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Your data matches 12 different statistics following compositions of up to 3 maps.
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Matching statistic: St001254
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001254: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001254: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 7 = 6 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 11 = 10 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 8 = 7 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 8 = 7 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 7 = 6 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 6 = 5 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 8 = 7 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 6 = 5 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 7 = 6 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 8 = 7 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 6 = 5 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 6 = 5 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 16 = 15 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 12 = 11 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 12 = 11 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 10 = 9 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 9 = 8 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 12 = 11 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 9 = 8 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 10 = 9 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 10 = 9 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 8 = 7 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 9 = 8 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 8 = 7 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 8 = 7 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 7 = 6 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 12 = 11 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 9 = 8 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 9 = 8 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 8 = 7 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 7 = 6 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 10 = 9 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 8 = 7 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 10 = 9 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 12 = 11 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 9 = 8 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 8 = 7 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 9 = 8 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 8 = 7 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 7 = 6 + 1
Description
The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J.
Matching statistic: St000448
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000448: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000448: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [[[]]]
=> ([(0,2),(1,2)],3)
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [[[],[]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 6
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 10
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[[],[],[[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 7
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[[],[[]],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 7
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[[],[[],[]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 5
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[[]],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 7
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 5
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[[[],[]],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 7
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 5
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 5
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[[[]],[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 5
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[[],[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 5
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[[[[]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 15
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [[[],[],[],[[]]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 11
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [[[],[],[[]],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 11
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [[[],[],[[],[]]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 9
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [[[],[],[[[]]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 8
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [[[],[[]],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 11
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [[[],[[]],[[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 8
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [[[],[[],[]],[]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 9
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [[[],[[],[],[]]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 9
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 7
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [[[],[[[]]],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 8
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [[[],[[[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 7
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [[[],[[[],[]]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 7
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[[],[[[[]]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[[[]],[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 11
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [[[[]],[],[[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 8
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [[[[]],[[]],[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 8
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [[[[]],[[],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 7
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[[[]],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[[[],[]],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 9
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [[[[],[]],[[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 7
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [[[[],[],[]],[]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 9
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [[[[],[],[],[]]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 11
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [[[[],[],[[]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 8
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [[[[],[[]]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 7
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [[[[],[[]],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 8
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [[[[],[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 7
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [[[[],[[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 6
Description
The number of pairs of vertices of a graph with distance 2.
This is the coefficient of the quadratic term of the Wiener polynomial.
Matching statistic: St001308
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St001308: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St001308: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [[[]]]
=> ([(0,2),(1,2)],3)
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [[[],[]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 6
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 10
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[[],[],[[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 7
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[[],[[]],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 7
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[[],[[],[]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 5
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[[]],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 7
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 5
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[[[],[]],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 7
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 5
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 5
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[[[]],[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 5
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[[],[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 5
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[[[[]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 15
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [[[],[],[],[[]]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 11
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [[[],[],[[]],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 11
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [[[],[],[[],[]]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 9
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [[[],[],[[[]]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 8
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [[[],[[]],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 11
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [[[],[[]],[[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 8
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [[[],[[],[]],[]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 9
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [[[],[[],[],[]]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 9
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 7
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [[[],[[[]]],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 8
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [[[],[[[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 7
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [[[],[[[],[]]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 7
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[[],[[[[]]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[[[]],[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 11
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [[[[]],[],[[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 8
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [[[[]],[[]],[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 8
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [[[[]],[[],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 7
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[[[]],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[[[],[]],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 9
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [[[[],[]],[[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 7
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [[[[],[],[]],[]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 9
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [[[[],[],[],[]]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 11
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [[[[],[],[[]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 8
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [[[[],[[]]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 7
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [[[[],[[]],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 8
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [[[[],[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 7
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [[[[],[[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 6
Description
The number of induced paths on three vertices in a graph.
Matching statistic: St001019
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001019: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 98%●distinct values known / distinct values provided: 92%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001019: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 98%●distinct values known / distinct values provided: 92%
Values
[1,0]
=> [1,0]
=> [1] => [1,0]
=> 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1] => [1,0,1,0]
=> 3
[1,1,0,0]
=> [1,1,0,0]
=> [2] => [1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> 6
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> 4
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> 4
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> 4
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 10
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 7
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 7
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 6
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 5
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 7
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 5
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 6
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 7
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 5
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 5
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 5
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 5
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 15
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 11
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 11
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 9
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 8
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 11
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 8
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 9
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 9
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 7
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 8
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 7
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 7
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 11
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 8
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 8
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 7
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 9
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 7
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 9
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 11
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 8
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 7
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 8
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 7
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 6
[]
=> []
=> [] => ?
