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Your data matches 45 different statistics following compositions of up to 3 maps.
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Matching statistic: St001060
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[[],[],[],[],[[]]]
=> [[[[[.,.],.],.],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[],[],[],[[],[]]]
=> [[[[.,.],.],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[],[],[],[[[]]]]
=> [[[[.,.],.],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[],[],[[]],[[]]]
=> [[[[.,.],.],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[],[[],[],[]]]
=> [[[.,.],.],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[],[],[[],[[]]]]
=> [[[.,.],.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[],[],[[[]],[]]]
=> [[[.,.],.],[[.,[.,.]],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[],[],[[[],[]]]]
=> [[[.,.],.],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[],[],[[[[]]]]]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[],[[]],[],[[]]]
=> [[[[.,.],[.,.]],.],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[]],[[],[]]]
=> [[[.,.],[.,.]],[[.,.],.]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[],[[]],[[[]]]]
=> [[[.,.],[.,.]],[.,[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[],[[],[]],[[]]]
=> [[[.,.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[[]]],[[]]]
=> [[[.,.],[.,[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[],[],[],[]]]
=> [[.,.],[[[[.,.],.],.],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[],[[],[],[[]]]]
=> [[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[],[[]],[]]]
=> [[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[],[[],[]]]]
=> [[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[],[[[]]]]]
=> [[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[[]],[],[]]]
=> [[.,.],[[[.,[.,.]],.],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[],[[[]],[[]]]]
=> [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[[],[]],[]]]
=> [[.,.],[[.,[[.,.],.]],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[],[[[[]]],[]]]
=> [[.,.],[[.,[.,[.,.]]],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[],[[[],[],[]]]]
=> [[.,.],[.,[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[],[[[],[[]]]]]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[],[[[[]],[]]]]
=> [[.,.],[.,[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[],[[[[],[]]]]]
=> [[.,.],[.,[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[],[[[[[]]]]]]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[[]],[],[],[[]]]
=> [[[[.,[.,.]],.],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[[]],[],[[],[]]]
=> [[[.,[.,.]],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[[]],[],[[[]]]]
=> [[[.,[.,.]],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[[]],[[]],[[]]]
=> [[[.,[.,.]],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St001668
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001668: Posets ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 67%
Mp00013: Binary trees —to poset⟶ Posets
St001668: Posets ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 67%
Values
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 3
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 3
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 3
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 3
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 3
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 3
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 3
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 3
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 3
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 3
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 3
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 3
[[],[],[],[],[[]]]
=> [[[[[.,.],.],.],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4
[[],[],[],[[],[]]]
=> [[[[.,.],.],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 2
[[],[],[],[[[]]]]
=> [[[[.,.],.],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 2
[[],[],[[]],[[]]]
=> [[[[.,.],.],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 2
[[],[],[[],[],[]]]
=> [[[.,.],.],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 2
[[],[],[[],[[]]]]
=> [[[.,.],.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 2
[[],[],[[[]],[]]]
=> [[[.,.],.],[[.,[.,.]],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 2
[[],[],[[[],[]]]]
=> [[[.,.],.],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 2
[[],[],[[[[]]]]]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 2
[[],[[]],[],[[]]]
=> [[[[.,.],[.,.]],.],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ? = 2
[[],[[]],[[],[]]]
=> [[[.,.],[.,.]],[[.,.],.]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 2
[[],[[]],[[[]]]]
=> [[[.,.],[.,.]],[.,[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 2
[[],[[],[]],[[]]]
=> [[[.,.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 2
[[],[[[]]],[[]]]
=> [[[.,.],[.,[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 2
[[],[[],[],[],[]]]
=> [[.,.],[[[[.,.],.],.],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4
[[],[[],[],[[]]]]
=> [[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 2
[[],[[],[[]],[]]]
=> [[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ? = 2
[[],[[],[[],[]]]]
=> [[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 2
[[],[[],[[[]]]]]
=> [[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 2
[[],[[[]],[],[]]]
=> [[.,.],[[[.,[.,.]],.],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4
[[],[[[]],[[]]]]
=> [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 2
[[],[[[],[]],[]]]
=> [[.,.],[[.,[[.,.],.]],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4
[[],[[[[]]],[]]]
=> [[.,.],[[.,[.,[.,.]]],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4
[[],[[[],[],[]]]]
=> [[.,.],[.,[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4
[[],[[[],[[]]]]]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ? = 2
[[],[[[[]],[]]]]
=> [[.,.],[.,[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4
[[],[[[[],[]]]]]
=> [[.,.],[.,[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4
[[],[[[[[]]]]]]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4
[[[]],[],[],[[]]]
=> [[[[.,[.,.]],.],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4
[[[]],[],[[],[]]]
=> [[[.,[.,.]],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 2
[[[]],[],[[[]]]]
=> [[[.,[.,.]],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 2
[[[]],[[]],[[]]]
=> [[[.,[.,.]],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 2
[[[]],[[],[],[]]]
=> [[.,[.,.]],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 2
[[[]],[[],[[]]]]
=> [[.,[.,.]],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 2
[[[]],[[[]],[]]]
=> [[.,[.,.]],[[.,[.,.]],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 2
[[[]],[[[],[]]]]
=> [[.,[.,.]],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 2
[[[]],[[[[]]]]]
=> [[.,[.,.]],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 2
[[[],[]],[],[[]]]
=> [[[.,[[.,.],.]],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4
[[[[]]],[],[[]]]
=> [[[.,[.,[.,.]]],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4
[[[],[]],[[],[]]]
=> [[.,[[.,.],.]],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 2
[[[],[]],[[[]]]]
=> [[.,[[.,.],.]],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 2
[[[[]]],[[],[]]]
=> [[.,[.,[.,.]]],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 2
[[[[]]],[[[]]]]
=> [[.,[.,[.,.]]],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 2
[[[],[],[]],[[]]]
=> [[.,[[[.,.],.],.]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4
[[[],[[]]],[[]]]
=> [[.,[[.,.],[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ? = 2
[[[[]],[]],[[]]]
=> [[.,[[.,[.,.]],.]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4
[[[[],[]]],[[]]]
=> [[.,[.,[[.,.],.]]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4
[[[[[]]]],[[]]]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4
Description
The number of points of the poset minus the width of the poset.
