Processing math: 54%

Your data matches 98 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001064
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001064: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => 1
[2]
 => [1,1,0,0,1,0]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 2
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 2
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
Description
Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules.
Matching statistic: St001964
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St001964: Posets ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 20%
Values
[1]
 => 1 => ([(0,1)],2)
 => 0 = 1 - 1
[2]
 => 0 => ([(0,1)],2)
 => 0 = 1 - 1
[1,1]
 => 11 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[3]
 => 1 => ([(0,1)],2)
 => 0 = 1 - 1
[2,1]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,1,1]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[4]
 => 0 => ([(0,1)],2)
 => 0 = 1 - 1
[3,1]
 => 11 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[2,2]
 => 00 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[2,1,1]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1 - 1
[1,1,1,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[5]
 => 1 => ([(0,1)],2)
 => 0 = 1 - 1
[4,1]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[3,2]
 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[3,1,1]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[2,2,1]
 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1 - 1
[2,1,1,1]
 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[1,1,1,1,1]
 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[6]
 => 0 => ([(0,1)],2)
 => 0 = 1 - 1
[5,1]
 => 11 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[4,2]
 => 00 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[4,1,1]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1 - 1
[3,3]
 => 11 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[3,2,1]
 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 2 - 1
[3,1,1,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[2,2,2]
 => 000 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[2,2,1,1]
 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 1 - 1
[2,1,1,1,1]
 => 01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 1 - 1
[1,1,1,1,1,1]
 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ? = 1 - 1
[6,1]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[5,2]
 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[5,1,1]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[4,3]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[4,2,1]
 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 2 - 1
[4,1,1,1]
 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[3,3,1]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[3,2,2]
 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1 - 1
[3,2,1,1]
 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 2 - 1
[3,1,1,1,1]
 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[2,2,2,1]
 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[2,2,1,1,1]
 => 00111 => ([(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)
 => ? = 1 - 1
[2,1,1,1,1,1]
 => 011111 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
 => ? = 1 - 1
[6,2]
 => 00 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[6,1,1]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1 - 1
[5,3]
 => 11 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[5,2,1]
 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 2 - 1
[5,1,1,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[4,4]
 => 00 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[4,3,1]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1 - 1
[4,2,2]
 => 000 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[4,2,1,1]
 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 - 1
[4,1,1,1,1]
 => 01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 1 - 1
[3,3,2]
 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1 - 1
[3,3,1,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[3,2,2,1]
 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 2 - 1
[3,2,1,1,1]
 => 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 2 - 1
[3,1,1,1,1,1]
 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ? = 1 - 1
[2,2,2,2]
 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[2,2,2,1,1]
 => 00011 => ([(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)
 => ? = 1 - 1
[2,2,1,1,1,1]
 => 001111 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
 => ? = 1 - 1
[6,3]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[6,2,1]
 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 2 - 1
[6,1,1,1]
 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[5,4]
 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[5,3,1]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[5,2,2]
 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1 - 1
[5,2,1,1]
 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 2 - 1
[5,1,1,1,1]
 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[4,4,1]
 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1 - 1
[4,3,2]
 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 2 - 1
[4,3,1,1]
 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[4,2,2,1]
 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[4,2,1,1,1]
 => 00111 => ([(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)
 => ? = 2 - 1
[4,1,1,1,1,1]
 => 011111 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
 => ? = 1 - 1
[3,3,3]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[3,3,2,1]
 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 2 - 1
[3,3,1,1,1]
 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[3,2,2,2]
 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[3,2,2,1,1]
 => 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 2 - 1
[3,2,1,1,1,1]
 => 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
 => ? = 2 - 1
[2,2,2,2,1]
 => 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 1 - 1
[2,2,2,1,1,1]
 => 000111 => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
 => ? = 1 - 1
[6,4]
 => 00 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[6,3,1]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1 - 1
[6,2,2]
 => 000 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[6,2,1,1]
 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 - 1
[6,1,1,1,1]
 => 01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 1 - 1
[5,5]
 => 11 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[5,4,1]
 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 1 - 1
[5,3,2]
 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 2 - 1
[5,3,1,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[5,2,2,1]
 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 1 - 1
[5,2,1,1,1]
 => 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 2 - 1
[5,1,1,1,1,1]
 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ? = 1 - 1
[4,4,2]
 => 000 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[4,2,2,2]
 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[3,3,3,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[2,2,2,2,2]
 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[6,5]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[5,5,1]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
Description
The interval resolution global dimension of a poset. This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Matching statistic: St000181
(load all 6 compositions to match this statistic)
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000181: Posets ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 20%
Values
[1]
 => 1 => ([(0,1)],2)
 => 1
[2]
 => 0 => ([(0,1)],2)
 => 1
[1,1]
 => 11 => ([(0,2),(2,1)],3)
 => 1
[3]
 => 1 => ([(0,1)],2)
 => 1
[2,1]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,1]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1
[4]
 => 0 => ([(0,1)],2)
 => 1
[3,1]
 => 11 => ([(0,2),(2,1)],3)
 => 1
[2,2]
 => 00 => ([(0,2),(2,1)],3)
 => 1
[2,1,1]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1
[1,1,1,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[5]
 => 1 => ([(0,1)],2)
 => 1
[4,1]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[3,2]
 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[3,1,1]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,2,1]
 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1
[2,1,1,1]
 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[1,1,1,1,1]
 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ? = 1
[6]
 => 0 => ([(0,1)],2)
 => 1
[5,1]
 => 11 => ([(0,2),(2,1)],3)
 => 1
[4,2]
 => 00 => ([(0,2),(2,1)],3)
 => 1
[4,1,1]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1
[3,3]
 => 11 => ([(0,2),(2,1)],3)
 => 1
[3,2,1]
 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 2
[3,1,1,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,2,2]
 => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,2,1,1]
 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 1
[2,1,1,1,1]
 => 01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 1
[1,1,1,1,1,1]
 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ? = 1
[6,1]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[5,2]
 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[5,1,1]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1
[4,3]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[4,2,1]
 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 2
[4,1,1,1]
 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[3,3,1]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,2,2]
 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1
[3,2,1,1]
 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 2
[3,1,1,1,1]
 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ? = 1
[2,2,2,1]
 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[2,2,1,1,1]
 => 00111 => ([(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)
 => ? = 1
[2,1,1,1,1,1]
 => 011111 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
 => ? = 1
[6,2]
 => 00 => ([(0,2),(2,1)],3)
 => 1
[6,1,1]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1
[5,3]
 => 11 => ([(0,2),(2,1)],3)
 => 1
[5,2,1]
 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 2
[5,1,1,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,4]
 => 00 => ([(0,2),(2,1)],3)
 => 1
[4,3,1]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1
[4,2,2]
 => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1
[4,2,1,1]
 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2
[4,1,1,1,1]
 => 01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 1
[3,3,2]
 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1
[3,3,1,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[3,2,2,1]
 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 2
[3,2,1,1,1]
 => 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 2
[3,1,1,1,1,1]
 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ? = 1
[2,2,2,2]
 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,2,2,1,1]
 => 00011 => ([(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)
 => ? = 1
[2,2,1,1,1,1]
 => 001111 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
 => ? = 1
[6,3]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[6,2,1]
 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 2
[6,1,1,1]
 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[5,4]
 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[5,3,1]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1
[5,2,2]
 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1
[5,2,1,1]
 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 2
[5,1,1,1,1]
 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ? = 1
[4,4,1]
 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1
[4,3,2]
 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 2
[4,3,1,1]
 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[4,2,2,1]
 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[4,2,1,1,1]
 => 00111 => ([(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)
 => ? = 2
[4,1,1,1,1,1]
 => 011111 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
 => ? = 1
[3,3,3]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,3,2,1]
 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 2
[3,3,1,1,1]
 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ? = 1
[3,2,2,2]
 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[3,2,2,1,1]
 => 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 2
[3,2,1,1,1,1]
 => 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
 => ? = 2
[2,2,2,2,1]
 => 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 1
[2,2,2,1,1,1]
 => 000111 => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
 => ? = 1
[6,4]
 => 00 => ([(0,2),(2,1)],3)
 => 1
[6,3,1]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1
[6,2,2]
 => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1
[6,2,1,1]
 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2
[6,1,1,1,1]
 => 01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 1
[5,5]
 => 11 => ([(0,2),(2,1)],3)
 => 1
[5,4,1]
 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 1
[5,3,1,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,4,2]
 => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1
[4,2,2,2]
 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[3,3,3,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[6,5]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[5,5,1]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1
[5,3,3]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1
[6,4,2]
 => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1
[6,2,2,2]
 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[5,5,1,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[5,3,3,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
Description
The number of connected components of the Hasse diagram for the poset.