=> ? = 0
Description
Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001228
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001228: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 98%●distinct values known / distinct values provided: 92%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001228: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 98%●distinct values known / distinct values provided: 92%
Values
[1,0]
=> [1,0]
=> [1] => [1,0]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [2] => [1,1,0,0]
=> 3
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> 6
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> 4
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> 4
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> 4
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> 10
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 7
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 7
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 6
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 5
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 7
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 5
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 6
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 7
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 5
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 5
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 5
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 5
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 15
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 11
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 11
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 9
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 8
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 11
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 8
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 9
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 9
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 7
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 8
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 7
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 7
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 11
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 8
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 8
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 7
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 9
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 7
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 9
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 11
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 8
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 7
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 8
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 7
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 6
[]
=> []
=> [] => ?
=> ? = 0
Description
The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St001003
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001003: Dyck paths ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 69%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001003: Dyck paths ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 69%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 4 = 1 + 3
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 6 = 3 + 3
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 5 = 2 + 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 9 = 6 + 3
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 7 = 4 + 3
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 7 = 4 + 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 7 = 4 + 3
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 13 = 10 + 3
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 10 = 7 + 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 10 = 7 + 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 9 = 6 + 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 8 = 5 + 3
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 10 = 7 + 3
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 8 = 5 + 3
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 9 = 6 + 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 10 = 7 + 3
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 8 = 5 + 3
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 8 = 5 + 3
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 8 = 5 + 3
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 8 = 5 + 3
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 7 = 4 + 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 15 + 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 11 + 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 11 + 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 9 + 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 8 + 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 11 + 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 8 + 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 9 + 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 9 + 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 7 + 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 8 + 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 7 + 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 7 + 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 + 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 11 + 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 8 + 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 8 + 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 7 + 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 6 + 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9 + 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 7 + 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 9 + 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 11 + 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 8 + 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 7 + 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 8 + 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 7 + 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6 + 3
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 8 + 3
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 6 + 3
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 7 + 3
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 8 + 3
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 6 + 3
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 7 + 3
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 7 + 3
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 8 + 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 3
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 6 + 3
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 6 + 3
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 6 + 3
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6 + 3
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5 + 3
[]
=> [1,0]
=> [1,0]
=> 3 = 0 + 3
Description
The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001821
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001821: Signed permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 46%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001821: Signed permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 46%
Values
[1,0]
=> [1,0]
=> [2,1] => [2,1] => 1
[1,0,1,0]
=> [1,0,1,0]
=> [3,1,2] => [3,1,2] => 3
[1,1,0,0]
=> [1,1,0,0]
=> [2,3,1] => [2,3,1] => 2
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => 6
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [3,1,4,2] => 4
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [2,4,1,3] => 4
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [3,1,4,2] => 4
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,4,1] => 3
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 10
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,2,5,3] => ? = 7
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,1,5,2,4] => ? = 7
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,2,5,3] => ? = 6
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,4,5,2] => ? = 5
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,5,1,3,4] => ? = 7
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [2,4,1,5,3] => ? = 5
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,1,5,2,4] => ? = 6
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [2,4,1,5,3] => ? = 7
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,4,5,2] => ? = 5
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [2,3,5,1,4] => ? = 5
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [2,4,1,5,3] => ? = 5
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,4,5,2] => ? = 5
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => ? = 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? = 15
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,2,3,6,4] => ? = 11
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,2,6,3,5] => ? = 11
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,2,3,6,4] => ? = 9
[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] => [4,1,2,5,6,3] => ? = 8
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,2,4,5] => ? = 11
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,5,2,6,4] => ? = 8
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,2,6,3,5] => ? = 9
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,5,2,6,4] => ? = 9
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,2,5,6,3] => ? = 7
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,4,6,2,5] => ? = 8
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,5,2,6,4] => ? = 7
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,2,5,6,3] => ? = 7
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,4,5,6,2] => ? = 6
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,6,1,3,4,5] => ? = 11
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,5,1,3,6,4] => ? = 8
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,4,1,6,3,5] => ? = 8
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,5,1,3,6,4] => ? = 7
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,4,1,5,6,3] => ? = 6
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,2,4,5] => ? = 9
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,5,2,6,4] => ? = 7
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,4,1,6,3,5] => ? = 9
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,5,2,6,4] => ? = 11
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,4,1,5,6,3] => ? = 8
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,4,6,2,5] => ? = 7
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,5,2,6,4] => ? = 8
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,4,1,5,6,3] => ? = 7
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,4,5,6,2] => ? = 6
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,3,6,1,4,5] => ? = 8
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,3,5,1,6,4] => ? = 6
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,4,1,6,3,5] => ? = 7
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,3,5,1,6,4] => ? = 8
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,4,1,5,6,3] => ? = 6
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,4,6,2,5] => ? = 7
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,3,5,1,6,4] => ? = 7
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,4,1,5,6,3] => ? = 8
[]
=> []
=> [1] => [1] => 0
Description
The sorting index of a signed permutation.