Matching statistic: St001515
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001515: Dyck paths ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 67%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001515: Dyck paths ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 67%
Values
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 3 + 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4 = 3 + 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 3 + 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 3 + 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4 = 3 + 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4 = 3 + 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4 = 3 + 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4 = 3 + 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]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 4 + 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]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 2 + 1
[[],[],[],[[[]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 2 + 1
[[],[],[[]],[[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 2 + 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]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 2 + 1
[[],[],[[],[[]]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 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]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 2 + 1
[[],[],[[[],[]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 + 1
[[],[],[[[[]]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 2 + 1
[[],[[]],[],[[]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 2 + 1
[[],[[]],[[],[]]]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2 + 1
[[],[[]],[[[]]]]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 2 + 1
[[],[[],[]],[[]]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 1
[[],[[[]]],[[]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 2 + 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]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[[],[[],[],[[]]]]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 2 + 1
[[],[[],[[]],[]]]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 1
[[],[[],[[],[]]]]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 2 + 1
[[],[[],[[[]]]]]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 2 + 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]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> ? = 4 + 1
[[],[[[]],[[]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 2 + 1
[[],[[[],[]],[]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 4 + 1
[[],[[[[]]],[]]]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 4 + 1
[[],[[[],[],[]]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 4 + 1
[[],[[[],[[]]]]]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[[],[[[[]],[]]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 4 + 1
[[],[[[[],[]]]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 4 + 1
[[],[[[[[]]]]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 4 + 1
[[[]],[],[],[[]]]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 4 + 1
[[[]],[],[[],[]]]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 + 1
[[[]],[],[[[]]]]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 + 1
[[[]],[[]],[[]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2 + 1
[[[]],[[],[],[]]]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[[[]],[[],[[]]]]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 2 + 1
[[[]],[[[]],[]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
[[[]],[[[],[]]]]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 2 + 1
[[[]],[[[[]]]]]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 + 1
[[[],[]],[],[[]]]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 4 + 1
[[[[]]],[],[[]]]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 4 + 1
[[[],[]],[[],[]]]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 2 + 1
[[[],[]],[[[]]]]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 2 + 1
[[[[]]],[[],[]]]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 2 + 1
[[[[]]],[[[]]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 2 + 1
[[[],[],[]],[[]]]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4 + 1
[[[],[[]]],[[]]]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 2 + 1
[[[[]],[]],[[]]]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
[[[[],[]]],[[]]]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 4 + 1
[[[[[]]]],[[]]]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 4 + 1
Description
The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule).
