Identifier
- St000879: Permutations ⟶ ℤ
Values
=>
[1]=>0
[1,2]=>0
[2,1]=>0
[1,2,3]=>0
[1,3,2]=>0
[2,1,3]=>0
[2,3,1]=>0
[3,1,2]=>0
[3,2,1]=>1
[1,2,3,4]=>0
[1,2,4,3]=>0
[1,3,2,4]=>0
[1,3,4,2]=>0
[1,4,2,3]=>0
[1,4,3,2]=>1
[2,1,3,4]=>0
[2,1,4,3]=>0
[2,3,1,4]=>0
[2,3,4,1]=>0
[2,4,1,3]=>0
[2,4,3,1]=>1
[3,1,2,4]=>0
[3,1,4,2]=>0
[3,2,1,4]=>1
[3,2,4,1]=>1
[3,4,1,2]=>0
[3,4,2,1]=>2
[4,1,2,3]=>0
[4,1,3,2]=>1
[4,2,1,3]=>1
[4,2,3,1]=>2
[4,3,1,2]=>2
[4,3,2,1]=>8
[1,2,3,4,5]=>0
[1,2,3,5,4]=>0
[1,2,4,3,5]=>0
[1,2,4,5,3]=>0
[1,2,5,3,4]=>0
[1,2,5,4,3]=>1
[1,3,2,4,5]=>0
[1,3,2,5,4]=>0
[1,3,4,2,5]=>0
[1,3,4,5,2]=>0
[1,3,5,2,4]=>0
[1,3,5,4,2]=>1
[1,4,2,3,5]=>0
[1,4,2,5,3]=>0
[1,4,3,2,5]=>1
[1,4,3,5,2]=>1
[1,4,5,2,3]=>0
[1,4,5,3,2]=>2
[1,5,2,3,4]=>0
[1,5,2,4,3]=>1
[1,5,3,2,4]=>1
[1,5,3,4,2]=>2
[1,5,4,2,3]=>2
[1,5,4,3,2]=>8
[2,1,3,4,5]=>0
[2,1,3,5,4]=>0
[2,1,4,3,5]=>0
[2,1,4,5,3]=>0
[2,1,5,3,4]=>0
[2,1,5,4,3]=>2
[2,3,1,4,5]=>0
[2,3,1,5,4]=>0
[2,3,4,1,5]=>0
[2,3,4,5,1]=>0
[2,3,5,1,4]=>0
[2,3,5,4,1]=>1
[2,4,1,3,5]=>0
[2,4,1,5,3]=>0
[2,4,3,1,5]=>1
[2,4,3,5,1]=>1
[2,4,5,1,3]=>0
[2,4,5,3,1]=>3
[2,5,1,3,4]=>0
[2,5,1,4,3]=>2
[2,5,3,1,4]=>2
[2,5,3,4,1]=>3
[2,5,4,1,3]=>5
[2,5,4,3,1]=>15
[3,1,2,4,5]=>0
[3,1,2,5,4]=>0
[3,1,4,2,5]=>0
[3,1,4,5,2]=>0
[3,1,5,2,4]=>0
[3,1,5,4,2]=>2
[3,2,1,4,5]=>1
[3,2,1,5,4]=>2
[3,2,4,1,5]=>1
[3,2,4,5,1]=>1
[3,2,5,1,4]=>2
[3,2,5,4,1]=>6
[3,4,1,2,5]=>0
[3,4,1,5,2]=>0
[3,4,2,1,5]=>2
[3,4,2,5,1]=>3
[3,4,5,1,2]=>0
[3,4,5,2,1]=>5
[3,5,1,2,4]=>0
[3,5,1,4,2]=>2
[3,5,2,1,4]=>5
[3,5,2,4,1]=>11
[3,5,4,1,2]=>5
[3,5,4,2,1]=>30
[4,1,2,3,5]=>0
[4,1,2,5,3]=>0
[4,1,3,2,5]=>1
[4,1,3,5,2]=>2
[4,1,5,2,3]=>0
[4,1,5,3,2]=>5
[4,2,1,3,5]=>1
[4,2,1,5,3]=>2
[4,2,3,1,5]=>2
[4,2,3,5,1]=>3
[4,2,5,1,3]=>2
[4,2,5,3,1]=>11
[4,3,1,2,5]=>2
