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Matching statistic: St001757

St001757: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => 1
([],2)
 => 2
([(0,1)],2)
 => 2
([],3)
 => 3
([(1,2)],3)
 => 2
([(0,2),(1,2)],3)
 => 3
([(0,1),(0,2),(1,2)],3)
 => 6
([],4)
 => 8
([(2,3)],4)
 => 8
([(1,3),(2,3)],4)
 => 4
([(0,3),(1,3),(2,3)],4)
 => 8
([(0,3),(1,2)],4)
 => 8
([(0,3),(1,2),(2,3)],4)
 => 8
([(1,2),(1,3),(2,3)],4)
 => 4
([(0,3),(1,2),(1,3),(2,3)],4)
 => 8
([(0,2),(0,3),(1,2),(1,3)],4)
 => 16
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 16
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 24
([],5)
 => 30
([(3,4)],5)
 => 24
([(2,4),(3,4)],5)
 => 18
([(1,4),(2,4),(3,4)],5)
 => 12
([(0,4),(1,4),(2,4),(3,4)],5)
 => 30
([(1,4),(2,3)],5)
 => 18
([(1,4),(2,3),(3,4)],5)
 => 12
([(0,1),(2,4),(3,4)],5)
 => 12
([(2,3),(2,4),(3,4)],5)
 => 18
([(0,4),(1,4),(2,3),(3,4)],5)
 => 30
([(1,4),(2,3),(2,4),(3,4)],5)
 => 12
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 30
([(1,3),(1,4),(2,3),(2,4)],5)
 => 10
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 20
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 10
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 30
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 20
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 40
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 50
([(0,4),(1,3),(2,3),(2,4)],5)
 => 30
([(0,1),(2,3),(2,4),(3,4)],5)
 => 12
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 30
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 40
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 30
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 40
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 50
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 20
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 12
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 30
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 60
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 50
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 80

Description

The number of orbits of toric promotion on a graph. Let

$(V, E)$ be a graph with

$n=|V|$ vertices, and let

$\sigma: V \to [n]$ be a labelling of its vertices. Let

$\tau_{i, j}(\sigma) = \begin{cases} \sigma & \text{if $\{\sigma^{-1}(i), \sigma^{-1}(j)\}\in E$}\\ (i, j)\circ\sigma & \text{otherwise}. \end{cases}$ The toric promotion operator is the product

$\tau_{n,1}\tau_{n-1,n}\dots\tau_{1,2}$ . This statistic records the number of orbits in the orbit decomposition of toric promotion.