Matching statistic: St001690
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001690: Graphs ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 67%
Mp00011: Binary trees —to graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001690: Graphs ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 67%
Values
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 5 = 3 + 2
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 5 = 3 + 2
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 5 = 3 + 2
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 5 = 3 + 2
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 5 = 3 + 2
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 5 = 3 + 2
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 5 = 3 + 2
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 5 = 3 + 2
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 5 = 3 + 2
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 5 = 3 + 2
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 5 = 3 + 2
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 5 = 3 + 2
[[],[],[],[],[[]]]
=> [[[[[.,.],.],.],.],[.,.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 + 2
[[],[],[],[[],[]]]
=> [[[[.,.],.],.],[[.,.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 2
[[],[],[],[[[]]]]
=> [[[[.,.],.],.],[.,[.,.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 2
[[],[],[[]],[[]]]
=> [[[[.,.],.],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[[],[],[[],[],[]]]
=> [[[.,.],.],[[[.,.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 2
[[],[],[[],[[]]]]
=> [[[.,.],.],[[.,.],[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[[],[],[[[]],[]]]
=> [[[.,.],.],[[.,[.,.]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 2
[[],[],[[[],[]]]]
=> [[[.,.],.],[.,[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 2
[[],[],[[[[]]]]]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 2
[[],[[]],[],[[]]]
=> [[[[.,.],[.,.]],.],[.,.]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[[],[[]],[[],[]]]
=> [[[.,.],[.,.]],[[.,.],.]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[[],[[]],[[[]]]]
=> [[[.,.],[.,.]],[.,[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[[],[[],[]],[[]]]
=> [[[.,.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[[],[[[]]],[[]]]
=> [[[.,.],[.,[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[[],[[],[],[],[]]]
=> [[.,.],[[[[.,.],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 + 2
[[],[[],[],[[]]]]
=> [[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[[],[[],[[]],[]]]
=> [[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[[],[[],[[],[]]]]
=> [[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[[],[[],[[[]]]]]
=> [[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[[],[[[]],[],[]]]
=> [[.,.],[[[.,[.,.]],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 + 2
[[],[[[]],[[]]]]
=> [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[[],[[[],[]],[]]]
=> [[.,.],[[.,[[.,.],.]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 + 2
[[],[[[[]]],[]]]
=> [[.,.],[[.,[.,[.,.]]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 + 2
[[],[[[],[],[]]]]
=> [[.,.],[.,[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 + 2
[[],[[[],[[]]]]]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[[],[[[[]],[]]]]
=> [[.,.],[.,[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 + 2
[[],[[[[],[]]]]]
=> [[.,.],[.,[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 + 2
[[],[[[[[]]]]]]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 + 2
[[[]],[],[],[[]]]
=> [[[[.,[.,.]],.],.],[.,.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 + 2
[[[]],[],[[],[]]]
=> [[[.,[.,.]],.],[[.,.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 2
[[[]],[],[[[]]]]
=> [[[.,[.,.]],.],[.,[.,.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 2
[[[]],[[]],[[]]]
=> [[[.,[.,.]],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[[[]],[[],[],[]]]
=> [[.,[.,.]],[[[.,.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 2
[[[]],[[],[[]]]]
=> [[.,[.,.]],[[.,.],[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[[[]],[[[]],[]]]
=> [[.,[.,.]],[[.,[.,.]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 2
[[[]],[[[],[]]]]
=> [[.,[.,.]],[.,[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 2
[[[]],[[[[]]]]]
=> [[.,[.,.]],[.,[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 2
[[[],[]],[],[[]]]
=> [[[.,[[.,.],.]],.],[.,.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 + 2
[[[[]]],[],[[]]]
=> [[[.,[.,[.,.]]],.],[.,.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 + 2
[[[],[]],[[],[]]]
=> [[.,[[.,.],.]],[[.,.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 2
[[[],[]],[[[]]]]
=> [[.,[[.,.],.]],[.,[.,.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 2
[[[[]]],[[],[]]]
=> [[.,[.,[.,.]]],[[.,.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 2
[[[[]]],[[[]]]]
=> [[.,[.,[.,.]]],[.,[.,.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 2
[[[],[],[]],[[]]]
=> [[.,[[[.,.],.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 + 2
[[[],[[]]],[[]]]
=> [[.,[[.,.],[.,.]]],[.,.]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[[[[]],[]],[[]]]
=> [[.,[[.,[.,.]],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 + 2
[[[[],[]]],[[]]]
=> [[.,[.,[[.,.],.]]],[.,.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 + 2
[[[[[]]]],[[]]]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 + 2
Description
The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path.
Put differently, for every vertex $v$ of such a path $P$, there is a vertex $w\in P$ and a vertex $u\not\in P$ such that $(v, u)$ and $(u, w)$ are edges.
The length of such a path is $0$ if the graph is a forest.
It is maximal, if and only if the graph is obtained from a graph $H$ with a Hamiltonian path by joining a new vertex to each of the vertices of $H$.
