Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001122
Mapping
Mp00054: Parking functions —to inverse des composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001122: 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 multiplicity of the sign representation in the Kronecker square corresponding to a partition. The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$: $$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$ This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{1^n}$, for $\lambda\vdash n$. It equals $1$ if and only if $\lambda$ is self-conjugate.
Matching statistic: St000454
(load all 2 compositions to match this statistic)
Mapping
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: 100%
Values
[2,1,1] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 1 + 2
[3,1,1] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 1 + 2
[2,1,2] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 1 + 2
[2,1,3] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 1 + 2
[3,1,2] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 1 + 2
[1,2,1,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 2
[2,1,1,1] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 0 + 2
[1,3,1,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 2
[3,1,1,1] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 0 + 2
[1,1,4,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 2
[4,1,1,1] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 0 + 2
[1,2,1,2] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 2
[2,1,1,2] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 0 + 2
[2,1,2,1] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[2,2,1,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 2
[1,2,1,3] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 2
[1,3,1,2] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 2
[2,1,1,3] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 0 + 2
[2,1,3,1] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[2,3,1,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 2
[3,1,1,2] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 0 + 2
[3,1,2,1] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[3,2,1,1] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[1,1,4,2] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 2
[1,2,4,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 2
[2,1,1,4] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[2,1,4,1] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 0 + 2
[2,4,1,1] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[4,1,1,2] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 0 + 2
[4,1,2,1] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[4,2,1,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 2
[1,1,3,3] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 2
[3,1,1,3] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 0 + 2
[3,1,3,1] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[3,3,1,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 2
[1,1,3,4] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 2
[1,4,3,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 2
[3,1,1,4] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[3,1,4,1] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 0 + 2
[3,4,1,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 2
[4,1,1,3] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[4,1,3,1] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 0 + 2
[4,3,1,1] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[2,1,2,2] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 0 + 2
[2,2,1,2] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 2
[1,3,2,2] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 2
[2,1,2,3] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 0 + 2
[2,1,3,2] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[2,2,1,3] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 2
[2,3,1,2] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 2
[2,1,1,1,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[3,1,1,1,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[4,1,1,1,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,1,1,1,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[2,1,1,1,2] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[2,1,1,1,3] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[3,1,1,1,2] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[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 = 1 + 2
[2,1,1,4,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[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 = 1 + 2
[4,1,1,1,2] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[2,1,5,1,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[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 = 1 + 2
[5,1,1,1,2] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[3,1,1,1,3] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[3,1,1,4,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[4,1,1,3,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[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 = 1 + 2
[3,1,5,1,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[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 = 1 + 2
[5,1,1,3,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[4,1,1,4,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[4,1,5,1,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,1,4,1,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[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 = 1 + 2
[2,1,1,2,2] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[2,1,1,2,3] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[3,1,1,2,2] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[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 = 1 + 2
[2,1,1,4,2] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[4,1,1,2,2] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[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 = 1 + 2
[2,1,5,1,2] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[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 = 1 + 2
[5,1,1,2,2] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[2,1,1,3,3] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[3,1,1,2,3] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[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 = 1 + 2
[2,1,1,3,4] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[3,1,1,2,4] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[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 = 1 + 2
[4,1,1,2,3] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[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 = 1 + 2
[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 = 1 + 2
[2,1,5,1,3] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[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 = 1 + 2
[3,1,5,1,2] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[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 = 1 + 2
[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 = 1 + 2
[5,1,1,2,3] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
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)
Mapping
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: 50%
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.