[4,3,1,5,2]=>5
[4,3,2,1,5]=>8
[4,3,2,5,1]=>15
[4,3,5,1,2]=>5
[4,3,5,2,1]=>30
[4,5,1,2,3]=>0
[4,5,1,3,2]=>5
[4,5,2,1,3]=>5
[4,5,2,3,1]=>16
[4,5,3,1,2]=>16
[4,5,3,2,1]=>77
[5,1,2,3,4]=>0
[5,1,2,4,3]=>1
[5,1,3,2,4]=>1
[5,1,3,4,2]=>3
[5,1,4,2,3]=>3
[5,1,4,3,2]=>15
[5,2,1,3,4]=>1
[5,2,1,4,3]=>6
[5,2,3,1,4]=>3
[5,2,3,4,1]=>6
[5,2,4,1,3]=>11
[5,2,4,3,1]=>36
[5,3,1,2,4]=>3
[5,3,1,4,2]=>11
[5,3,2,1,4]=>15
[5,3,2,4,1]=>36
[5,3,4,1,2]=>16
[5,3,4,2,1]=>91
[5,4,1,2,3]=>5
[5,4,1,3,2]=>30
[5,4,2,1,3]=>30
[5,4,2,3,1]=>91
[5,4,3,1,2]=>77
[5,4,3,2,1]=>384
[1,2,3,4,5,6]=>0
[1,2,3,4,6,5]=>0
[1,2,3,5,4,6]=>0
[1,2,3,5,6,4]=>0
[1,2,3,6,4,5]=>0
[1,2,3,6,5,4]=>1
[1,2,4,3,5,6]=>0
[1,2,4,3,6,5]=>0
[1,2,4,5,3,6]=>0
[1,2,4,5,6,3]=>0
[1,2,4,6,3,5]=>0
[1,2,4,6,5,3]=>1
[1,2,5,3,4,6]=>0
[1,2,5,3,6,4]=>0
[1,2,5,4,3,6]=>1
[1,2,5,4,6,3]=>1
[1,2,5,6,3,4]=>0
[1,2,5,6,4,3]=>2
[1,2,6,3,4,5]=>0
[1,2,6,3,5,4]=>1
[1,2,6,4,3,5]=>1
[1,2,6,4,5,3]=>2
[1,2,6,5,3,4]=>2
[1,2,6,5,4,3]=>8
[1,3,2,4,5,6]=>0
[1,3,2,4,6,5]=>0
[1,3,2,5,4,6]=>0
[1,3,2,5,6,4]=>0
[1,3,2,6,4,5]=>0
[1,3,2,6,5,4]=>2
[1,3,4,2,5,6]=>0
[1,3,4,2,6,5]=>0
[1,3,4,5,2,6]=>0
[1,3,4,5,6,2]=>0
[1,3,4,6,2,5]=>0
[1,3,4,6,5,2]=>1
[1,3,5,2,4,6]=>0
[1,3,5,2,6,4]=>0
[1,3,5,4,2,6]=>1
[1,3,5,4,6,2]=>1
[1,3,5,6,2,4]=>0
[1,3,5,6,4,2]=>3
[1,3,6,2,4,5]=>0
[1,3,6,2,5,4]=>2
[1,3,6,4,2,5]=>2
[1,3,6,4,5,2]=>3
[1,3,6,5,2,4]=>5
[1,3,6,5,4,2]=>15
[1,4,2,3,5,6]=>0
[1,4,2,3,6,5]=>0
[1,4,2,5,3,6]=>0
[1,4,2,5,6,3]=>0
[1,4,2,6,3,5]=>0
[1,4,2,6,5,3]=>2
[1,4,3,2,5,6]=>1
[1,4,3,2,6,5]=>2
[1,4,3,5,2,6]=>1
[1,4,3,5,6,2]=>1
[1,4,3,6,2,5]=>2
[1,4,3,6,5,2]=>6
[1,4,5,2,3,6]=>0
[1,4,5,2,6,3]=>0
[1,4,5,3,2,6]=>2
[1,4,5,3,6,2]=>3
[1,4,5,6,2,3]=>0
[1,4,5,6,3,2]=>5
[1,4,6,2,3,5]=>0
[1,4,6,2,5,3]=>2
[1,4,6,3,2,5]=>5
[1,4,6,3,5,2]=>11