Matching statistic: St001890
(load all 4 compositions to match this statistic)
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St001890: Posets ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 20%
Values
[1]
 => 1 => ([(0,1)],2)
 => 1
[2]
 => 0 => ([(0,1)],2)
 => 1
[1,1]
 => 11 => ([(0,2),(2,1)],3)
 => 1
[3]
 => 1 => ([(0,1)],2)
 => 1
[2,1]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,1]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1
[4]
 => 0 => ([(0,1)],2)
 => 1
[3,1]
 => 11 => ([(0,2),(2,1)],3)
 => 1
[2,2]
 => 00 => ([(0,2),(2,1)],3)
 => 1
[2,1,1]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1
[1,1,1,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[5]
 => 1 => ([(0,1)],2)
 => 1
[4,1]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[3,2]
 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[3,1,1]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,2,1]
 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1
[2,1,1,1]
 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[1,1,1,1,1]
 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ? = 1
[6]
 => 0 => ([(0,1)],2)
 => 1
[5,1]
 => 11 => ([(0,2),(2,1)],3)
 => 1
[4,2]
 => 00 => ([(0,2),(2,1)],3)
 => 1
[4,1,1]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1
[3,3]
 => 11 => ([(0,2),(2,1)],3)
 => 1
[3,2,1]
 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 2
[3,1,1,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,2,2]
 => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,2,1,1]
 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 1
[2,1,1,1,1]
 => 01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 1
[1,1,1,1,1,1]
 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ? = 1
[6,1]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[5,2]
 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[5,1,1]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1
[4,3]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[4,2,1]
 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 2
[4,1,1,1]
 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[3,3,1]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,2,2]
 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1
[3,2,1,1]
 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 2
[3,1,1,1,1]
 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ? = 1
[2,2,2,1]
 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[2,2,1,1,1]
 => 00111 => ([(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)
 => ? = 1
[2,1,1,1,1,1]
 => 011111 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
 => ? = 1
[6,2]
 => 00 => ([(0,2),(2,1)],3)
 => 1
[6,1,1]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1
[5,3]
 => 11 => ([(0,2),(2,1)],3)
 => 1
[5,2,1]
 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 2
[5,1,1,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,4]
 => 00 => ([(0,2),(2,1)],3)
 => 1
[4,3,1]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1
[4,2,2]
 => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1
[4,2,1,1]
 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2
[4,1,1,1,1]
 => 01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 1
[3,3,2]
 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1
[3,3,1,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[3,2,2,1]
 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 2
[3,2,1,1,1]
 => 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 2
[3,1,1,1,1,1]
 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ? = 1
[2,2,2,2]
 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,2,2,1,1]
 => 00011 => ([(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)
 => ? = 1
[2,2,1,1,1,1]
 => 001111 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
 => ? = 1
[6,3]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[6,2,1]
 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 2
[6,1,1,1]
 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[5,4]
 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[5,3,1]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1
[5,2,2]
 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1
[5,2,1,1]
 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 2
[5,1,1,1,1]
 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ? = 1
[4,4,1]
 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1
[4,3,2]
 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 2
[4,3,1,1]
 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[4,2,2,1]
 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[4,2,1,1,1]
 => 00111 => ([(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)
 => ? = 2
[4,1,1,1,1,1]
 => 011111 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
 => ? = 1
[3,3,3]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,3,2,1]
 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 2
[3,3,1,1,1]
 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ? = 1
[3,2,2,2]
 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[3,2,2,1,1]
 => 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 2
[3,2,1,1,1,1]
 => 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
 => ? = 2
[2,2,2,2,1]
 => 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 1
[2,2,2,1,1,1]
 => 000111 => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
 => ? = 1
[6,4]
 => 00 => ([(0,2),(2,1)],3)
 => 1
[6,3,1]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1
[6,2,2]
 => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1
[6,2,1,1]
 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2
[6,1,1,1,1]
 => 01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 1
[5,5]
 => 11 => ([(0,2),(2,1)],3)
 => 1
[5,4,1]
 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 1
[5,3,1,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,4,2]
 => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1
[4,2,2,2]
 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[3,3,3,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[6,5]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[5,5,1]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1
[5,3,3]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1
[6,4,2]
 => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1
[6,2,2,2]
 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[5,5,1,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[5,3,3,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
Description
The maximum magnitude of the Möbius function of a poset. The '''Möbius function''' of a poset is the multiplicative inverse of the zeta function in the incidence algebra. The Möbius value μ(x,y) is equal to the signed sum of chains from x to y, where odd-length chains are counted with a minus sign, so this statistic is bounded above by the total number of chains in the poset.