A signed permutation $\sigma = [\sigma(1),\ldots,\sigma(n)]$ can be sorted $[1,\ldots,n]$ by signed transpositions in the following way:
First move $\pm n$ to its position and swap the sign if needed, then $\pm (n-1), \pm (n-2)$ and so on.
For example for $[2,-4,5,-1,-3]$ we have the swaps
$$
[2,-4,5,-1,-3] \rightarrow [2,-4,-3,-1,5] \rightarrow [2,1,-3,4,5] \rightarrow [2,1,3,4,5] \rightarrow [1,2,3,4,5]
$$
given by the signed transpositions $(3,5), (-2,4), (-3,3), (1,2)$.
If $(i_1,j_1),\ldots,(i_n,j_n)$ is the decomposition of $\sigma$ obtained this way (including trivial transpositions) then the sorting index of $\sigma$ is defined as
$$
\operatorname{sor}_B(\sigma) = \sum_{k=1}^{n-1} j_k - i_k - \chi(i_k < 0),
$$
where $\chi(i_k < 0)$ is 1 if $i_k$ is negative and 0 otherwise.
For $\sigma = [2,-4,5,-1,-3]$ we have
$$
\operatorname{sor}_B(\sigma) = (5-3) + (4-(-2)-1) + (3-(-3)-1) + (2-1) = 13.
$$
Matching statistic: St000558
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00092: Perfect matchings —to set partition⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000558: Set partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 38%
Mp00092: Perfect matchings —to set partition⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000558: Set partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 38%
Values
[1,0]
=> [(1,2)]
=> {{1,2}}
=> {{1,2}}
=> 1
[1,0,1,0]
=> [(1,2),(3,4)]
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> 3
[1,1,0,0]
=> [(1,4),(2,3)]
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 2
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> {{1,2},{3,4},{5,6}}
=> {{1,2,4,6},{3},{5}}
=> 6
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> {{1,2},{3,6},{4,5}}
=> {{1,2,5},{3,6},{4}}
=> 4
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> {{1,4},{2,3},{5,6}}
=> {{1,3,6},{2,4},{5}}
=> 4
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> {{1,6},{2,3},{4,5}}
=> {{1,3,5},{2},{4,6}}
=> 4
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> {{1,6},{2,5},{3,4}}
=> {{1,4},{2,5},{3,6}}
=> 3
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> {{1,2},{3,4},{5,6},{7,8}}
=> {{1,2,4,6,8},{3},{5},{7}}
=> ? = 10
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> {{1,2},{3,4},{5,8},{6,7}}
=> {{1,2,4,7},{3,8},{5},{6}}
=> ? = 7
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> {{1,2},{3,6},{4,5},{7,8}}
=> {{1,2,5,8},{3,6},{4},{7}}
=> ? = 7
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> {{1,2},{3,8},{4,5},{6,7}}
=> {{1,2,5,7},{3},{4,8},{6}}
=> ? = 6
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> {{1,2},{3,8},{4,7},{5,6}}
=> {{1,2,6},{3,7},{4,8},{5}}
=> ? = 5
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> {{1,4},{2,3},{5,6},{7,8}}
=> {{1,3,6,8},{2,4},{5},{7}}
=> ? = 7
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> {{1,4},{2,3},{5,8},{6,7}}