[1,4,6,5,2,3]=>5
[1,4,6,5,3,2]=>30
[1,5,2,3,4,6]=>0
[1,5,2,3,6,4]=>0
[1,5,2,4,3,6]=>1
[1,5,2,4,6,3]=>2
[1,5,2,6,3,4]=>0
[1,5,2,6,4,3]=>5
[1,5,3,2,4,6]=>1
[1,5,3,2,6,4]=>2
[1,5,3,4,2,6]=>2
[1,5,3,4,6,2]=>3
[1,5,3,6,2,4]=>2
[1,5,3,6,4,2]=>11
[1,5,4,2,3,6]=>2
[1,5,4,2,6,3]=>5
[1,5,4,3,2,6]=>8
[1,5,4,3,6,2]=>15
[1,5,4,6,2,3]=>5
[1,5,4,6,3,2]=>30
[1,5,6,2,3,4]=>0
[1,5,6,2,4,3]=>5
[1,5,6,3,2,4]=>5
[1,5,6,3,4,2]=>16
[1,5,6,4,2,3]=>16
[1,5,6,4,3,2]=>77
[1,6,2,3,4,5]=>0
[1,6,2,3,5,4]=>1
[1,6,2,4,3,5]=>1
[1,6,2,4,5,3]=>3
[1,6,2,5,3,4]=>3
[1,6,2,5,4,3]=>15
[1,6,3,2,4,5]=>1
[1,6,3,2,5,4]=>6
[1,6,3,4,2,5]=>3
[1,6,3,4,5,2]=>6
[1,6,3,5,2,4]=>11
[1,6,3,5,4,2]=>36
[1,6,4,2,3,5]=>3
[1,6,4,2,5,3]=>11
[1,6,4,3,2,5]=>15
[1,6,4,3,5,2]=>36
[1,6,4,5,2,3]=>16
[1,6,4,5,3,2]=>91
[1,6,5,2,3,4]=>5
[1,6,5,2,4,3]=>30
[1,6,5,3,2,4]=>30
[1,6,5,3,4,2]=>91
[1,6,5,4,2,3]=>77
[1,6,5,4,3,2]=>384
[2,1,3,4,5,6]=>0
[2,1,3,4,6,5]=>0
[2,1,3,5,4,6]=>0
[2,1,3,5,6,4]=>0
[2,1,3,6,4,5]=>0
[2,1,3,6,5,4]=>2
[2,1,4,3,5,6]=>0
[2,1,4,3,6,5]=>0
[2,1,4,5,3,6]=>0
[2,1,4,5,6,3]=>0
[2,1,4,6,3,5]=>0
[2,1,4,6,5,3]=>3
[2,1,5,3,4,6]=>0
[2,1,5,3,6,4]=>0
[2,1,5,4,3,6]=>2
[2,1,5,4,6,3]=>3
[2,1,5,6,3,4]=>0
[2,1,5,6,4,3]=>8
[2,1,6,3,4,5]=>0
[2,1,6,3,5,4]=>3
[2,1,6,4,3,5]=>3
[2,1,6,4,5,3]=>8
[2,1,6,5,3,4]=>8
[2,1,6,5,4,3]=>40
[2,3,1,4,5,6]=>0
[2,3,1,4,6,5]=>0
[2,3,1,5,4,6]=>0
[2,3,1,5,6,4]=>0
[2,3,1,6,4,5]=>0
[2,3,1,6,5,4]=>3
[2,3,4,1,5,6]=>0
[2,3,4,1,6,5]=>0
[2,3,4,5,1,6]=>0
[2,3,4,5,6,1]=>0
[2,3,4,6,1,5]=>0
[2,3,4,6,5,1]=>1
[2,3,5,1,4,6]=>0
[2,3,5,1,6,4]=>0
[2,3,5,4,1,6]=>1
[2,3,5,4,6,1]=>1
[2,3,5,6,1,4]=>0
[2,3,5,6,4,1]=>4
[2,3,6,1,4,5]=>0
[2,3,6,1,5,4]=>3
[2,3,6,4,1,5]=>3
[2,3,6,4,5,1]=>4
[2,3,6,5,1,4]=>9
[2,3,6,5,4,1]=>24
[2,4,1,3,5,6]=>0
[2,4,1,3,6,5]=>0