Matching statistic: St000408
(load all 2 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000408: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 40%
Values
[1]
 => [1,0]
 => [1,1,0,0]
 => [1,2] => 0 = 1 - 1
[2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,1,3] => 0 = 1 - 1
[1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => 0 = 1 - 1
[3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [3,2,1,4] => 0 = 1 - 1
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,3,1,4] => 0 = 1 - 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => 0 = 1 - 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => 0 = 1 - 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => 0 = 1 - 1
[2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0 = 1 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => 0 = 1 - 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => 0 = 1 - 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,1,6] => 0 = 1 - 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [4,5,3,2,1,6] => 0 = 1 - 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => 0 = 1 - 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [4,3,5,2,1,6] => 0 = 1 - 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => 0 = 1 - 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,1,6] => 0 = 1 - 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [4,3,2,1,5,6] => 0 = 1 - 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,2,1,7] => 0 = 1 - 1
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,4,3,2,1,7] => 0 = 1 - 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [3,4,5,2,1,6] => 0 = 1 - 1
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [5,4,6,3,2,1,7] => 0 = 1 - 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => 0 = 1 - 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,1,6] => 1 = 2 - 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [5,4,3,6,2,1,7] => 0 = 1 - 1
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0 = 1 - 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [4,3,1,2,5,6] => 0 = 1 - 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [5,4,3,2,6,1,7] => 0 = 1 - 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [5,4,3,2,1,6,7] => 0 = 1 - 1
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [6,7,5,4,3,2,1,8] => 0 = 1 - 1
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [4,5,6,3,2,1,7] => 0 = 1 - 1
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [6,5,7,4,3,2,1,8] => 0 = 1 - 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,1,6] => 0 = 1 - 1
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [5,3,4,6,2,1,7] => ? = 2 - 1
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [6,5,4,7,3,2,1,8] => 0 = 1 - 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [4,2,1,3,5,6] => 0 = 1 - 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,1,6] => 0 = 1 - 1
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [5,4,2,3,6,1,7] => ? = 2 - 1
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [6,5,4,3,7,2,1,8] => ? = 1 - 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [4,1,2,3,5,6] => 0 = 1 - 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [5,4,3,1,2,6,7] => ? = 1 - 1
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [6,5,4,3,2,7,1,8] => ? = 1 - 1
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [5,6,7,4,3,2,1,8] => 0 = 1 - 1
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [7,6,8,5,4,3,2,1,9] => 0 = 1 - 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,2,1,7] => ? = 1 - 1
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [6,4,5,7,3,2,1,8] => ? = 2 - 1
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [7,6,5,8,4,3,2,1,9] => ? = 1 - 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [3,2,1,4,5,6] => 0 = 1 - 1
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [5,3,2,4,6,1,7] => ? = 1 - 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,2,1,7] => ? = 1 - 1
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [6,5,3,4,7,2,1,8] => ? = 2 - 1
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [7,6,5,4,8,3,2,1,9] => ? = 1 - 1
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [2,3,1,4,5,6] => 0 = 1 - 1
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [5,4,2,1,3,6,7] => ? = 1 - 1
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [5,2,3,4,6,1,7] => ? = 2 - 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [6,5,4,2,3,7,1,8] => ? = 2 - 1
[3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [7,6,5,4,3,8,2,1,9] => ? = 1 - 1
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => 0 = 1 - 1
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [5,4,1,2,3,6,7] => ? = 1 - 1
[2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
 => [6,5,4,3,1,2,7,8] => ? = 1 - 1
[6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [5,4,6,7,3,2,1,8] => ? = 1 - 1
[6,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [7,5,6,8,4,3,2,1,9] => ? = 2 - 1
[6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [8,7,6,9,5,4,3,2,1,10] => ? = 1 - 1
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [4,3,2,5,6,1,7] => ? = 1 - 1
[5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [6,4,3,5,7,2,1,8] => ? = 1 - 1
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,3,2,1,8] => ? = 1 - 1
[5,2,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [7,6,4,5,8,3,2,1,9] => ? = 2 - 1
[5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [8,7,6,5,9,4,3,2,1,10] => ? = 1 - 1
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [5,3,2,1,4,6,7] => ? = 1 - 1
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [3,4,2,5,6,1,7] => ? = 2 - 1
[4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [6,5,3,2,4,7,1,8] => ? = 1 - 1
[4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [6,3,4,5,7,2,1,8] => ? = 1 - 1
[4,2,1,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [7,6,5,3,4,8,2,1,9] => ? = 2 - 1
[4,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [8,7,6,5,4,9,3,2,1,10] => ? = 1 - 1
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => 0 = 1 - 1
[3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [5,2,3,1,4,6,7] => ? = 2 - 1
[3,3,1,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
 => [6,5,4,2,1,3,7,8] => ? = 1 - 1
[3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,1,7] => ? = 1 - 1
[3,2,2,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [6,5,2,3,4,7,1,8] => ? = 2 - 1
[3,2,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
 => [7,6,5,4,2,3,8,1,9] => ? = 2 - 1
[2,2,2,2,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [5,2,1,3,4,6,7] => ? = 1 - 1
[2,2,2,1,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3,7,8] => ? = 1 - 1
[6,4]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [5,4,3,6,7,2,1,8] => ? = 1 - 1
[6,3,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [7,5,4,6,8,3,2,1,9] => ? = 1 - 1
[6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [5,6,7,8,4,3,2,1,9] => ? = 1 - 1
[6,2,1,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [8,7,5,6,9,4,3,2,1,10] => ? = 2 - 1
[6,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [9,8,7,6,10,5,4,3,2,1,11] => ? = 1 - 1
[5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [4,3,2,1,5,6,7] => 0 = 1 - 1
[5,4,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [6,4,3,2,5,7,1,8] => ? = 1 - 1
[5,3,2]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [4,5,3,6,7,2,1,8] => ? = 2 - 1
[5,3,1,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [7,6,4,3,5,8,2,1,9] => ? = 1 - 1
[5,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [7,4,5,6,8,3,2,1,9] => ? = 1 - 1
[5,2,1,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [8,7,6,4,5,9,3,2,1,10] => ? = 2 - 1
[5,1,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [9,8,7,6,5,10,4,3,2,1,11] => ? = 1 - 1
[4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,1,7] => 0 = 1 - 1
[3,3,3,1]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [5,1,2,3,4,6,7] => 0 = 1 - 1
[2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1,4,5,6,7] => 0 = 1 - 1
[4,4,3]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [2,3,4,1,5,6,7] => 0 = 1 - 1
[3,3,3,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [3,1,2,4,5,6,7] => 0 = 1 - 1
[4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => 0 = 1 - 1
Description
The number of occurrences of the pattern 4231 in a permutation. It is a necessary condition that a permutation π avoids this pattern for the Schubert variety associated to π to be smooth [2].
Matching statistic: St000440
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000440: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 40%
Values
[1]
 => [1,0]
 => [1,1,0,0]
 => [1,2] => 0 = 1 - 1
[2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,1,3] => 0 = 1 - 1
[1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => 0 = 1 - 1
[3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [3,2,1,4] => 0 = 1 - 1
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,3,1,4] => 0 = 1 - 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => 0 = 1 - 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => 0 = 1 - 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => 0 = 1 - 1
[2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0 = 1 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => 0 = 1 - 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => 0 = 1 - 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,1,6] => 0 = 1 - 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [4,5,3,2,1,6] => 0 = 1 - 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => 0 = 1 - 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [4,3,5,2,1,6] => 0 = 1 - 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => 0 = 1 - 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,1,6] => 0 = 1 - 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [4,3,2,1,5,6] => 0 = 1 - 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,2,1,7] => 0 = 1 - 1
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,4,3,2,1,7] => 0 = 1 - 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [3,4,5,2,1,6] => 0 = 1 - 1
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [5,4,6,3,2,1,7] => 0 = 1 - 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => 0 = 1 - 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,1,6] => 1 = 2 - 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [5,4,3,6,2,1,7] => 0 = 1 - 1
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0 = 1 - 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [4,3,1,2,5,6] => 0 = 1 - 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [5,4,3,2,6,1,7] => 0 = 1 - 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [5,4,3,2,1,6,7] => 0 = 1 - 1
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [6,7,5,4,3,2,1,8] => 0 = 1 - 1
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [4,5,6,3,2,1,7] => 0 = 1 - 1
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [6,5,7,4,3,2,1,8] => 0 = 1 - 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,1,6] => 0 = 1 - 1
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [5,3,4,6,2,1,7] => ? = 2 - 1
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [6,5,4,7,3,2,1,8] => 0 = 1 - 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [4,2,1,3,5,6] => 0 = 1 - 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,1,6] => 0 = 1 - 1
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [5,4,2,3,6,1,7] => ? = 2 - 1
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [6,5,4,3,7,2,1,8] => ? = 1 - 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [4,1,2,3,5,6] => 0 = 1 - 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [5,4,3,1,2,6,7] => ? = 1 - 1
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [6,5,4,3,2,7,1,8] => ? = 1 - 1
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [5,6,7,4,3,2,1,8] => 0 = 1 - 1
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [7,6,8,5,4,3,2,1,9] => 0 = 1 - 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,2,1,7] => ? = 1 - 1
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [6,4,5,7,3,2,1,8] => ? = 2 - 1