=> {{1,3,7},{2,4,8},{5},{6}}
=> ? = 5
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> {{1,6},{2,3},{4,5},{7,8}}
=> {{1,3,5,8},{2},{4,6},{7}}
=> ? = 6
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> {{1,8},{2,3},{4,5},{6,7}}
=> {{1,3,5,7},{2},{4},{6,8}}
=> ? = 7
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> {{1,8},{2,3},{4,7},{5,6}}
=> {{1,3,6},{2,7},{4},{5,8}}
=> ? = 5
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> {{1,6},{2,5},{3,4},{7,8}}
=> {{1,4,8},{2,5},{3,6},{7}}
=> ? = 5
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> {{1,8},{2,5},{3,4},{6,7}}
=> {{1,4,7},{2,5},{3},{6,8}}
=> ? = 5
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> {{1,8},{2,7},{3,4},{5,6}}
=> {{1,4,6},{2},{3,7},{5,8}}
=> ? = 5
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> {{1,8},{2,7},{3,6},{4,5}}
=> {{1,5},{2,6},{3,7},{4,8}}
=> ? = 4
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> {{1,2},{3,4},{5,6},{7,8},{9,10}}
=> {{1,2,4,6,8,10},{3},{5},{7},{9}}
=> ? = 15
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> {{1,2},{3,4},{5,6},{7,10},{8,9}}
=> {{1,2,4,6,9},{3,10},{5},{7},{8}}
=> ? = 11
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> {{1,2},{3,4},{5,8},{6,7},{9,10}}
=> {{1,2,4,7,10},{3,8},{5},{6},{9}}
=> ? = 11
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> {{1,2},{3,4},{5,10},{6,7},{8,9}}
=> {{1,2,4,7,9},{3},{5,10},{6},{8}}
=> ? = 9
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> {{1,2},{3,4},{5,10},{6,9},{7,8}}
=> {{1,2,4,8},{3,9},{5,10},{6},{7}}
=> ? = 8
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> {{1,2},{3,6},{4,5},{7,8},{9,10}}
=> {{1,2,5,8,10},{3,6},{4},{7},{9}}
=> ? = 11
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> {{1,2},{3,6},{4,5},{7,10},{8,9}}
=> {{1,2,5,9},{3,6,10},{4},{7},{8}}
=> ? = 8
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> {{1,2},{3,8},{4,5},{6,7},{9,10}}
=> {{1,2,5,7,10},{3},{4,8},{6},{9}}
=> ? = 9
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> {{1,2},{3,10},{4,5},{6,7},{8,9}}
=> {{1,2,5,7,9},{3},{4},{6,10},{8}}
=> ? = 9
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> {{1,2},{3,10},{4,5},{6,9},{7,8}}
=> {{1,2,5,8},{3,9},{4},{6,10},{7}}
=> ? = 7
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> {{1,2},{3,8},{4,7},{5,6},{9,10}}
=> {{1,2,6,10},{3,7},{4,8},{5},{9}}
=> ? = 8
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> {{1,2},{3,10},{4,7},{5,6},{8,9}}
=> {{1,2,6,9},{3,7},{4},{5,10},{8}}
=> ? = 7
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> {{1,2},{3,10},{4,9},{5,6},{7,8}}
=> {{1,2,6,8},{3},{4,9},{5,10},{7}}
=> ? = 7
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> {{1,2},{3,10},{4,9},{5,8},{6,7}}
=> {{1,2,7},{3,8},{4,9},{5,10},{6}}
=> ? = 6
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> {{1,4},{2,3},{5,6},{7,8},{9,10}}
=> {{1,3,6,8,10},{2,4},{5},{7},{9}}
=> ? = 11
