(8,11),(10,12),(11,9),(11,12)],13)
=> ? = 15015
[5,4,2,2]
=> [[5,4,2,2],[]]
=> ([(0,6),(0,7),(2,10),(3,5),(3,11),(4,2),(4,12),(5,1),(5,9),(6,3),(6,8),(7,4),(7,8),(8,11),(8,12),(11,9),(12,10)],13)
=> ? = 12870
[5,4,2,1,1]
=> [[5,4,2,1,1],[]]
=> ([(0,7),(0,8),(3,2),(4,6),(4,12),(5,3),(5,11),(6,1),(6,10),(7,4),(7,9),(8,5),(8,9),(9,11),(9,12),(12,10)],13)
=> ? = 21450
[5,3,3,2]
=> [[5,3,3,2],[]]
=> ([(0,6),(0,7),(2,9),(3,1),(4,3),(4,10),(5,2),(5,11),(6,4),(6,8),(7,5),(7,8),(8,10),(8,11),(10,12),(11,9),(11,12)],13)
=> ? = 11583
[5,3,3,1,1]
=> [[5,3,3,1,1],[]]
=> ([(0,7),(0,8),(3,2),(4,1),(5,3),(5,10),(6,4),(6,11),(7,5),(7,9),(8,6),(8,9),(9,10),(9,11),(10,12),(11,12)],13)
=> ? = 16016
[5,3,2,2,1]
=> [[5,3,2,2,1],[]]
=> ([(0,7),(0,8),(3,2),(4,6),(4,12),(5,3),(5,11),(6,1),(6,10),(7,4),(7,9),(8,5),(8,9),(9,11),(9,12),(12,10)],13)
=> ? = 21450
[4,4,3,2]
=> [[4,4,3,2],[]]
=> ([(0,5),(0,6),(1,9),(2,8),(3,2),(3,10),(4,1),(4,11),(5,3),(5,7),(6,4),(6,7),(7,10),(7,11),(10,8),(10,12),(11,9),(11,12)],13)
=> ? = 8580
[4,4,3,1,1]
=> [[4,4,3,1,1],[]]
=> ([(0,6),(0,7),(2,9),(3,1),(4,3),(4,10),(5,2),(5,11),(6,4),(6,8),(7,5),(7,8),(8,10),(8,11),(10,12),(11,9),(11,12)],13)
=> ? = 11583
[4,4,2,2,1]
=> [[4,4,2,2,1],[]]
=> ([(0,6),(0,7),(2,10),(3,5),(3,11),(4,2),(4,12),(5,1),(5,9),(6,3),(6,8),(7,4),(7,8),(8,11),(8,12),(11,9),(12,10)],13)
=> ? = 12870
[4,3,3,3]
=> [[4,3,3,3],[]]
=> ([(0,5),(0,6),(2,8),(3,2),(3,10),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ? = 3432
[4,3,3,2,1]
=> [[4,3,3,2,1],[]]
=> ([(0,6),(0,7),(3,4),(3,11),(4,1),(4,9),(5,2),(5,10),(6,5),(6,8),(7,3),(7,8),(8,10),(8,11),(10,12),(11,9),(11,12)],13)
=> ? = 15015
[6,4,4]
=> [[6,4,4],[]]
=> ?
=> ? = 9009
[5,5,2,2]
=> [[5,5,2,2],[]]
=> ?
=> ? = 21450
[5,4,3,2]
=> [[5,4,3,2],[]]
=> ([(0,6),(0,7),(2,10),(3,5),(3,11),(4,2),(4,12),(5,1),(5,9),(6,3),(6,8),(7,4),(7,8),(8,11),(8,12),(11,9),(11,13),(12,10),(12,13)],14)
=> ? = 48048
[5,4,3,1,1]
=> [[5,4,3,1,1],[]]
=> ([(0,7),(0,8),(3,2),(4,6),(4,12),(5,3),(5,11),(6,1),(6,10),(7,4),(7,9),(8,5),(8,9),(9,11),(9,12),(11,13),(12,10),(12,13)],14)
=> ? = 64064
[5,4,2,2,1]
=> [[5,4,2,2,1],[]]
=> ([(0,7),(0,8),(3,5),(3,12),(4,6),(4,13),(5,2),(5,10),(6,1),(6,11),(7,3),(7,9),(8,4),(8,9),(9,12),(9,13),(12,10),(13,11)],14)
=> ? = 68640
Description
The number of linear extensions of a poset.