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [7,6,5,8,4,3,2,1,9] => ? = 1 - 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [3,2,1,4,5,6] => 0 = 1 - 1
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [5,3,2,4,6,1,7] => ? = 1 - 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,2,1,7] => ? = 1 - 1
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [6,5,3,4,7,2,1,8] => ? = 2 - 1
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [7,6,5,4,8,3,2,1,9] => ? = 1 - 1
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [2,3,1,4,5,6] => 0 = 1 - 1
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [5,4,2,1,3,6,7] => ? = 1 - 1
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [5,2,3,4,6,1,7] => ? = 2 - 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [6,5,4,2,3,7,1,8] => ? = 2 - 1
[3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [7,6,5,4,3,8,2,1,9] => ? = 1 - 1
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => 0 = 1 - 1
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [5,4,1,2,3,6,7] => ? = 1 - 1
[2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
 => [6,5,4,3,1,2,7,8] => ? = 1 - 1
[6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [5,4,6,7,3,2,1,8] => ? = 1 - 1
[6,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [7,5,6,8,4,3,2,1,9] => ? = 2 - 1
[6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [8,7,6,9,5,4,3,2,1,10] => ? = 1 - 1
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [4,3,2,5,6,1,7] => ? = 1 - 1
[5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [6,4,3,5,7,2,1,8] => ? = 1 - 1
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,3,2,1,8] => ? = 1 - 1
[5,2,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [7,6,4,5,8,3,2,1,9] => ? = 2 - 1
[5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [8,7,6,5,9,4,3,2,1,10] => ? = 1 - 1
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [5,3,2,1,4,6,7] => ? = 1 - 1
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [3,4,2,5,6,1,7] => ? = 2 - 1
[4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [6,5,3,2,4,7,1,8] => ? = 1 - 1
[4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [6,3,4,5,7,2,1,8] => ? = 1 - 1
[4,2,1,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [7,6,5,3,4,8,2,1,9] => ? = 2 - 1
[4,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [8,7,6,5,4,9,3,2,1,10] => ? = 1 - 1
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => 0 = 1 - 1
[3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [5,2,3,1,4,6,7] => ? = 2 - 1
[3,3,1,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
 => [6,5,4,2,1,3,7,8] => ? = 1 - 1
[3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,1,7] => ? = 1 - 1
[3,2,2,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [6,5,2,3,4,7,1,8] => ? = 2 - 1
[3,2,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
 => [7,6,5,4,2,3,8,1,9] => ? = 2 - 1
[2,2,2,2,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [5,2,1,3,4,6,7] => ? = 1 - 1
[2,2,2,1,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3,7,8] => ? = 1 - 1
[6,4]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [5,4,3,6,7,2,1,8] => ? = 1 - 1
[6,3,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [7,5,4,6,8,3,2,1,9] => ? = 1 - 1
[6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [5,6,7,8,4,3,2,1,9] => ? = 1 - 1
[6,2,1,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [8,7,5,6,9,4,3,2,1,10] => ? = 2 - 1
[6,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [9,8,7,6,10,5,4,3,2,1,11] => ? = 1 - 1
[5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [4,3,2,1,5,6,7] => 0 = 1 - 1
[5,4,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [6,4,3,2,5,7,1,8] => ? = 1 - 1
[5,3,2]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [4,5,3,6,7,2,1,8] => ? = 2 - 1
[5,3,1,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [7,6,4,3,5,8,2,1,9] => ? = 1 - 1
[5,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [7,4,5,6,8,3,2,1,9] => ? = 1 - 1
[5,2,1,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [8,7,6,4,5,9,3,2,1,10] => ? = 2 - 1
[5,1,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [9,8,7,6,5,10,4,3,2,1,11] => ? = 1 - 1
[4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,1,7] => 0 = 1 - 1
[3,3,3,1]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [5,1,2,3,4,6,7] => 0 = 1 - 1
[2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1,4,5,6,7] => 0 = 1 - 1
[4,4,3]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [2,3,4,1,5,6,7] => 0 = 1 - 1
[3,3,3,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [3,1,2,4,5,6,7] => 0 = 1 - 1
[4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => 0 = 1 - 1
Description
The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. There is a bijection between permutations avoiding these two pattern and Schröder paths [1,2].
Matching statistic: St000036
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000036: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 40%
Values
[1]
 => [1,0]
 => [1,1,0,0]
 => [1,2] => 1
[2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,1,3] => 1
[1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => 1
[3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [3,2,1,4] => 1
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,3,1,4] => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => 1
[2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,1,6] => 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [4,5,3,2,1,6] => 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [4,3,5,2,1,6] => 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,1,6] => 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [4,3,2,1,5,6] => 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,2,1,7] => 1
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,4,3,2,1,7] => 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [3,4,5,2,1,6] => 1
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [5,4,6,3,2,1,7] => 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,1,6] => 2
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [5,4,3,6,2,1,7] => 1
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [4,3,1,2,5,6] => 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [5,4,3,2,6,1,7] => 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [5,4,3,2,1,6,7] => 1
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [6,7,5,4,3,2,1,8] => 1
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [4,5,6,3,2,1,7] => 1
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [6,5,7,4,3,2,1,8] => 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,1,6] => 1
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [5,3,4,6,2,1,7] => ? = 2
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [6,5,4,7,3,2,1,8] => 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [4,2,1,3,5,6] => 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,1,6] => 1
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [5,4,2,3,6,1,7] => ? = 2
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [6,5,4,3,7,2,1,8] => ? = 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [4,1,2,3,5,6] => 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [5,4,3,1,2,6,7] => ? = 1
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [6,5,4,3,2,7,1,8] => ? = 1
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [5,6,7,4,3,2,1,8] => 1
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [7,6,8,5,4,3,2,1,9] => ? = 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,2,1,7] => ? = 1
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [6,4,5,7,3,2,1,8] => ? = 2
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [7,6,5,8,4,3,2,1,9] => ? = 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [3,2,1,4,5,6] => 1
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [5,3,2,4,6,1,7] => ? = 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,2,1,7] => ? = 1
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [6,5,3,4,7,2,1,8] => ? = 2
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [7,6,5,4,8,3,2,1,9] => ? = 1
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [2,3,1,4,5,6] => 1
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [5,4,2,1,3,6,7] => ? = 1
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [5,2,3,4,6,1,7] => ? = 2
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [6,5,4,2,3,7,1,8] => ? = 2
[3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [7,6,5,4,3,8,2,1,9] => ? = 1
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => 1
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [5,4,1,2,3,6,7] => ? = 1
[2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
 => [6,5,4,3,1,2,7,8] => ? = 1
[6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [5,4,6,7,3,2,1,8] => ? = 1
[6,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [7,5,6,8,4,3,2,1,9] => ? = 2
[6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [8,7,6,9,5,4,3,2,1,10] => ? = 1
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [4,3,2,5,6,1,7] => ? = 1
[5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [6,4,3,5,7,2,1,8] => ? = 1
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,3,2,1,8] => ? = 1
[5,2,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [7,6,4,5,8,3,2,1,9] => ? = 2
[5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [8,7,6,5,9,4,3,2,1,10] => ? = 1
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [5,3,2,1,4,6,7] => ? = 1
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [3,4,2,5,6,1,7] => ? = 2
[4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [6,5,3,2,4,7,1,8] => ? = 1
[4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [6,3,4,5,7,2,1,8] => ? = 1
[4,2,1,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [7,6,5,3,4,8,2,1,9] => ? = 2
[4,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [8,7,6,5,4,9,3,2,1,10] => ? = 1
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => 1
[3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [5,2,3,1,4,6,7] => ? = 2
[3,3,1,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
 => [6,5,4,2,1,3,7,8] => ? = 1
[3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,1,7] => ? = 1
[3,2,2,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [6,5,2,3,4,7,1,8] => ? = 2
[3,2,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
 => [7,6,5,4,2,3,8,1,9] => ? = 2
[2,2,2,2,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [5,2,1,3,4,6,7] => ? = 1
[2,2,2,1,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3,7,8] => ? = 1
[6,4]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [5,4,3,6,7,2,1,8] => ? = 1
[6,3,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [7,5,4,6,8,3,2,1,9] => ? = 1
[6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [5,6,7,8,4,3,2,1,9] => ? = 1
[6,2,1,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [8,7,5,6,9,4,3,2,1,10] => ? = 2
[6,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [9,8,7,6,10,5,4,3,2,1,11] => ? = 1
[5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [4,3,2,1,5,6,7] => 1
[5,4,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [6,4,3,2,5,7,1,8] => ? = 1
[5,3,2]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [4,5,3,6,7,2,1,8] => ? = 2
[5,3,1,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [7,6,4,3,5,8,2,1,9] => ? = 1
[5,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [7,4,5,6,8,3,2,1,9] => ? = 1
[5,2,1,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [8,7,6,4,5,9,3,2,1,10] => ? = 2
[4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,1,7] => 1
[3,3,3,1]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [5,1,2,3,4,6,7] => 1
[2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1,4,5,6,7] => 1
[4,4,3]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [2,3,4,1,5,6,7] => 1
[3,3,3,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [3,1,2,4,5,6,7] => 1
[4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => 1
[3,3,3,3]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7] => 1
Description
The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. These are multiplicities of Verma modules.