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> {{1,4},{2,3},{5,6},{7,10},{8,9}}
=> {{1,3,6,9},{2,4,10},{5},{7},{8}}
=> ? = 8
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> {{1,4},{2,3},{5,8},{6,7},{9,10}}
=> {{1,3,7,10},{2,4,8},{5},{6},{9}}
=> ? = 8
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> {{1,4},{2,3},{5,10},{6,7},{8,9}}
=> {{1,3,7,9},{2,4},{5,10},{6},{8}}
=> ? = 7
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> {{1,4},{2,3},{5,10},{6,9},{7,8}}
=> {{1,3,8},{2,4,9},{5,10},{6},{7}}
=> ? = 6
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> {{1,6},{2,3},{4,5},{7,8},{9,10}}
=> {{1,3,5,8,10},{2},{4,6},{7},{9}}
=> ? = 9
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> {{1,6},{2,3},{4,5},{7,10},{8,9}}
=> {{1,3,5,9},{2,10},{4,6},{7},{8}}
=> ? = 7
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> {{1,8},{2,3},{4,5},{6,7},{9,10}}
=> {{1,3,5,7,10},{2},{4},{6,8},{9}}
=> ? = 9
[1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> {{1,10},{2,3},{4,5},{6,7},{8,9}}
=> {{1,3,5,7,9},{2},{4},{6},{8,10}}
=> ? = 11
[1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> {{1,10},{2,3},{4,5},{6,9},{7,8}}
=> {{1,3,5,8},{2,9},{4},{6},{7,10}}
=> ? = 8
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> {{1,8},{2,3},{4,7},{5,6},{9,10}}
=> {{1,3,6,10},{2,7},{4},{5,8},{9}}
=> ? = 7
[1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> {{1,10},{2,3},{4,7},{5,6},{8,9}}
=> {{1,3,6,9},{2,7},{4},{5},{8,10}}
=> ? = 8
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> {{1,10},{2,3},{4,9},{5,6},{7,8}}
=> {{1,3,6,8},{2},{4,9},{5},{7,10}}
=> ? = 7
[1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> {{1,10},{2,3},{4,9},{5,8},{6,7}}
=> {{1,3,7},{2,8},{4,9},{5},{6,10}}
=> ? = 6
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> {{1,6},{2,5},{3,4},{7,8},{9,10}}
=> {{1,4,8,10},{2,5},{3,6},{7},{9}}
=> ? = 8
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> {{1,6},{2,5},{3,4},{7,10},{8,9}}
=> {{1,4,9},{2,5,10},{3,6},{7},{8}}
=> ? = 6
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> {{1,8},{2,5},{3,4},{6,7},{9,10}}
=> {{1,4,7,10},{2,5},{3},{6,8},{9}}
=> ? = 7
[1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> {{1,10},{2,5},{3,4},{6,7},{8,9}}
=> {{1,4,7,9},{2,5},{3},{6},{8,10}}
=> ? = 8
[1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> {{1,10},{2,5},{3,4},{6,9},{7,8}}
=> {{1,4,8},{2,5,9},{3},{6},{7,10}}
=> ? = 6
[1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> {{1,8},{2,7},{3,4},{5,6},{9,10}}
=> {{1,4,6,10},{2},{3,7},{5,8},{9}}
=> ? = 7
[1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> {{1,10},{2,7},{3,4},{5,6},{8,9}}
=> {{1,4,6,9},{2},{3,7},{5},{8,10}}
=> ? = 7
[1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> {{1,10},{2,9},{3,4},{5,6},{7,8}}
=> {{1,4,6,8},{2},{3},{5,9},{7,10}}
=> ? = 8
Description
The number of occurrences of the pattern {{1,2}} in a set partition.