Matching statistic: St001170
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
St001170: Dyck paths ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 67%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
St001170: Dyck paths ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 67%
Values
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 5 = 2 + 3
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5 = 2 + 3
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 5 = 2 + 3
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 5 = 2 + 3
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 6 = 3 + 3
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 6 = 3 + 3
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 6 = 3 + 3
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 5 = 2 + 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 6 = 3 + 3
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 5 = 2 + 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 6 = 3 + 3
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 6 = 3 + 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 6 = 3 + 3
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 6 = 3 + 3
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 6 = 3 + 3
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 6 = 3 + 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 6 = 3 + 3
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 6 = 3 + 3
[[],[],[],[],[[]]]
=> [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]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4 + 3
[[],[],[],[[],[]]]
=> [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]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 3
[[],[],[],[[[]]]]
=> [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]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 3
[[],[],[[]],[[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 2 + 3
[[],[],[[],[],[]]]
=> [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]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 3
[[],[],[[],[[]]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 2 + 3
[[],[],[[[]],[]]]
=> [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]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 2 + 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]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 2 + 3
[[],[],[[[[]]]]]
=> [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]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 3
[[],[[]],[],[[]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 2 + 3
[[],[[]],[[],[]]]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 2 + 3
[[],[[]],[[[]]]]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 2 + 3
[[],[[],[]],[[]]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 2 + 3
[[],[[[]]],[[]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 2 + 3
[[],[[],[],[],[]]]
=> [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]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 4 + 3
[[],[[],[],[[]]]]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 2 + 3
[[],[[],[[]],[]]]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 2 + 3
[[],[[],[[],[]]]]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 2 + 3
[[],[[],[[[]]]]]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 3
[[],[[[]],[],[]]]
=> [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]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 4 + 3
[[],[[[]],[[]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 2 + 3
[[],[[[],[]],[]]]
=> [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]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 4 + 3
[[],[[[[]]],[]]]
=> [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]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 4 + 3
[[],[[[],[],[]]]]
=> [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]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 4 + 3
[[],[[[],[[]]]]]
=> [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]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 + 3
[[],[[[[]],[]]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 4 + 3
[[],[[[[],[]]]]]
=> [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]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 + 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]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 4 + 3
[[[]],[],[],[[]]]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4 + 3
[[[]],[],[[],[]]]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 3
[[[]],[],[[[]]]]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 3
[[[]],[[]],[[]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 2 + 3
[[[]],[[],[],[]]]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 3
[[[]],[[],[[]]]]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 2 + 3
[[[]],[[[]],[]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 2 + 3
[[[]],[[[],[]]]]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 2 + 3
[[[]],[[[[]]]]]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 3
[[[],[]],[],[[]]]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 4 + 3
[[[[]]],[],[[]]]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 4 + 3
[[[],[]],[[],[]]]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 2 + 3
[[[],[]],[[[]]]]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 2 + 3
[[[[]]],[[],[]]]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 2 + 3
[[[[]]],[[[]]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 2 + 3
[[[],[],[]],[[]]]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 4 + 3
[[[],[[]]],[[]]]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 2 + 3
[[[[]],[]],[[]]]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 4 + 3
[[[[],[]]],[[]]]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> ? = 4 + 3
[[[[[]]]],[[]]]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 4 + 3
Description
Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra.
Matching statistic: St001237
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001237: Dyck paths ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 67%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001237: Dyck paths ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 67%
Values
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5 = 2 + 3
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 5 = 2 + 3
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 5 = 2 + 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]
=> 5 = 2 + 3
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 6 = 3 + 3
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 6 = 3 + 3
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 6 = 3 + 3
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 5 = 2 + 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 6 = 3 + 3
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 5 = 2 + 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 6 = 3 + 3
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 6 = 3 + 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 6 = 3 + 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]
=> 6 = 3 + 3
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 6 = 3 + 3
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 6 = 3 + 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 6 = 3 + 3
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 6 = 3 + 3
[[],[],[],[],[[]]]
=> [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]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 4 + 3
[[],[],[],[[],[]]]
=> [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]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2 + 3
[[],[],[],[[[]]]]
=> [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]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 2 + 3
[[],[],[[]],[[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> ? = 2 + 3
[[],[],[[],[],[]]]
=> [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]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 2 + 3
[[],[],[[],[[]]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 2 + 3
[[],[],[[[]],[]]]
=> [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]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 + 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]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2 + 3
[[],[],[[[[]]]]]
=> [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]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 + 3
[[],[[]],[],[[]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 2 + 3
[[],[[]],[[],[]]]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 2 + 3
[[],[[]],[[[]]]]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 2 + 3
[[],[[],[]],[[]]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> ? = 2 + 3
[[],[[[]]],[[]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2 + 3
[[],[[],[],[],[]]]
=> [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]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 4 + 3
[[],[[],[],[[]]]]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 2 + 3
[[],[[],[[]],[]]]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 2 + 3
[[],[[],[[],[]]]]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 2 + 3
[[],[[],[[[]]]]]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 2 + 3
[[],[[[]],[],[]]]
=> [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]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 4 + 3
[[],[[[]],[[]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> ? = 2 + 3
[[],[[[],[]],[]]]
=> [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]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 4 + 3
[[],[[[[]]],[]]]
=> [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]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 4 + 3
[[],[[[],[],[]]]]
=> [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]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 4 + 3
[[],[[[],[[]]]]]
=> [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]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 2 + 3
[[],[[[[]],[]]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 4 + 3
[[],[[[[],[]]]]]
=> [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]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 4 + 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]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 3
[[[]],[],[],[[]]]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 4 + 3
[[[]],[],[[],[]]]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 + 3
[[[]],[],[[[]]]]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 2 + 3
[[[]],[[]],[[]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 2 + 3
[[[]],[[],[],[]]]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 2 + 3
[[[]],[[],[[]]]]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> ? = 2 + 3
[[[]],[[[]],[]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 2 + 3
[[[]],[[[],[]]]]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 + 3
[[[]],[[[[]]]]]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 2 + 3
[[[],[]],[],[[]]]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 4 + 3
[[[[]]],[],[[]]]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 4 + 3
[[[],[]],[[],[]]]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 + 3
[[[],[]],[[[]]]]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 2 + 3
[[[[]]],[[],[]]]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 2 + 3
[[[[]]],[[[]]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 2 + 3
[[[],[],[]],[[]]]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> ? = 4 + 3
[[[],[[]]],[[]]]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 2 + 3
[[[[]],[]],[[]]]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 4 + 3
[[[[],[]]],[[]]]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 4 + 3
[[[[[]]]],[[]]]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 4 + 3
Description
The number of simple modules with injective dimension at most one or dominant dimension at least one.