[2,4,1,5,3,6]=>0
[2,4,1,5,6,3]=>0
[2,4,1,6,3,5]=>0
[2,4,1,6,5,3]=>5
[2,4,3,1,5,6]=>1
[2,4,3,1,6,5]=>3
[2,4,3,5,1,6]=>1
[2,4,3,5,6,1]=>1
[2,4,3,6,1,5]=>3
[2,4,3,6,5,1]=>8
[2,4,5,1,3,6]=>0
[2,4,5,1,6,3]=>0
[2,4,5,3,1,6]=>3
[2,4,5,3,6,1]=>4
[2,4,5,6,1,3]=>0
[2,4,5,6,3,1]=>9
[2,4,6,1,3,5]=>0
[2,4,6,1,5,3]=>5
[2,4,6,3,1,5]=>11
[2,4,6,3,5,1]=>19
[2,4,6,5,1,3]=>14
[2,4,6,5,3,1]=>63
[2,5,1,3,4,6]=>0
[2,5,1,3,6,4]=>0
[2,5,1,4,3,6]=>2
[2,5,1,4,6,3]=>5
[2,5,1,6,3,4]=>0
[2,5,1,6,4,3]=>16
[2,5,3,1,4,6]=>2
[2,5,3,1,6,4]=>5
[2,5,3,4,1,6]=>3
[2,5,3,4,6,1]=>4
[2,5,3,6,1,4]=>5
[2,5,3,6,4,1]=>19
[2,5,4,1,3,6]=>5
[2,5,4,1,6,3]=>14
[2,5,4,3,1,6]=>15
[2,5,4,3,6,1]=>24
[2,5,4,6,1,3]=>14
[2,5,4,6,3,1]=>63
[2,5,6,1,3,4]=>0
[2,5,6,1,4,3]=>16
[2,5,6,3,1,4]=>16
[2,5,6,3,4,1]=>35
[2,5,6,4,1,3]=>56
[2,5,6,4,3,1]=>190
[2,6,1,3,4,5]=>0
[2,6,1,3,5,4]=>3
[2,6,1,4,3,5]=>3
[2,6,1,4,5,3]=>11
[2,6,1,5,3,4]=>11
[2,6,1,5,4,3]=>66
[2,6,3,1,4,5]=>3
[2,6,3,1,5,4]=>17
[2,6,3,4,1,5]=>6
[2,6,3,4,5,1]=>10
[2,6,3,5,1,4]=>26
[2,6,3,5,4,1]=>70
[2,6,4,1,3,5]=>11
[2,6,4,1,5,3]=>42
[2,6,4,3,1,5]=>36
[2,6,4,3,5,1]=>70
[2,6,4,5,1,3]=>56
[2,6,4,5,3,1]=>224
[2,6,5,1,3,4]=>21
[2,6,5,1,4,3]=>133
[2,6,5,3,1,4]=>91
[2,6,5,3,4,1]=>210
[2,6,5,4,1,3]=>314
[2,6,5,4,3,1]=>1071
[3,1,2,4,5,6]=>0
[3,1,2,4,6,5]=>0
[3,1,2,5,4,6]=>0
[3,1,2,5,6,4]=>0
[3,1,2,6,4,5]=>0
[3,1,2,6,5,4]=>3
[3,1,4,2,5,6]=>0
[3,1,4,2,6,5]=>0
[3,1,4,5,2,6]=>0
[3,1,4,5,6,2]=>0
[3,1,4,6,2,5]=>0
[3,1,4,6,5,2]=>3
[3,1,5,2,4,6]=>0
[3,1,5,2,6,4]=>0
[3,1,5,4,2,6]=>2
[3,1,5,4,6,2]=>3
[3,1,5,6,2,4]=>0
[3,1,5,6,4,2]=>11
[3,1,6,2,4,5]=>0
[3,1,6,2,5,4]=>5
[3,1,6,4,2,5]=>5
[3,1,6,4,5,2]=>11
[3,1,6,5,2,4]=>16
[3,1,6,5,4,2]=>66
[3,2,1,4,5,6]=>1
[3,2,1,4,6,5]=>2
[3,2,1,5,4,6]=>2
[3,2,1,5,6,4]=>3
[3,2,1,6,4,5]=>3
[3,2,1,6,5,4]=>16