Matching statistic: St000371
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 40%
Values
[1]
 => [[1]]
 => [1] => [1] => 0 = 1 - 1
[2]
 => [[1,2]]
 => [1,2] => [1,2] => 0 = 1 - 1
[1,1]
 => [[1],[2]]
 => [2,1] => [1,2] => 0 = 1 - 1
[3]
 => [[1,2,3]]
 => [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,1]
 => [[1,3],[2]]
 => [2,1,3] => [1,3,2] => 0 = 1 - 1
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => [1,2,3] => 0 = 1 - 1
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => [1,3,4,2] => 0 = 1 - 1
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => [1,2,3,4] => 0 = 1 - 1
[2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => [1,4,2,3] => 0 = 1 - 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => [1,3,4,5,2] => 0 = 1 - 1
[3,2]
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => [1,2,5,3,4] => 0 = 1 - 1
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [1,4,5,2,3] => 0 = 1 - 1
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [1,3,2,5,4] => 0 = 1 - 1
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [1,5,2,3,4] => 0 = 1 - 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [1,3,4,5,6,2] => 0 = 1 - 1
[4,2]
 => [[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => [1,2,5,6,3,4] => 0 = 1 - 1
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => [1,4,5,6,2,3] => 0 = 1 - 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => [1,2,3,4,5,6] => 0 = 1 - 1
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [4,2,5,1,3,6] => [1,3,6,2,5,4] => 1 = 2 - 1
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => [1,5,6,2,3,4] => 0 = 1 - 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => [1,2,3,4,5,6] => 0 = 1 - 1
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [5,3,2,6,1,4] => [1,4,2,6,3,5] => 0 = 1 - 1
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [5,4,3,2,1,6] => [1,6,2,3,4,5] => 0 = 1 - 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0 = 1 - 1
[6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => [1,3,4,5,6,7,2] => 0 = 1 - 1
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => [1,2,5,6,7,3,4] => 0 = 1 - 1
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => [1,4,5,6,7,2,3] => 0 = 1 - 1
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => [1,2,3,7,4,5,6] => 0 = 1 - 1
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [4,2,5,1,3,6,7] => [1,3,6,7,2,5,4] => 1 = 2 - 1
[4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => [4,3,2,1,5,6,7] => [1,5,6,7,2,3,4] => 0 = 1 - 1
[3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [5,2,6,7,1,3,4] => [1,3,4,2,6,7,5] => 0 = 1 - 1
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [5,6,3,4,1,2,7] => [1,2,7,3,4,5,6] => 0 = 1 - 1
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => [5,3,2,6,1,4,7] => [1,4,7,2,6,3,5] => ? = 2 - 1
[3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => [5,4,3,2,1,6,7] => [1,6,7,2,3,4,5] => 0 = 1 - 1
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3] => [1,3,2,5,4,7,6] => 0 = 1 - 1
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => [6,4,3,2,7,1,5] => [1,5,2,7,3,4,6] => 0 = 1 - 1
[2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1,7] => [1,7,2,3,4,5,6] => 0 = 1 - 1
[6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [3,4,1,2,5,6,7,8] => [1,2,5,6,7,8,3,4] => ? = 1 - 1
[6,1,1]
 => [[1,4,5,6,7,8],[2],[3]]
 => [3,2,1,4,5,6,7,8] => [1,4,5,6,7,8,2,3] => ? = 1 - 1
[5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => [4,5,6,1,2,3,7,8] => [1,2,3,7,8,4,5,6] => ? = 1 - 1
[5,2,1]
 => [[1,3,6,7,8],[2,5],[4]]
 => [4,2,5,1,3,6,7,8] => [1,3,6,7,8,2,5,4] => ? = 2 - 1
[5,1,1,1]
 => [[1,5,6,7,8],[2],[3],[4]]
 => [4,3,2,1,5,6,7,8] => [1,5,6,7,8,2,3,4] => ? = 1 - 1
[4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[4,3,1]
 => [[1,3,4,8],[2,6,7],[5]]
 => [5,2,6,7,1,3,4,8] => [1,3,4,8,2,6,7,5] => ? = 1 - 1
[4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => [5,6,3,4,1,2,7,8] => [1,2,7,8,3,4,5,6] => ? = 1 - 1
[4,2,1,1]
 => [[1,4,7,8],[2,6],[3],[5]]
 => [5,3,2,6,1,4,7,8] => [1,4,7,8,2,6,3,5] => ? = 2 - 1
[4,1,1,1,1]
 => [[1,6,7,8],[2],[3],[4],[5]]
 => [5,4,3,2,1,6,7,8] => [1,6,7,8,2,3,4,5] => 0 = 1 - 1
[3,3,2]
 => [[1,2,5],[3,4,8],[6,7]]
 => [6,7,3,4,8,1,2,5] => [1,2,5,3,4,8,6,7] => ? = 1 - 1
[3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => [6,3,2,7,8,1,4,5] => [1,4,5,2,7,8,3,6] => ? = 1 - 1
[3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3,8] => [1,3,8,2,5,4,7,6] => ? = 2 - 1
[3,2,1,1,1]
 => [[1,5,8],[2,7],[3],[4],[6]]
 => [6,4,3,2,7,1,5,8] => [1,5,8,2,7,3,4,6] => ? = 2 - 1
[3,1,1,1,1,1]
 => [[1,7,8],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1,7,8] => [1,7,8,2,3,4,5,6] => 0 = 1 - 1
[2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2] => [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5],[7]]
 => [7,5,3,8,2,6,1,4] => [1,4,2,6,3,8,5,7] => 0 = 1 - 1
[2,2,1,1,1,1]