Matching statistic: St001879
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00009: Binary trees —left rotate⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Mp00009: Binary trees —left rotate⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Values
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 2 + 1
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3 + 1
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 1
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 2 + 1
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3 + 1
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 2 + 1
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3 + 1
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3 + 1
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3 + 1
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3 + 1
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 1
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[[],[],[],[],[[]]]
=> [[[[[.,.],.],.],.],[.,.]]
=> [[[[[[.,.],.],.],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[[],[],[],[[],[]]]
=> [[[[.,.],.],.],[[.,.],.]]
=> [[[[[.,.],.],.],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 2 + 1
[[],[],[],[[[]]]]
=> [[[[.,.],.],.],[.,[.,.]]]
=> [[[[[.,.],.],.],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 2 + 1
[[],[],[[]],[[]]]
=> [[[[.,.],.],[.,.]],[.,.]]
=> [[[[[.,.],.],[.,.]],.],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 2 + 1
[[],[],[[],[],[]]]
=> [[[.,.],.],[[[.,.],.],.]]
=> [[[[.,.],.],[[.,.],.]],.]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 2 + 1
[[],[],[[],[[]]]]
=> [[[.,.],.],[[.,.],[.,.]]]
=> [[[[.,.],.],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 2 + 1
[[],[],[[[]],[]]]
=> [[[.,.],.],[[.,[.,.]],.]]
=> [[[[.,.],.],[.,[.,.]]],.]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 2 + 1
[[],[],[[[],[]]]]
=> [[[.,.],.],[.,[[.,.],.]]]
=> [[[[.,.],.],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 2 + 1
[[],[],[[[[]]]]]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 2 + 1
[[],[[]],[],[[]]]
=> [[[[.,.],[.,.]],.],[.,.]]
=> [[[[[.,.],[.,.]],.],.],.]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? = 2 + 1
[[],[[]],[[],[]]]
=> [[[.,.],[.,.]],[[.,.],.]]
=> [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ? = 2 + 1
[[],[[]],[[[]]]]
=> [[[.,.],[.,.]],[.,[.,.]]]
=> [[[[.,.],[.,.]],.],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ? = 2 + 1
[[],[[],[]],[[]]]
=> [[[.,.],[[.,.],.]],[.,.]]
=> [[[[.,.],[[.,.],.]],.],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 2 + 1
[[],[[[]]],[[]]]
=> [[[.,.],[.,[.,.]]],[.,.]]
=> [[[[.,.],[.,[.,.]]],.],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 2 + 1
[[],[[],[],[],[]]]
=> [[.,.],[[[[.,.],.],.],.]]
=> [[[.,.],[[[.,.],.],.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4 + 1
[[],[[],[],[[]]]]
=> [[.,.],[[[.,.],.],[.,.]]]
=> [[[.,.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 2 + 1
[[],[[],[[]],[]]]
=> [[.,.],[[[.,.],[.,.]],.]]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ? = 2 + 1
[[],[[],[[],[]]]]
=> [[.,.],[[.,.],[[.,.],.]]]
=> [[[.,.],[.,.]],[[.,.],.]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 2 + 1
[[],[[],[[[]]]]]
=> [[.,.],[[.,.],[.,[.,.]]]]
=> [[[.,.],[.,.]],[.,[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 2 + 1
[[],[[[]],[],[]]]
=> [[.,.],[[[.,[.,.]],.],.]]
=> [[[.,.],[[.,[.,.]],.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4 + 1
[[],[[[]],[[]]]]
=> [[.,.],[[.,[.,.]],[.,.]]]
=> [[[.,.],[.,[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 2 + 1
[[],[[[],[]],[]]]
=> [[.,.],[[.,[[.,.],.]],.]]
=> [[[.,.],[.,[[.,.],.]]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4 + 1
[[],[[[[]]],[]]]
=> [[.,.],[[.,[.,[.,.]]],.]]
=> [[[.,.],[.,[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4 + 1
[[],[[[],[],[]]]]
=> [[.,.],[.,[[[.,.],.],.]]]
=> [[[.,.],.],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 4 + 1
[[],[[[],[[]]]]]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> [[[.,.],.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 2 + 1
[[],[[[[]],[]]]]
=> [[.,.],[.,[[.,[.,.]],.]]]
=> [[[.,.],.],[[.,[.,.]],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 4 + 1
[[],[[[[],[]]]]]
=> [[.,.],[.,[.,[[.,.],.]]]]
=> [[[.,.],.],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 4 + 1
[[],[[[[[]]]]]]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 4 + 1
[[[]],[],[],[[]]]
=> [[[[.,[.,.]],.],.],[.,.]]
=> [[[[[.,[.,.]],.],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[[[]],[],[[],[]]]
=> [[[.,[.,.]],.],[[.,.],.]]
=> [[[[.,[.,.]],.],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 2 + 1
[[[]],[],[[[]]]]
=> [[[.,[.,.]],.],[.,[.,.]]]