[3,2,4,1,5,6]=>1
[3,2,4,1,6,5]=>3
[3,2,4,5,1,6]=>1
[3,2,4,5,6,1]=>1
[3,2,4,6,1,5]=>3
[3,2,4,6,5,1]=>8
[3,2,5,1,4,6]=>2
[3,2,5,1,6,4]=>5
[3,2,5,4,1,6]=>6
[3,2,5,4,6,1]=>8
[3,2,5,6,1,4]=>5
[3,2,5,6,4,1]=>28
[3,2,6,1,4,5]=>3
[3,2,6,1,5,4]=>22
[3,2,6,4,1,5]=>17
[3,2,6,4,5,1]=>29
[3,2,6,5,1,4]=>56
[3,2,6,5,4,1]=>168
[3,4,1,2,5,6]=>0
[3,4,1,2,6,5]=>0
[3,4,1,5,2,6]=>0
[3,4,1,5,6,2]=>0
[3,4,1,6,2,5]=>0
[3,4,1,6,5,2]=>5
[3,4,2,1,5,6]=>2
[3,4,2,1,6,5]=>8
[3,4,2,5,1,6]=>3
[3,4,2,5,6,1]=>4
[3,4,2,6,1,5]=>11
[3,4,2,6,5,1]=>28
[3,4,5,1,2,6]=>0
[3,4,5,1,6,2]=>0
[3,4,5,2,1,6]=>5
[3,4,5,2,6,1]=>9
[3,4,5,6,1,2]=>0
[3,4,5,6,2,1]=>14
[3,4,6,1,2,5]=>0
[3,4,6,1,5,2]=>5
[3,4,6,2,1,5]=>21
[3,4,6,2,5,1]=>49
[3,4,6,5,1,2]=>14
[3,4,6,5,2,1]=>112
[3,5,1,2,4,6]=>0
[3,5,1,2,6,4]=>0
[3,5,1,4,2,6]=>2
[3,5,1,4,6,2]=>5
[3,5,1,6,2,4]=>0
[3,5,1,6,4,2]=>21
[3,5,2,1,4,6]=>5
[3,5,2,1,6,4]=>16
[3,5,2,4,1,6]=>11
[3,5,2,4,6,1]=>19
[3,5,2,6,1,4]=>21
[3,5,2,6,4,1]=>84
[3,5,4,1,2,6]=>5
[3,5,4,1,6,2]=>14
[3,5,4,2,1,6]=>30
[3,5,4,2,6,1]=>63
[3,5,4,6,1,2]=>14
[3,5,4,6,2,1]=>112
[3,5,6,1,2,4]=>0
[3,5,6,1,4,2]=>21
[3,5,6,2,1,4]=>42
[3,5,6,2,4,1]=>126
[3,5,6,4,1,2]=>70
[3,5,6,4,2,1]=>414
[3,6,1,2,4,5]=>0
[3,6,1,2,5,4]=>5
[3,6,1,4,2,5]=>5
[3,6,1,4,5,2]=>16
[3,6,1,5,2,4]=>21
[3,6,1,5,4,2]=>112
[3,6,2,1,4,5]=>9
[3,6,2,1,5,4]=>56
[3,6,2,4,1,5]=>26
[3,6,2,4,5,1]=>55
[3,6,2,5,1,4]=>112
[3,6,2,5,4,1]=>370
[3,6,4,1,2,5]=>16
[3,6,4,1,5,2]=>56
[3,6,4,2,1,5]=>91
[3,6,4,2,5,1]=>224
[3,6,4,5,1,2]=>70
[3,6,4,5,2,1]=>462
[3,6,5,1,2,4]=>42
[3,6,5,1,4,2]=>224
[3,6,5,2,1,4]=>294
[3,6,5,2,4,1]=>882
[3,6,5,4,1,2]=>468
[3,6,5,4,2,1]=>2625
[4,1,2,3,5,6]=>0
[4,1,2,3,6,5]=>0
[4,1,2,5,3,6]=>0
[4,1,2,5,6,3]=>0
[4,1,2,6,3,5]=>0
[4,1,2,6,5,3]=>3
[4,1,3,2,5,6]=>1
[4,1,3,2,6,5]=>3