 => [[1,6],[2,8],[3],[4],[5],[7]]
 => [7,5,4,3,2,8,1,6] => [1,6,2,8,3,4,5,7] => 0 = 1 - 1
[6,3]
 => [[1,2,3,7,8,9],[4,5,6]]
 => [4,5,6,1,2,3,7,8,9] => [1,2,3,7,8,9,4,5,6] => ? = 1 - 1
[6,2,1]
 => [[1,3,6,7,8,9],[2,5],[4]]
 => [4,2,5,1,3,6,7,8,9] => [1,3,6,7,8,9,2,5,4] => ? = 2 - 1
[6,1,1,1]
 => [[1,5,6,7,8,9],[2],[3],[4]]
 => [4,3,2,1,5,6,7,8,9] => [1,5,6,7,8,9,2,3,4] => ? = 1 - 1
[5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4,9] => [1,2,3,4,9,5,6,7,8] => ? = 1 - 1
[5,3,1]
 => [[1,3,4,8,9],[2,6,7],[5]]
 => [5,2,6,7,1,3,4,8,9] => [1,3,4,8,9,2,6,7,5] => ? = 1 - 1
[5,2,2]
 => [[1,2,7,8,9],[3,4],[5,6]]
 => [5,6,3,4,1,2,7,8,9] => [1,2,7,8,9,3,4,5,6] => ? = 1 - 1
[5,2,1,1]
 => [[1,4,7,8,9],[2,6],[3],[5]]
 => [5,3,2,6,1,4,7,8,9] => [1,4,7,8,9,2,6,3,5] => ? = 2 - 1
[5,1,1,1,1]
 => [[1,6,7,8,9],[2],[3],[4],[5]]
 => [5,4,3,2,1,6,7,8,9] => [1,6,7,8,9,2,3,4,5] => ? = 1 - 1
[4,4,1]
 => [[1,3,4,5],[2,7,8,9],[6]]
 => [6,2,7,8,9,1,3,4,5] => [1,3,4,5,2,7,8,9,6] => ? = 1 - 1
[4,3,2]
 => [[1,2,5,9],[3,4,8],[6,7]]
 => [6,7,3,4,8,1,2,5,9] => [1,2,5,9,3,4,8,6,7] => ? = 2 - 1
[4,3,1,1]
 => [[1,4,5,9],[2,7,8],[3],[6]]
 => [6,3,2,7,8,1,4,5,9] => [1,4,5,9,2,7,8,3,6] => ? = 1 - 1
[4,2,2,1]
 => [[1,3,8,9],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3,8,9] => [1,3,8,9,2,5,4,7,6] => ? = 1 - 1
[4,2,1,1,1]
 => [[1,5,8,9],[2,7],[3],[4],[6]]
 => [6,4,3,2,7,1,5,8,9] => [1,5,8,9,2,7,3,4,6] => ? = 2 - 1
[4,1,1,1,1,1]
 => [[1,7,8,9],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1,7,8,9] => [1,7,8,9,2,3,4,5,6] => ? = 1 - 1
[3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => [7,8,9,4,5,6,1,2,3] => [1,2,3,4,5,6,7,8,9] => 0 = 1 - 1
[3,3,2,1]
 => [[1,3,6],[2,5,9],[4,8],[7]]
 => [7,4,8,2,5,9,1,3,6] => [1,3,6,2,5,9,4,8,7] => ? = 2 - 1
[3,3,1,1,1]
 => [[1,5,6],[2,8,9],[3],[4],[7]]
 => [7,4,3,2,8,9,1,5,6] => [1,5,6,2,8,9,3,4,7] => ? = 1 - 1
[3,2,2,2]
 => [[1,2,9],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2,9] => [1,2,9,3,4,5,6,7,8] => ? = 1 - 1
[3,2,2,1,1]
 => [[1,4,9],[2,6],[3,8],[5],[7]]
 => [7,5,3,8,2,6,1,4,9] => [1,4,9,2,6,3,8,5,7] => ? = 2 - 1
[3,2,1,1,1,1]
 => [[1,6,9],[2,8],[3],[4],[5],[7]]
 => [7,5,4,3,2,8,1,6,9] => [1,6,9,2,8,3,4,5,7] => ? = 2 - 1
[2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8]]
 => [8,6,9,4,7,2,5,1,3] => [1,3,2,5,4,7,6,9,8] => ? = 1 - 1
[2,2,2,1,1,1]
 => [[1,5],[2,7],[3,9],[4],[6],[8]]
 => [8,6,4,3,9,2,7,1,5] => [1,5,2,7,3,9,4,6,8] => ? = 1 - 1
[6,4]
 => [[1,2,3,4,9,10],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4,9,10] => [1,2,3,4,9,10,5,6,7,8] => ? = 1 - 1
[6,3,1]
 => [[1,3,4,8,9,10],[2,6,7],[5]]
 => [5,2,6,7,1,3,4,8,9,10] => [1,3,4,8,9,10,2,6,7,5] => ? = 1 - 1
[6,2,2]
 => [[1,2,7,8,9,10],[3,4],[5,6]]
 => [5,6,3,4,1,2,7,8,9,10] => [1,2,7,8,9,10,3,4,5,6] => ? = 1 - 1
[6,2,1,1]
 => [[1,4,7,8,9,10],[2,6],[3],[5]]
 => [5,3,2,6,1,4,7,8,9,10] => [1,4,7,8,9,10,2,6,3,5] => ? = 2 - 1
[6,1,1,1,1]
 => [[1,6,7,8,9,10],[2],[3],[4],[5]]
 => [5,4,3,2,1,6,7,8,9,10] => [1,6,7,8,9,10,2,3,4,5] => ? = 1 - 1
[5,5]
 => [[1,2,3,4,5],[6,7,8,9,10]]
 => [6,7,8,9,10,1,2,3,4,5] => [1,2,3,4,5,6,7,8,9,10] => 0 = 1 - 1
[5,4,1]
 => [[1,3,4,5,10],[2,7,8,9],[6]]
 => [6,2,7,8,9,1,3,4,5,10] => [1,3,4,5,10,2,7,8,9,6] => ? = 1 - 1
[5,3,2]
 => [[1,2,5,9,10],[3,4,8],[6,7]]
 => [6,7,3,4,8,1,2,5,9,10] => [1,2,5,9,10,3,4,8,6,7] => ? = 2 - 1
[5,3,1,1]
 => [[1,4,5,9,10],[2,7,8],[3],[6]]
 => [6,3,2,7,8,1,4,5,9,10] => [1,4,5,9,10,2,7,8,3,6] => ? = 1 - 1
[5,2,2,1]
 => [[1,3,8,9,10],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3,8,9,10] => [1,3,8,9,10,2,5,4,7,6] => ? = 1 - 1
[5,2,1,1,1]
 => [[1,5,8,9,10],[2,7],[3],[4],[6]]
 => [6,4,3,2,7,1,5,8,9,10] => [1,5,8,9,10,2,7,3,4,6] => ? = 2 - 1
[5,1,1,1,1,1]
 => [[1,7,8,9,10],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1,7,8,9,10] => [1,7,8,9,10,2,3,4,5,6] => ? = 1 - 1
[4,4,2]
 => [[1,2,5,6],[3,4,9,10],[7,8]]
 => [7,8,3,4,9,10,1,2,5,6] => [1,2,5,6,3,4,9,10,7,8] => ? = 1 - 1
[4,4,1,1]
 => [[1,4,5,6],[2,8,9,10],[3],[7]]
 => [7,3,2,8,9,10,1,4,5,6] => [1,4,5,6,2,8,9,10,3,7] => ? = 1 - 1
[4,3,3]
 => [[1,2,3,10],[4,5,6],[7,8,9]]
 => [7,8,9,4,5,6,1,2,3,10] => [1,2,3,10,4,5,6,7,8,9] => ? = 1 - 1
[4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => [7,4,8,2,5,9,1,3,6,10] => [1,3,6,10,2,5,9,4,8,7] => ? = 3 - 1
[4,3,1,1,1]
 => [[1,5,6,10],[2,8,9],[3],[4],[7]]
 => [7,4,3,2,8,9,1,5,6,10] => [1,5,6,10,2,8,9,3,4,7] => ? = 1 - 1
[2,2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8],[9,10]]
 => [9,10,7,8,5,6,3,4,1,2] => [1,2,3,4,5,6,7,8,9,10] => 0 = 1 - 1
Description
The number of mid points of decreasing subsequences of length 3 in a permutation. For a permutation π of {1,…,n}, this is the number of indices j such that there exist indices i,k with i<j<k and π(i)>π(j)>π(k). In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima. This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence. See also [[St000119]].