=> [[[[.,[.,.]],.],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 2 + 1
[[[]],[[]],[[]]]
=> [[[.,[.,.]],[.,.]],[.,.]]
=> [[[[.,[.,.]],[.,.]],.],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 2 + 1
[[[]],[[],[],[]]]
=> [[.,[.,.]],[[[.,.],.],.]]
=> [[[.,[.,.]],[[.,.],.]],.]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 2 + 1
[[[]],[[],[[]]]]
=> [[.,[.,.]],[[.,.],[.,.]]]
=> [[[.,[.,.]],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 2 + 1
[[[]],[[[]],[]]]
=> [[.,[.,.]],[[.,[.,.]],.]]
=> [[[.,[.,.]],[.,[.,.]]],.]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 2 + 1
[[[]],[[[],[]]]]
=> [[.,[.,.]],[.,[[.,.],.]]]
=> [[[.,[.,.]],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 2 + 1
[[[]],[[[[]]]]]
=> [[.,[.,.]],[.,[.,[.,.]]]]
=> [[[.,[.,.]],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 2 + 1
[[[],[]],[],[[]]]
=> [[[.,[[.,.],.]],.],[.,.]]
=> [[[[.,[[.,.],.]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[[[[]]],[],[[]]]
=> [[[.,[.,[.,.]]],.],[.,.]]
=> [[[[.,[.,[.,.]]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[[[],[]],[[],[]]]
=> [[.,[[.,.],.]],[[.,.],.]]
=> [[[.,[[.,.],.]],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 2 + 1
[[[],[]],[[[]]]]
=> [[.,[[.,.],.]],[.,[.,.]]]
=> [[[.,[[.,.],.]],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 2 + 1
[[[[]]],[[],[]]]
=> [[.,[.,[.,.]]],[[.,.],.]]
=> [[[.,[.,[.,.]]],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 2 + 1
[[[],[],[]],[[]]]
=> [[.,[[[.,.],.],.]],[.,.]]
=> [[[.,[[[.,.],.],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[[[[]],[]],[[]]]
=> [[.,[[.,[.,.]],.]],[.,.]]
=> [[[.,[[.,[.,.]],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[[[[],[]]],[[]]]
=> [[.,[.,[[.,.],.]]],[.,.]]
=> [[[.,[.,[[.,.],.]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[[[[[]]]],[[]]]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> [[[.,[.,[.,[.,.]]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001880
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00009: Binary trees —left rotate⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Mp00009: Binary trees —left rotate⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Values
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 2 + 2
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 2
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3 + 2
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 2
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 2 + 2
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3 + 2
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 2 + 2
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3 + 2
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3 + 2
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3 + 2
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3 + 2
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 2
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[[],[],[],[],[[]]]
=> [[[[[.,.],.],.],.],[.,.]]
=> [[[[[[.,.],.],.],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[[],[],[],[[],[]]]
=> [[[[.,.],.],.],[[.,.],.]]
=> [[[[[.,.],.],.],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 2 + 2
[[],[],[],[[[]]]]
=> [[[[.,.],.],.],[.,[.,.]]]
=> [[[[[.,.],.],.],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 2 + 2
[[],[],[[]],[[]]]
=> [[[[.,.],.],[.,.]],[.,.]]
=> [[[[[.,.],.],[.,.]],.],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 2 + 2
[[],[],[[],[],[]]]
=> [[[.,.],.],[[[.,.],.],.]]
=> [[[[.,.],.],[[.,.],.]],.]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 2 + 2
[[],[],[[],[[]]]]
=> [[[.,.],.],[[.,.],[.,.]]]
=> [[[[.,.],.],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 2 + 2
[[],[],[[[]],[]]]
=> [[[.,.],.],[[.,[.,.]],.]]
=> [[[[.,.],.],[.,[.,.]]],.]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 2 + 2
[[],[],[[[],[]]]]
=> [[[.,.],.],[.,[[.,.],.]]]
=> [[[[.,.],.],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 2 + 2
[[],[],[[[[]]]]]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 2 + 2
[[],[[]],[],[[]]]
=> [[[[.,.],[.,.]],.],[.,.]]
=> [[[[[.,.],[.,.]],.],.],.]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? = 2 + 2
[[],[[]],[[],[]]]
=> [[[.,.],[.,.]],[[.,.],.]]
=> [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ? = 2 + 2
[[],[[]],[[[]]]]
=> [[[.,.],[.,.]],[.,[.,.]]]
=> [[[[.,.],[.,.]],.],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ? = 2 + 2
[[],[[],[]],[[]]]
=> [[[.,.],[[.,.],.]],[.,.]]
=> [[[[.,.],[[.,.],.]],.],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 2 + 2
[[],[[[]]],[[]]]
=> [[[.,.],[.,[.,.]]],[.,.]]
=> [[[[.,.],[.,[.,.]]],.],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 2 + 2
[[],[[],[],[],[]]]
=> [[.,.],[[[[.,.],.],.],.]]
=> [[[.,.],[[[.,.],.],.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4 + 2
[[],[[],[],[[]]]]
=> [[.,.],[[[.,.],.],[.,.]]]