Matching statistic: St000809
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000809: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 38%
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000809: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 38%
Values
[1,0]
=> [(1,2)]
=> [2,1] => [2,1] => 1
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,4,1,3] => 3
[1,1,0,0]
=> [(1,4),(2,3)]
=> [4,3,2,1] => [4,3,2,1] => 2
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,4,6,1,3,5] => 6
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [6,5,2,4,1,3] => 4
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [4,3,2,6,1,5] => 4
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [3,6,2,5,4,1] => 4
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => 3
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => ? = 10
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,8,7,6,5] => [8,7,2,4,6,1,3,5] => ? = 7
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [6,5,2,4,8,1,3,7] => ? = 7
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [2,8,5,7,4,6,1,3] => ? = 6
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => [8,7,6,5,2,4,1,3] => ? = 5
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [4,3,2,6,8,1,5,7] => ? = 7
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => [4,8,3,7,2,6,1,5] => ? = 5
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [3,6,2,5,4,8,1,7] => ? = 6
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [3,5,8,2,4,7,6,1] => ? = 7
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [8,3,2,7,6,5,4,1] => [8,7,3,6,2,5,4,1] => ? = 5
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [6,5,4,3,2,8,1,7] => ? = 5
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [5,4,3,8,2,7,6,1] => ? = 5
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [4,8,3,7,6,5,2,1] => ? = 5
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => ? = 4
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => ? = 15
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,10,9,8,7] => [10,9,2,4,6,8,1,3,5,7] => ? = 11
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,8,7,6,5,10,9] => [8,7,2,4,6,10,1,3,5,9] => ? = 11
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,10,7,6,9,8,5] => [2,10,7,9,4,6,8,1,3,5] => ? = 9
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,10,9,8,7,6,5] => [10,9,8,7,2,4,6,1,3,5] => ? = 8
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,6,5,4,3,8,7,10,9] => [6,5,2,4,8,10,1,3,7,9] => ? = 11
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [2,1,6,5,4,3,10,9,8,7] => [6,10,5,9,2,4,8,1,3,7] => ? = 8
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,8,5,4,7,6,3,10,9] => [2,8,5,7,4,6,10,1,3,9] => ? = 9
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,10,5,4,7,6,9,8,3] => [10,5,2,4,7,9,6,8,1,3] => ? = 9
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,10,5,4,9,8,7,6,3] => [10,9,2,8,5,7,4,6,1,3] => ? = 7
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,8,7,6,5,4,3,10,9] => [8,7,6,5,2,4,10,1,3,9] => ? = 8
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,10,7,6,5,4,9,8,3] => [10,7,2,6,5,9,4,8,1,3] => ? = 7
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,10,9,6,5,8,7,4,3] => [6,10,5,9,8,7,2,4,1,3] => ? = 7
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,10,9,8,7,6,5,4,3] => [10,9,8,7,6,5,2,4,1,3] => ? = 6
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [4,3,2,1,6,5,8,7,10,9] => [4,3,2,6,8,10,1,5,7,9] => ? = 11
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [4,3,2,1,6,5,10,9,8,7] => [4,10,3,9,2,6,8,1,5,7] => ? = 8
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [4,3,2,1,8,7,6,5,10,9] => [4,8,3,7,2,6,10,1,5,9] => ? = 8
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [4,3,2,1,10,7,6,9,8,5] => [4,3,10,7,9,2,6,8,1,5] => ? = 7
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [4,3,2,1,10,9,8,7,6,5] => [10,9,4,8,3,7,2,6,1,5] => ? = 6
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [6,3,2,5,4,1,8,7,10,9] => [3,6,2,5,4,8,10,1,7,9] => ? = 9
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [6,3,2,5,4,1,10,9,8,7] => [3,6,10,2,5,9,4,8,1,7] => ? = 7
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [8,3,2,5,4,7,6,1,10,9] => [3,5,8,2,4,7,6,10,1,9] => ? = 9
[1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [10,3,2,5,4,7,6,9,8,1] => [3,5,7,10,2,4,6,9,8,1] => ? = 11
[1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [10,3,2,5,4,9,8,7,6,1] => [10,9,3,5,8,2,4,7,6,1] => ? = 8
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [8,3,2,7,6,5,4,1,10,9] => [8,7,3,6,2,5,4,10,1,9] => ? = 7
[1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [10,3,2,7,6,5,4,9,8,1] => [7,6,3,5,10,2,4,9,8,1] => ? = 8
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [10,3,2,9,6,5,8,7,4,1] => [3,6,10,9,8,2,5,7,4,1] => ? = 7
[1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> [10,3,2,9,8,7,6,5,4,1] => [10,9,8,7,3,6,2,5,4,1] => ? = 6
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [6,5,4,3,2,1,8,7,10,9] => [6,5,4,3,2,8,10,1,7,9] => ? = 8
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [6,5,4,3,2,1,10,9,8,7] => [6,5,4,10,3,9,2,8,1,7] => ? = 6
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [8,5,4,3,2,7,6,1,10,9] => [5,4,3,8,2,7,6,10,1,9] => ? = 7
[1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [10,5,4,3,2,7,6,9,8,1] => [5,4,3,7,10,2,6,9,8,1] => ? = 8
[1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> [10,5,4,3,2,9,8,7,6,1] => [5,10,4,9,3,8,2,7,6,1] => ? = 6
[1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [8,7,4,3,6,5,2,1,10,9] => [4,8,3,7,6,5,2,10,1,9] => ? = 7
[1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [10,7,4,3,6,5,2,9,8,1] => [4,7,3,6,5,10,2,9,8,1] => ? = 7
[1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [10,9,4,3,6,5,8,7,2,1] => [4,6,10,3,5,9,8,7,2,1] => ? = 8
Description
The reduced reflection length of the permutation.