Matching statistic: St000405
(load all 5 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000405: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 40%
Values
[1]
 => [1,0]
 => [1,0]
 => [2,1] => 0 = 1 - 1
[2]
 => [1,0,1,0]
 => [1,1,0,0]
 => [2,3,1] => 0 = 1 - 1
[1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => [3,1,2] => 0 = 1 - 1
[3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => 0 = 1 - 1
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 0 = 1 - 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 0 = 1 - 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 0 = 1 - 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 0 = 1 - 1
[2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => 0 = 1 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0 = 1 - 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 0 = 1 - 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 0 = 1 - 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => 0 = 1 - 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => 0 = 1 - 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => 0 = 1 - 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => 0 = 1 - 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 0 = 1 - 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => 0 = 1 - 1
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => 0 = 1 - 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [2,3,6,5,1,4] => 0 = 1 - 1
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,3,4,7,1,5,6] => 0 = 1 - 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => 0 = 1 - 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => 1 = 2 - 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,3,7,1,4,5,6] => ? = 1 - 1
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => 0 = 1 - 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => 0 = 1 - 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,7,1,3,4,5,6] => 0 = 1 - 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 0 = 1 - 1
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,8,1,7] => 0 = 1 - 1
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [2,3,4,7,6,1,5] => ? = 1 - 1
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [2,3,4,5,8,1,6,7] => ? = 1 - 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2,6,4,5,1,3] => 0 = 1 - 1
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [2,3,7,5,1,4,6] => ? = 2 - 1
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [2,3,4,8,1,5,6,7] => ? = 1 - 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => 0 = 1 - 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => 0 = 1 - 1
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [2,7,4,1,3,5,6] => ? = 2 - 1
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [2,3,8,1,4,5,6,7] => ? = 1 - 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => 0 = 1 - 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [7,3,1,2,4,5,6] => 0 = 1 - 1
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,8,1,3,4,5,6,7] => 0 = 1 - 1
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [2,3,4,5,8,7,1,6] => ? = 1 - 1
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [2,3,4,5,6,9,1,7,8] => ? = 1 - 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [2,3,7,5,6,1,4] => ? = 1 - 1
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [2,3,4,8,6,1,5,7] => ? = 2 - 1
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [2,3,4,5,9,1,6,7,8] => ? = 1 - 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [6,3,4,5,1,2] => 0 = 1 - 1
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [2,7,4,5,1,3,6] => ? = 1 - 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [2,3,7,6,1,4,5] => ? = 1 - 1
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [2,3,8,5,1,4,6,7] => ? = 2 - 1
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [2,3,4,9,1,5,6,7,8] => ? = 1 - 1
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => 0 = 1 - 1
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [7,3,4,1,2,5,6] => 0 = 1 - 1
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [2,7,5,1,3,4,6] => ? = 2 - 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => [2,8,4,1,3,5,6,7] => ? = 2 - 1
[3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,9,1,4,5,6,7,8] => ? = 1 - 1
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => 0 = 1 - 1
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [7,4,1,2,3,5,6] => 0 = 1 - 1
[2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [8,3,1,2,4,5,6,7] => 0 = 1 - 1
[6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [2,3,4,8,6,7,1,5] => ? = 1 - 1
[6,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
 => [2,3,4,5,9,7,1,6,8] => ? = 2 - 1
[6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
 => [2,3,4,5,6,10,1,7,8,9] => ? = 1 - 1
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [2,7,4,5,6,1,3] => ? = 1 - 1
[5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [2,3,8,5,6,1,4,7] => ? = 1 - 1
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [2,3,4,8,7,1,5,6] => ? = 1 - 1
[5,2,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
 => [2,3,4,9,6,1,5,7,8] => ? = 2 - 1
[5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,10,1,6,7,8,9] => ? = 1 - 1
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => ? = 1 - 1
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,7,4,6,1,3,5] => ? = 2 - 1
[4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [2,8,4,5,1,3,6,7] => ? = 1 - 1
[4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
 => [2,3,8,6,1,4,5,7] => ? = 1 - 1
[4,2,1,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
 => [2,3,9,5,1,4,6,7,8] => ? = 2 - 1
[4,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,10,1,5,6,7,8,9] => ? = 1 - 1
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => 0 = 1 - 1
[3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [7,3,5,1,2,4,6] => ? = 2 - 1
[3,3,1,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => [8,3,4,1,2,5,6,7] => ? = 1 - 1
[3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,6,7,1,3,4,5] => 0 = 1 - 1
[3,2,2,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
 => [2,8,5,1,3,4,6,7] => ? = 2 - 1
[3,2,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => [2,9,4,1,3,5,6,7,8] => ? = 2 - 1
[2,2,2,2,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,7,1,2,3,4,6] => ? = 1 - 1
[2,2,2,1,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [8,4,1,2,3,5,6,7] => 0 = 1 - 1
[6,4]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [2,3,8,5,6,7,1,4] => ? = 1 - 1
[6,3,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => [2,3,4,9,6,7,1,5,8] => ? = 1 - 1
[6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => [2,3,4,5,9,8,1,6,7] => ? = 1 - 1
[6,2,1,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,1,0]
 => [2,3,4,5,10,7,1,6,8,9] => ? = 2 - 1
[6,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,11,1,7,8,9,10] => ? = 1 - 1
[5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [7,3,4,5,6,1,2] => ? = 1 - 1
[5,4,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [2,8,4,5,6,1,3,7] => ? = 1 - 1
[5,3,2]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
 => [2,3,8,5,7,1,4,6] => ? = 2 - 1
[5,3,1,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
 => [2,3,9,5,6,1,4,7,8] => ? = 1 - 1
[5,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
 => [2,3,4,9,7,1,5,6,8] => ? = 1 - 1
[5,2,1,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0]
 => [2,3,4,10,6,1,5,7,8,9] => ? = 2 - 1
[5,1,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,11,1,6,7,8,9,10] => ? = 1 - 1
[2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,1,2,3,4,5] => 0 = 1 - 1
[4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [7,6,4,5,1,2,3] => 0 = 1 - 1
Description
The number of occurrences of the pattern 1324 in a permutation. There is no explicit formula known for the number of permutations avoiding this pattern (denoted by Sn(1324)), but it is shown in [1], improving bounds in [2] and [3] that lim
Matching statistic: St000404
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000404: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 40%
Values
[1]
 => [1,0]
 => [1,0]
 => [2,1] => 0 = 1 - 1
[2]
 => [1,0,1,0]
 => [1,1,0,0]
 => [2,3,1] => 0 = 1 - 1
[1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => [3,1,2] => 0 = 1 - 1
[3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => 0 = 1 - 1
[2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 0 = 1 - 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 0 = 1 - 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 0 = 1 - 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 0 = 1 - 1
[2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 0 = 1 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 0 = 1 - 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 0 = 1 - 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 0 = 1 - 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 0 = 1 - 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0 = 1 - 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 0 = 1 - 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => 0 = 1 - 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => 0 = 1 - 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 0 = 1 - 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => 0 = 1 - 1