=> [[[.,.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 2 + 2
[[],[[],[[]],[]]]
=> [[.,.],[[[.,.],[.,.]],.]]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ? = 2 + 2
[[],[[],[[],[]]]]
=> [[.,.],[[.,.],[[.,.],.]]]
=> [[[.,.],[.,.]],[[.,.],.]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 2 + 2
[[],[[],[[[]]]]]
=> [[.,.],[[.,.],[.,[.,.]]]]
=> [[[.,.],[.,.]],[.,[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 2 + 2
[[],[[[]],[],[]]]
=> [[.,.],[[[.,[.,.]],.],.]]
=> [[[.,.],[[.,[.,.]],.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4 + 2
[[],[[[]],[[]]]]
=> [[.,.],[[.,[.,.]],[.,.]]]
=> [[[.,.],[.,[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 2 + 2
[[],[[[],[]],[]]]
=> [[.,.],[[.,[[.,.],.]],.]]
=> [[[.,.],[.,[[.,.],.]]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4 + 2
[[],[[[[]]],[]]]
=> [[.,.],[[.,[.,[.,.]]],.]]
=> [[[.,.],[.,[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4 + 2
[[],[[[],[],[]]]]
=> [[.,.],[.,[[[.,.],.],.]]]
=> [[[.,.],.],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 4 + 2
[[],[[[],[[]]]]]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> [[[.,.],.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 2 + 2
[[],[[[[]],[]]]]
=> [[.,.],[.,[[.,[.,.]],.]]]
=> [[[.,.],.],[[.,[.,.]],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 4 + 2
[[],[[[[],[]]]]]
=> [[.,.],[.,[.,[[.,.],.]]]]
=> [[[.,.],.],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 4 + 2
[[],[[[[[]]]]]]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 4 + 2
[[[]],[],[],[[]]]
=> [[[[.,[.,.]],.],.],[.,.]]
=> [[[[[.,[.,.]],.],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[[[]],[],[[],[]]]
=> [[[.,[.,.]],.],[[.,.],.]]
=> [[[[.,[.,.]],.],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 2 + 2
[[[]],[],[[[]]]]
=> [[[.,[.,.]],.],[.,[.,.]]]
=> [[[[.,[.,.]],.],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 2 + 2
[[[]],[[]],[[]]]
=> [[[.,[.,.]],[.,.]],[.,.]]
=> [[[[.,[.,.]],[.,.]],.],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 2 + 2
[[[]],[[],[],[]]]
=> [[.,[.,.]],[[[.,.],.],.]]
=> [[[.,[.,.]],[[.,.],.]],.]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 2 + 2
[[[]],[[],[[]]]]
=> [[.,[.,.]],[[.,.],[.,.]]]
=> [[[.,[.,.]],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 2 + 2
[[[]],[[[]],[]]]
=> [[.,[.,.]],[[.,[.,.]],.]]
=> [[[.,[.,.]],[.,[.,.]]],.]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 2 + 2
[[[]],[[[],[]]]]
=> [[.,[.,.]],[.,[[.,.],.]]]
=> [[[.,[.,.]],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 2 + 2
[[[]],[[[[]]]]]
=> [[.,[.,.]],[.,[.,[.,.]]]]
=> [[[.,[.,.]],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 2 + 2
[[[],[]],[],[[]]]
=> [[[.,[[.,.],.]],.],[.,.]]
=> [[[[.,[[.,.],.]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[[[[]]],[],[[]]]
=> [[[.,[.,[.,.]]],.],[.,.]]
=> [[[[.,[.,[.,.]]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[[[],[]],[[],[]]]
=> [[.,[[.,.],.]],[[.,.],.]]
=> [[[.,[[.,.],.]],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 2 + 2
[[[],[]],[[[]]]]
=> [[.,[[.,.],.]],[.,[.,.]]]
=> [[[.,[[.,.],.]],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 2 + 2
[[[[]]],[[],[]]]
=> [[.,[.,[.,.]]],[[.,.],.]]
=> [[[.,[.,[.,.]]],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 2 + 2
[[[],[],[]],[[]]]
=> [[.,[[[.,.],.],.]],[.,.]]
=> [[[.,[[[.,.],.],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[[[[]],[]],[[]]]
=> [[.,[[.,[.,.]],.]],[.,.]]
=> [[[.,[[.,[.,.]],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[[[[],[]]],[[]]]
=> [[.,[.,[[.,.],.]]],[.,.]]