[4,1,3,5,2,6]=>2
[4,1,3,5,6,2]=>3
[4,1,3,6,2,5]=>5
[4,1,3,6,5,2]=>17
[4,1,5,2,3,6]=>0
[4,1,5,2,6,3]=>0
[4,1,5,3,2,6]=>5
[4,1,5,3,6,2]=>11
[4,1,5,6,2,3]=>0
[4,1,5,6,3,2]=>21
[4,1,6,2,3,5]=>0
[4,1,6,2,5,3]=>5
[4,1,6,3,2,5]=>14
[4,1,6,3,5,2]=>42
[4,1,6,5,2,3]=>16
[4,1,6,5,3,2]=>133
[4,2,1,3,5,6]=>1
[4,2,1,3,6,5]=>3
[4,2,1,5,3,6]=>2
[4,2,1,5,6,3]=>3
[4,2,1,6,3,5]=>5
[4,2,1,6,5,3]=>22
[4,2,3,1,5,6]=>2
[4,2,3,1,6,5]=>8
[4,2,3,5,1,6]=>3
[4,2,3,5,6,1]=>4
[4,2,3,6,1,5]=>11
[4,2,3,6,5,1]=>29
[4,2,5,1,3,6]=>2
[4,2,5,1,6,3]=>5
[4,2,5,3,1,6]=>11
[4,2,5,3,6,1]=>19
[4,2,5,6,1,3]=>5
[4,2,5,6,3,1]=>49
[4,2,6,1,3,5]=>5
[4,2,6,1,5,3]=>32
[4,2,6,3,1,5]=>42
[4,2,6,3,5,1]=>90
[4,2,6,5,1,3]=>77
[4,2,6,5,3,1]=>350
[4,3,1,2,5,6]=>2
[4,3,1,2,6,5]=>8
[4,3,1,5,2,6]=>5
[4,3,1,5,6,2]=>9
[4,3,1,6,2,5]=>16
[4,3,1,6,5,2]=>56
[4,3,2,1,5,6]=>8
[4,3,2,1,6,5]=>40
[4,3,2,5,1,6]=>15
[4,3,2,5,6,1]=>24
[4,3,2,6,1,5]=>66
[4,3,2,6,5,1]=>168
[4,3,5,1,2,6]=>5
[4,3,5,1,6,2]=>14
[4,3,5,2,1,6]=>30
[4,3,5,2,6,1]=>63
[4,3,5,6,1,2]=>14
[4,3,5,6,2,1]=>112
[4,3,6,1,2,5]=>16
[4,3,6,1,5,2]=>77
[4,3,6,2,1,5]=>133
[4,3,6,2,5,1]=>350
[4,3,6,5,1,2]=>140
[4,3,6,5,2,1]=>876
[4,5,1,2,3,6]=>0
[4,5,1,2,6,3]=>0
[4,5,1,3,2,6]=>5
[4,5,1,3,6,2]=>16
[4,5,1,6,2,3]=>0
[4,5,1,6,3,2]=>42
[4,5,2,1,3,6]=>5
[4,5,2,1,6,3]=>16
[4,5,2,3,1,6]=>16
[4,5,2,3,6,1]=>35
[4,5,2,6,1,3]=>21
[4,5,2,6,3,1]=>126
[4,5,3,1,2,6]=>16
[4,5,3,1,6,2]=>56
[4,5,3,2,1,6]=>77
[4,5,3,2,6,1]=>190
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Description
The number of long braid edges in the graph of braid moves of a permutation.