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 1 - 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => 0 = 1 - 1
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => 0 = 1 - 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0 = 1 - 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => 1 = 2 - 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => ? = 1 - 1
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => 0 = 1 - 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,3,4,6,1,7,5] => 0 = 1 - 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 1 - 1
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [3,1,4,5,6,7,8,2] => ? = 1 - 1
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => ? = 1 - 1
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,8,1,7] => 0 = 1 - 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => 0 = 1 - 1
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [4,3,1,5,7,2,6] => ? = 2 - 1
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [2,4,1,5,6,7,8,3] => ? = 1 - 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => 0 = 1 - 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => 0 = 1 - 1
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,6,1,5,3,7,4] => ? = 2 - 1
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [2,3,4,5,7,1,8,6] => ? = 1 - 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => 0 = 1 - 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [6,3,4,1,2,7,5] => ? = 1 - 1
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [2,3,5,1,6,7,8,4] => ? = 1 - 1
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [4,1,2,5,6,7,8,3] => ? = 1 - 1
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,9,1,8] => 0 = 1 - 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,3,4,7,1,5,6] => 0 = 1 - 1
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [4,3,1,5,6,8,2,7] => ? = 2 - 1
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,4,1,5,6,7,8,9,3] => ? = 1 - 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 0 = 1 - 1
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [5,3,1,2,6,7,4] => ? = 1 - 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => 0 = 1 - 1
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [2,6,1,5,3,7,8,4] => ? = 2 - 1
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [2,3,4,5,6,8,1,9,7] => ? = 1 - 1
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => 0 = 1 - 1
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [2,3,7,1,6,4,5] => ? = 1 - 1
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,4,1,2,7,3,6] => ? = 2 - 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,1,0,0]
 => [5,3,4,1,7,2,8,6] => ? = 2 - 1
[3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,5,1,6,7,8,9,4] => ? = 1 - 1
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => 0 = 1 - 1
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,7,1,6,3,4,5] => ? = 1 - 1
[2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
 => [2,3,8,1,6,7,4,5] => ? = 1 - 1
[6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [2,3,4,5,8,1,6,7] => ? = 1 - 1
[6,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [4,3,1,5,6,7,9,2,8] => ? = 2 - 1
[6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,4,1,5,6,7,8,9,10,3] => ? = 1 - 1
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ? = 1 - 1
[5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [5,3,1,2,6,7,8,4] => ? = 1 - 1
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [5,1,2,3,6,7,8,4] => ? = 1 - 1
[5,2,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,6,1,5,3,7,8,9,4] => ? = 2 - 1
[5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
 => [2,3,4,5,6,7,9,1,10,8] => ? = 1 - 1
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [2,6,4,1,3,7,5] => ? = 1 - 1
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => ? = 2 - 1
[4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
 => [2,3,4,8,1,7,5,6] => ? = 1 - 1
[4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [5,4,1,2,6,8,3,7] => ? = 1 - 1
[4,2,1,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0]
 => [5,3,4,1,6,8,2,9,7] => ? = 2 - 1
[4,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,5,1,6,7,8,9,10,4] => ? = 1 - 1
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 0 = 1 - 1
[3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [2,7,5,1,3,4,6] => ? = 2 - 1
[3,3,1,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
 => [6,3,4,1,2,7,8,5] => ? = 1 - 1
[3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,3,7,1,4,5,6] => ? = 1 - 1
[3,2,2,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [2,7,1,6,3,4,8,5] => ? = 2 - 1
[3,2,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0]
 => [2,3,8,1,6,7,4,9,5] => ? = 2 - 1
[2,2,2,2,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [6,4,1,2,3,7,5] => ? = 1 - 1
[2,2,2,1,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,1,0,0]
 => [7,5,4,1,2,3,8,6] => ? = 1 - 1
[6,4]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [2,4,1,5,6,8,3,7] => ? = 1 - 1
[6,3,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [5,3,1,2,6,7,8,9,4] => ? = 1 - 1
[6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,6,7,8,9,4] => ? = 1 - 1
[6,2,1,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,6,1,5,3,7,8,9,10,4] => ? = 2 - 1
[6,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
 => [2,3,4,5,6,7,8,10,1,11,9] => ? = 1 - 1
[5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 1 - 1
[4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => 0 = 1 - 1
[3,3,3,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [7,3,1,2,4,5,6] => 0 = 1 - 1
[5,5,2]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,7,4,8,6] => 0 = 1 - 1
[4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,7,1,3,4,5,6] => 0 = 1 - 1
[3,3,3,3]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 0 = 1 - 1
[4,3,3,3]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [7,1,2,3,4,5,8,6] => 0 = 1 - 1
[3,3,3,3,2]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [8,4,1,2,3,5,6,7] => 0 = 1 - 1
Description
The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. A permutation avoids these two pattern if and only if it is an ''input-restricted deques'', see [1].
The following 88 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000842The breadth of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000908The length of the shortest maximal antichain in a poset. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001532The leading coefficient of the Poincare polynomial of the poset cone. St000516The number of stretching pairs of a permutation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. 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. St001881The number of factors of a lattice as a Cartesian product of lattices. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001616The number of neutral elements in a lattice. St001715The number of non-records in a permutation. St001720The minimal length of a chain of small intervals in a lattice. St001846The number of elements which do not have a complement in the lattice. St001208The number of connected components of the quiver of A/T when T is the 1-tilting module corresponding to the permutation in the Auslander algebra A of K[x]/(x^n). St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000335The difference of lower and upper interactions. St000570The Edelman-Greene number of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001162The minimum jump of a permutation. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. 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. St001344The neighbouring number of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000406The number of occurrences of the pattern 3241 in a permutation. St000666The number of right tethers of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000951The dimension of Ext^{1}(D(A),A) of the corresponding LNakayama algebra. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001705The number of occurrences of the pattern 2413 in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000056The decomposition (or block) number of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St001256Number of simple reflexive modules that are 2-stable reflexive. St001461The number of topologically connected components of the chord diagram of a permutation. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length 3. 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. St001381The fertility of a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001960The number of descents of a permutation minus one if its first entry is not one. St000095The number of triangles of a graph. St000623The number of occurrences of the pattern 52341 in a permutation. St000664The number of right ropes of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. 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. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001330The hat guessing number of a graph. St000768The number of peaks in an integer composition. St000895The number of ones on the main diagonal of an alternating sign matrix. St001434The number of negative sum pairs of a signed permutation. St000022The number of fixed points of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000455The second largest eigenvalue of a graph if it is integral.