Matching statistic: St001595
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001595: Skew partitions ⟶ ℤResult quality: 18% ●values known / values provided: 25%●distinct values known / distinct values provided: 18%
St001595: Skew partitions ⟶ ℤResult quality: 18% ●values known / values provided: 25%●distinct values known / distinct values provided: 18%
Values
[1]
=> [[1],[]]
=> 1
[2]
=> [[2],[]]
=> 1
[1,1]
=> [[1,1],[]]
=> 1
[3]
=> [[3],[]]
=> 1
[2,1]
=> [[2,1],[]]
=> 2
[1,1,1]
=> [[1,1,1],[]]
=> 1
[4]
=> [[4],[]]
=> 1
[3,1]
=> [[3,1],[]]
=> 3
[2,2]
=> [[2,2],[]]
=> 2
[2,1,1]
=> [[2,1,1],[]]
=> 3
[1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[5]
=> [[5],[]]
=> 1
[4,1]
=> [[4,1],[]]
=> 4
[3,2]
=> [[3,2],[]]
=> 5
[3,1,1]
=> [[3,1,1],[]]
=> 6
[2,2,1]
=> [[2,2,1],[]]
=> 5
[2,1,1,1]
=> [[2,1,1,1],[]]
=> 4
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1
[6]
=> [[6],[]]
=> 1
[5,1]
=> [[5,1],[]]
=> 5
[4,2]
=> [[4,2],[]]
=> 9
[4,1,1]
=> [[4,1,1],[]]
=> 10
[3,3]
=> [[3,3],[]]
=> 5
[3,2,1]
=> [[3,2,1],[]]
=> 16
[3,1,1,1]
=> [[3,1,1,1],[]]
=> 10
[2,2,2]
=> [[2,2,2],[]]
=> 5
[2,2,1,1]
=> [[2,2,1,1],[]]
=> 9
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> 5
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> 1
[6,1]
=> [[6,1],[]]
=> 6
[5,2]
=> [[5,2],[]]
=> 14
[5,1,1]
=> [[5,1,1],[]]
=> 15
[4,3]
=> [[4,3],[]]
=> 14
[4,2,1]
=> [[4,2,1],[]]
=> 35
[4,1,1,1]
=> [[4,1,1,1],[]]
=> 20
[3,3,1]
=> [[3,3,1],[]]
=> 21
[3,2,2]
=> [[3,2,2],[]]
=> 21
[3,2,1,1]
=> [[3,2,1,1],[]]
=> 35
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> 15
[2,2,2,1]
=> [[2,2,2,1],[]]
=> 14
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> 14
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> 6
[6,2]
=> [[6,2],[]]
=> ? = 20
[6,1,1]
=> [[6,1,1],[]]
=> ? = 21
[5,3]
=> [[5,3],[]]
=> ? = 28
[5,2,1]
=> [[5,2,1],[]]
=> ? = 64
[5,1,1,1]
=> [[5,1,1,1],[]]
=> ? = 35
[4,4]
=> [[4,4],[]]
=> ? = 14
[4,3,1]
=> [[4,3,1],[]]
=> ? = 70
[4,2,2]
=> [[4,2,2],[]]
=> ? = 56
[4,2,1,1]
=> [[4,2,1,1],[]]
=> ? = 90
[4,1,1,1,1]
=> [[4,1,1,1,1],[]]
=> ? = 35
[3,3,2]
=> [[3,3,2],[]]
=> ? = 42
[3,3,1,1]
=> [[3,3,1,1],[]]
=> ? = 56
[3,2,2,1]
=> [[3,2,2,1],[]]
=> ? = 70
[3,2,1,1,1]
=> [[3,2,1,1,1],[]]
=> ? = 64
[3,1,1,1,1,1]
=> [[3,1,1,1,1,1],[]]
=> ? = 21
[2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 14
[2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ? = 28
[2,2,1,1,1,1]
=> [[2,2,1,1,1,1],[]]
=> ? = 20
[6,3]
=> [[6,3],[]]
=> ? = 48
[6,2,1]
=> [[6,2,1],[]]
=> ? = 105
[6,1,1,1]
=> [[6,1,1,1],[]]
=> ? = 56
[5,4]
=> [[5,4],[]]
=> ? = 42
[5,3,1]
=> [[5,3,1],[]]
=> ? = 162
[5,2,2]
=> [[5,2,2],[]]
=> ? = 120
[5,2,1,1]
=> [[5,2,1,1],[]]
=> ? = 189
[5,1,1,1,1]
=> [[5,1,1,1,1],[]]
=> ? = 70
[4,4,1]
=> [[4,4,1],[]]
=> ? = 84
[4,3,2]
=> [[4,3,2],[]]
=> ? = 168
[4,3,1,1]
=> [[4,3,1,1],[]]
=> ? = 216
[4,2,2,1]
=> [[4,2,2,1],[]]
=> ? = 216
[4,2,1,1,1]
=> [[4,2,1,1,1],[]]
=> ? = 189
[4,1,1,1,1,1]
=> [[4,1,1,1,1,1],[]]
=> ? = 56
[3,3,3]
=> [[3,3,3],[]]
=> ? = 42
[3,3,2,1]
=> [[3,3,2,1],[]]
=> ? = 168
[3,3,1,1,1]
=> [[3,3,1,1,1],[]]
=> ? = 120
[3,2,2,2]
=> [[3,2,2,2],[]]
=> ? = 84
[3,2,2,1,1]
=> [[3,2,2,1,1],[]]
=> ? = 162
[3,2,1,1,1,1]
=> [[3,2,1,1,1,1],[]]
=> ? = 105
[2,2,2,2,1]
=> [[2,2,2,2,1],[]]
=> ? = 42
[2,2,2,1,1,1]
=> [[2,2,2,1,1,1],[]]
=> ? = 48
[6,4]
=> [[6,4],[]]
=> ? = 90
[6,3,1]
=> [[6,3,1],[]]
=> ? = 315
[6,2,2]
=> [[6,2,2],[]]
=> ? = 225
[6,2,1,1]
=> [[6,2,1,1],[]]
=> ? = 350
[6,1,1,1,1]
=> [[6,1,1,1,1],[]]
=> ? = 126
[5,5]
=> [[5,5],[]]
=> ? = 42
[5,4,1]
=> [[5,4,1],[]]
=> ? = 288
[5,3,2]
=> [[5,3,2],[]]
=> ? = 450
[5,3,1,1]
=> [[5,3,1,1],[]]
=> ? = 567
[5,2,2,1]
=> [[5,2,2,1],[]]
=> ? = 525
Description
The number of standard Young tableaux of the skew 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!