=> [[[.,[.,[[.,.],.]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[[[[[]]]],[[]]]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> [[[.,[.,[.,[.,.]]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000741
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[],[[[]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[[]],[[]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 3
[[],[[[],[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 3
[[[]],[[],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[[[]],[[[]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[[[],[]],[[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[[[[]]],[[]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[[],[],[],[],[[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[],[],[],[[],[]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[],[],[],[[[]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[],[],[[]],[[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[],[],[[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[],[],[[],[[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[],[],[[[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[],[],[[[],[]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[],[],[[[[]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(0,6),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,6),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[],[[]],[],[[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[],[[]],[[],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[],[[]],[[[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[],[[],[]],[[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[],[[[]]],[[]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[],[[],[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[],[[],[],[[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[],[[],[[]],[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[],[[],[[],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[],[[],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[],[[[]],[],[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[],[[[],[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[],[[[[]]],[]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[],[[[],[],[]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[],[[[],[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[],[[[[]],[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[],[[[[],[]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(0,6),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,6),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[]],[],[],[[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[]],[],[[],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[[]],[],[[[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[[]],[[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[[]],[[],[[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[[]],[[[]],[]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[[]],[[[],[]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(0,6),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,6),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[[]],[[[[]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[[],[]],[],[[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[[]]],[],[[]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[],[]],[[],[]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[[],[]],[[[]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(0,6),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,6),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[[[]]],[[],[]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(0,6),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,6),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[[[]]],[[[]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[[],[],[]],[[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[],[[]]],[[]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
Description
The Colin de Verdière graph invariant.
Matching statistic: St001719
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 67%
Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 67%
Values
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 3
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 2
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 3
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 2
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 3
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 3
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 3
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 3
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 3
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 3
[[],[],[],[],[[]]]
=> [[[[[.,.],.],.],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4
[[],[],[],[[],[]]]
=> [[[[.,.],.],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 2
[[],[],[],[[[]]]]
=> [[[[.,.],.],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 2
[[],[],[[]],[[]]]
=> [[[[.,.],.],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 2
[[],[],[[],[],[]]]
=> [[[.,.],.],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 2
[[],[],[[],[[]]]]
=> [[[.,.],.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ? = 2
[[],[],[[[]],[]]]
=> [[[.,.],.],[[.,[.,.]],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 2
[[],[],[[[],[]]]]
=> [[[.,.],.],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 2
[[],[],[[[[]]]]]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 2
[[],[[]],[],[[]]]
=> [[[[.,.],[.,.]],.],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ? = 2
[[],[[]],[[],[]]]
=> [[[.,.],[.,.]],[[.,.],.]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ? = 2
[[],[[]],[[[]]]]
=> [[[.,.],[.,.]],[.,[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ? = 2
[[],[[],[]],[[]]]
=> [[[.,.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 2
[[],[[[]]],[[]]]
=> [[[.,.],[.,[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 2
[[],[[],[],[],[]]]
=> [[.,.],[[[[.,.],.],.],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4
[[],[[],[],[[]]]]
=> [[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 2
[[],[[],[[]],[]]]
=> [[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ? = 2
[[],[[],[[],[]]]]
=> [[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 2
[[],[[],[[[]]]]]
=> [[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 2
[[],[[[]],[],[]]]
=> [[.,.],[[[.,[.,.]],.],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4
[[],[[[]],[[]]]]
=> [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 2
[[],[[[],[]],[]]]
=> [[.,.],[[.,[[.,.],.]],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4
[[],[[[[]]],[]]]
=> [[.,.],[[.,[.,[.,.]]],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4
[[],[[[],[],[]]]]
=> [[.,.],[.,[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4
[[],[[[],[[]]]]]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ? = 2
[[],[[[[]],[]]]]
=> [[.,.],[.,[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4
[[],[[[[],[]]]]]
=> [[.,.],[.,[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4
[[],[[[[[]]]]]]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4
[[[]],[],[],[[]]]
=> [[[[.,[.,.]],.],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4
[[[]],[],[[],[]]]
=> [[[.,[.,.]],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 2
[[[]],[],[[[]]]]
=> [[[.,[.,.]],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 2
[[[]],[[]],[[]]]
=> [[[.,[.,.]],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 2
[[[]],[[],[],[]]]
=> [[.,[.,.]],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 2
[[[]],[[],[[]]]]
=> [[.,[.,.]],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ? = 2
[[[]],[[[]],[]]]
=> [[.,[.,.]],[[.,[.,.]],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 2
[[[]],[[[],[]]]]
=> [[.,[.,.]],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 2
[[[]],[[[[]]]]]
=> [[.,[.,.]],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 2
[[[],[]],[],[[]]]
=> [[[.,[[.,.],.]],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4
[[[[]]],[],[[]]]
=> [[[.,[.,[.,.]]],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4
[[[],[]],[[],[]]]
=> [[.,[[.,.],.]],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 2
[[[],[]],[[[]]]]
=> [[.,[[.,.],.]],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 2
[[[[]]],[[],[]]]
=> [[.,[.,[.,.]]],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 2
[[[[]]],[[[]]]]
=> [[.,[.,[.,.]]],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 2
[[[],[],[]],[[]]]
=> [[.,[[[.,.],.],.]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4
Description
The number of shortest chains of small intervals from the bottom to the top in a lattice.
An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.
The following 35 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001820The size of the image of the pop stack sorting operator. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001812The biclique partition number of a graph. St001644The dimension of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001723The differential of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001949The rigidity index of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001286The annihilation number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000778The metric dimension of a graph. St001637The number of (upper) dissectors of a poset. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St000098The chromatic number of a graph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001494The Alon-Tarsi number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001118The acyclic chromatic index of a graph. St001622The number of join-irreducible elements of a lattice. St001345The Hamming dimension of a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral.
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