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Matching statistic: St000137
Mp00054: Parking functions —to inverse des composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000137: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000137: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[3,1,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[2,1,2] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[2,1,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[3,1,2] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,1,1,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,3,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[3,1,1,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,4,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[4,1,1,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,2,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,1,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[2,1,2,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[2,2,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,2,1,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,3,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,1,1,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[2,1,3,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[2,3,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[3,1,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[3,1,2,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[3,2,1,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,4,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,2,4,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,1,1,4] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[2,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[2,4,1,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[4,1,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[4,1,2,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[4,2,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,1,3,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[3,1,1,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[3,1,3,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[3,3,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,1,3,4] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,4,3,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[3,1,1,4] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[3,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[3,4,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[4,1,1,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[4,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[4,3,1,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[2,1,2,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[2,2,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,3,2,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[2,1,3,2] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[2,2,1,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,3,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
Description
The Grundy value of an integer partition.
Consider the two-player game on an integer partition.
In each move, a player removes either a box, or a 2x2-configuration of boxes such that the resulting diagram is still a partition.
The first player that cannot move lose. This happens exactly when the empty partition is reached.
The grundy value of an integer partition is defined as the grundy value of this two-player game as defined in [1].
This game was described to me during Norcom 2013, by Urban Larsson, and it seems to be quite difficult to give a good description of the partitions with Grundy value 0.
Matching statistic: St000454
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00054: Parking functions —to inverse des composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 67%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 67%
Values
[2,1,1] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 1 + 1
[3,1,1] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 1 + 1
[2,1,2] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 1 + 1
[2,1,3] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 1 + 1
[3,1,2] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 1 + 1
[1,2,1,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[2,1,1,1] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 1
[1,3,1,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[3,1,1,1] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 1
[1,1,4,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[4,1,1,1] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 1
[1,2,1,2] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[2,1,1,2] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 1
[2,1,2,1] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[2,2,1,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[1,2,1,3] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[1,3,1,2] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[2,1,1,3] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 1
[2,1,3,1] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[2,3,1,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[3,1,1,2] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 1
[3,1,2,1] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[3,2,1,1] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[1,1,4,2] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[1,2,4,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[2,1,1,4] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[2,1,4,1] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 1
[2,4,1,1] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[4,1,1,2] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 1
[4,1,2,1] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[4,2,1,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[1,1,3,3] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[3,1,1,3] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 1
[3,1,3,1] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[3,3,1,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[1,1,3,4] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[1,4,3,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[3,1,1,4] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[3,1,4,1] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 1
[3,4,1,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[4,1,1,3] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[4,1,3,1] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 1
[4,3,1,1] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[2,1,2,2] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 1
[2,2,1,2] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[1,3,2,2] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[2,1,2,3] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 1
[2,1,3,2] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[2,2,1,3] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[2,3,1,2] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[2,1,1,1,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,1,1,1,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[4,1,1,1,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[5,1,1,1,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,1,1,2] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,1,1,3] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,1,1,1,2] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,2,1,1,1] => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,1,4,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,4,1,1,1] => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,1,1,1,2] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,5,1,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,5,1,1,1] => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[5,1,1,1,2] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,1,1,1,3] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,1,1,4,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[4,1,1,3,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[4,3,1,1,1] => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,1,5,1,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,5,1,1,1] => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[5,1,1,3,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[4,1,1,4,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[4,1,5,1,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[5,1,4,1,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[5,4,1,1,1] => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,1,2,2] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,1,2,3] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,1,1,2,2] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,2,1,1,2] => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,1,4,2] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[4,1,1,2,2] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[4,2,1,1,2] => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,5,1,2] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,5,1,1,2] => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[5,1,1,2,2] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,1,3,3] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,1,1,2,3] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,2,1,1,3] => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,1,3,4] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,1,1,2,4] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,2,1,1,4] => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,1,1,2,3] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[4,2,1,1,3] => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,3,1,1,2] => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,5,1,3] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,5,1,1,3] => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,1,5,1,2] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,2,1,5,1] => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,5,1,1,2] => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[5,1,1,2,3] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
Description
The largest eigenvalue of a graph if it is integral.
If a graph is d-regular, then its largest eigenvalue equals d. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001645
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00054: Parking functions —to inverse des composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[2,1,1] => [2,1] => [1,2] => ([(1,2)],3)
=> ? = 1 + 5
[3,1,1] => [2,1] => [1,2] => ([(1,2)],3)
=> ? = 1 + 5
[2,1,2] => [2,1] => [1,2] => ([(1,2)],3)
=> ? = 1 + 5
[2,1,3] => [2,1] => [1,2] => ([(1,2)],3)
=> ? = 1 + 5
[3,1,2] => [2,1] => [1,2] => ([(1,2)],3)
=> ? = 1 + 5
[1,2,1,1] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 5
[2,1,1,1] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 5
[1,3,1,1] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 5
[3,1,1,1] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 5
[1,1,4,1] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 5
[4,1,1,1] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 5
[1,2,1,2] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 5
[2,1,1,2] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 5
[2,1,2,1] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 5
[2,2,1,1] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 5
[1,2,1,3] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 5
[1,3,1,2] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 5
[2,1,1,3] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 5
[2,1,3,1] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 5
[2,3,1,1] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 5
[3,1,1,2] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 5
[3,1,2,1] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 5
[3,2,1,1] => [3,1] => [1,3] => ([(2,3)],4)
=> ? = 0 + 5
[1,1,4,2] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 5
[1,2,4,1] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 5
[2,1,1,4] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 5
[2,1,4,1] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 5
[2,4,1,1] => [3,1] => [1,3] => ([(2,3)],4)
=> ? = 0 + 5
[4,1,1,2] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 5
[4,1,2,1] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 5
[4,2,1,1] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 5
[1,1,3,3] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 5
[3,1,1,3] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 5
[3,1,3,1] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 5
[3,3,1,1] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 5
[1,1,3,4] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 5
[1,4,3,1] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 5
[3,1,1,4] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 5
[3,1,4,1] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 5
[3,4,1,1] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 5
[4,1,1,3] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 5
[4,1,3,1] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 5
[4,3,1,1] => [3,1] => [1,3] => ([(2,3)],4)
=> ? = 0 + 5
[2,1,2,2] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 5
[2,2,1,2] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 5
[1,3,2,2] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 5
[2,1,2,3] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 5
[2,1,3,2] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 5
[2,2,1,3] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 5
[2,3,1,2] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 5
[1,1,1,2,1,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,3,1,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,1,4,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,1,5,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,1,6,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,2,1,2] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,2,2,1,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,2,1,3] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,3,1,2] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,2,3,1,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,1,4,2] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,2,4,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,4,2,1,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,1,5,2] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,2,5,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,5,2,1,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,1,6,2] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,2,6,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,6,1,2,1,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,1,3,3] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,3,3,1,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,1,3,4] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,4,3,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,3,4,1,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,1,5,3] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,3,5,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,5,3,1,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,1,6,3] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,3,6,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,6,1,3,1,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,1,4,4] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,4,4,1,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,1,4,5] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,4,1,5,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,5,1,4,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,1,6,4] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,4,1,6,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,6,1,1,4,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,1,5,5] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,5,1,1,5,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,1,5,6] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,5,1,1,6,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,6,1,1,5,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,2,2,1,2] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,2,2,2,1,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,3,2,2] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,2,2,1,3] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,2,3,1,2] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,2,2,3,1,1] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
[1,1,1,4,2,2] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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)
=> 6 = 1 + 5
Description
The pebbling number of a connected graph.
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