Let $T$ be the set of reflections in a Coxeter group and let $\ell(w)$ be the usual length function. Then the reduced reflection length of $w$ is
$$\min\{r\in\mathbb N \mid w = t_1\cdots t_r,\quad t_1,\dots,t_r \in T,\quad \ell(w)=\sum \ell(t_i)\}.$$
In the case of the symmetric group, this is twice the depth [[St000029]] minus the usual length [[St000018]].
Matching statistic: St001874
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St001874: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 38%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St001874: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 38%
Values
[1,0]
=> [1,0]
=> [(1,2)]
=> [2,1] => 1
[1,0,1,0]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => 3
[1,1,0,0]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => 6
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 4
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 4
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 4
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => ? = 10
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 7
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? = 7
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => ? = 6
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 5
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ? = 7
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 5
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => ? = 6
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => ? = 7
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => ? = 5
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? = 5
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => ? = 5
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => ? = 5
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => ? = 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [6,7,8,9,10,5,4,3,2,1] => ? = 15
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [5,7,8,9,4,10,6,3,2,1] => ? = 11
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [5,6,8,9,4,3,10,7,2,1] => ? = 11
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [4,7,8,3,9,10,6,5,2,1] => ? = 9
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,2,1] => ? = 8
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [5,6,7,9,4,3,2,10,8,1] => ? = 11
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,2,10,8,1] => ? = 8
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> [4,5,8,3,2,9,10,7,6,1] => ? = 9
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> [3,7,2,8,9,10,6,5,4,1] => ? = 9
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [3,6,2,8,9,5,10,7,4,1] => ? = 7
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [4,5,7,3,2,9,6,10,8,1] => ? = 8
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [3,6,2,7,9,5,4,10,8,1] => ? = 7
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [3,5,2,8,4,9,10,7,6,1] => ? = 7
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [3,5,2,7,4,9,6,10,8,1] => ? = 6
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [5,6,7,8,4,3,2,1,10,9] => ? = 11
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,2,1,10,9] => ? = 8
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => ? = 8
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [3,6,2,7,8,5,4,1,10,9] => ? = 7
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [3,5,2,7,4,8,6,1,10,9] => ? = 6
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [4,5,6,3,2,1,9,10,8,7] => ? = 9
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [3,5,2,6,4,1,9,10,8,7] => ? = 7
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [3,4,2,1,8,9,10,7,6,5] => ? = 9
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => ? = 11
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => ? = 8
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [3,4,2,1,7,9,6,10,8,5] => ? = 7
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => ? = 8
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,5,8,4,9,10,7,6,3] => ? = 7
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => ? = 6
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => ? = 8
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [3,5,2,6,4,1,8,7,10,9] => ? = 6
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [3,4,2,1,7,8,6,5,10,9] => ? = 7
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,6,7,8,5,4,3,10,9] => ? = 8
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,5,7,4,8,6,3,10,9] => ? = 6
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [3,4,2,1,6,5,9,10,8,7] => ? = 7
[1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [2,1,5,6,4,3,9,10,8,7] => ? = 7
[1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => ? = 8
Description
Lusztig's a-function for the symmetric group.
Let $x$ be a permutation corresponding to the pair of tableaux $(P(x),Q(x))$
by the Robinson-Schensted correspondence and
$\operatorname{shape}(Q(x)')=( \lambda_1,...,\lambda_k)$
where $Q(x)'$ is the transposed tableau.
Then $a(x)=\sum\limits_{i=1}^{k}{\binom{\lambda_i}{2}}$.
See exercise 10 on page 198 in the book by Björner and Brenti "Combinatorics of Coxeter Groups" for equivalent characterisations and properties.
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