Given a permutation $\pi$, let $\operatorname{Red}(\pi)$ denote the set of reduced words for $\pi$ in terms of simple transpositions $s_i = (i,i+1)$. We now say that two reduced words are connected by a long braid move if they are obtained from each other by a modification of the form $s_i s_{i+1} s_i \leftrightarrow s_{i+1} s_i s_{i+1}$ as a consecutive subword of a reduced word.
For example, the two reduced words $s_1s_3s_2s_3$ and $s_1s_2s_3s_2$ for
$$(124) = (12)(34)(23)(34) = (12)(23)(34)(23)$$
share an edge because they are obtained from each other by interchanging $s_3s_2s_3 \leftrightarrow s_3s_2s_3$.
This statistic counts the number of such short braid moves among all reduced words.
Given a permutation $\pi$, let $\operatorname{Red}(\pi)$ denote the set of reduced words for $\pi$ in terms of simple transpositions $s_i = (i,i+1)$. We now say that two reduced words are connected by a long braid move if they are obtained from each other by a modification of the form $s_i s_{i+1} s_i \leftrightarrow s_{i+1} s_i s_{i+1}$ as a consecutive subword of a reduced word.
For example, the two reduced words $s_1s_3s_2s_3$ and $s_1s_2s_3s_2$ for
$$(124) = (12)(34)(23)(34) = (12)(23)(34)(23)$$
share an edge because they are obtained from each other by interchanging $s_3s_2s_3 \leftrightarrow s_3s_2s_3$.
This statistic counts the number of such short braid moves among all reduced words.
Code
def long_braid_move_graph(pi):
V = [ tuple(w) for w in pi.reduced_words() ]
is_edge = lambda w1,w2: w1 != w2 and any( w1[:i] == w2[:i] and w1[i+3:] == w2[i+3:] and w1[i] != w2[i] and w1[i+2] != w2[i+2] and w1[i] == w1[i+2]for i in range(len(w1)-2) )
return Graph([V,is_edge])
def statistic(pi):
return len( long_braid_move_graph(pi).edges() )
Created
Jul 03, 2017 at 16:58 by Christian Stump
Updated
Apr 01, 2018 at 22:25 by Martin Rubey
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