Values
([],1) => 1
([],2) => 1
([(0,1)],2) => 2
([],3) => 2
([(1,2)],3) => 2
([(0,2),(1,2)],3) => 4
([(0,1),(0,2),(1,2)],3) => 6
([],4) => 6
([(2,3)],4) => 6
([(1,3),(2,3)],4) => 4
([(0,3),(1,3),(2,3)],4) => 10
([(0,3),(1,2)],4) => 8
([(0,3),(1,2),(2,3)],4) => 10
([(1,2),(1,3),(2,3)],4) => 6
([(0,3),(1,2),(1,3),(2,3)],4) => 12
([(0,2),(0,3),(1,2),(1,3)],4) => 16
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 18
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 24
([],5) => 24
([(3,4)],5) => 24
([(2,4),(3,4)],5) => 16
([(1,4),(2,4),(3,4)],5) => 16
([(0,4),(1,4),(2,4),(3,4)],5) => 40
([(1,4),(2,3)],5) => 24
([(1,4),(2,3),(3,4)],5) => 12
([(0,1),(2,4),(3,4)],5) => 16
([(2,3),(2,4),(3,4)],5) => 24
([(0,4),(1,4),(2,3),(3,4)],5) => 24
([(1,4),(2,3),(2,4),(3,4)],5) => 16
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 40
([(1,3),(1,4),(2,3),(2,4)],5) => 24
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 32
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 24
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 28
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 40
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 56
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 60
([(0,4),(1,3),(2,3),(2,4)],5) => 28
([(0,1),(2,3),(2,4),(3,4)],5) => 24
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 32
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => 48
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 40
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 48
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 60
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 40
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 24
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 48
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 72
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 64
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 84
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 96
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 120
([],6) => 120
([(4,5)],6) => 120
([(3,5),(4,5)],6) => 80
([(2,5),(3,5),(4,5)],6) => 80
([(1,5),(2,5),(3,5),(4,5)],6) => 40
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 148
([(2,5),(3,4)],6) => 120
([(2,5),(3,4),(4,5)],6) => 60
([(1,2),(3,5),(4,5)],6) => 80
([(3,4),(3,5),(4,5)],6) => 120
([(1,5),(2,5),(3,4),(4,5)],6) => 32
([(0,1),(2,5),(3,5),(4,5)],6) => 96
([(2,5),(3,4),(3,5),(4,5)],6) => 80
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 112
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 40
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 152
([(2,4),(2,5),(3,4),(3,5)],6) => 120
([(0,5),(1,5),(2,4),(3,4)],6) => 80
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 32
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 96
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 120
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 28
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 96
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 128
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 40
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 108
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 140
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 56
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 116
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 156
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 60
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 120
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 164
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 260
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 276
([(0,5),(1,4),(2,3)],6) => 128
([(1,5),(2,4),(3,4),(3,5)],6) => 44
([(0,1),(2,5),(3,4),(4,5)],6) => 72
([(1,2),(3,4),(3,5),(4,5)],6) => 120
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 92
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 48
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => 96
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 120
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 48
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 160
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 80
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 116
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 56
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 104
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Description
The number of orbits of 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 promotion operator is the product $\tau_{n-1,n}\dots\tau_{1,2}$.
This statistic records the number of orbits in the orbit decomposition of promotion.
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 promotion operator is the product $\tau_{n-1,n}\dots\tau_{1,2}$.
This statistic records the number of orbits in the orbit decomposition of promotion.
References
[1] Defant, C. Toric Promotion arXiv:2112.06843
Code
from sage.combinat.cyclic_sieving_phenomenon import orbit_decomposition
def toggle_labelling(G, pi, i, j):
if G.has_edge(pi.index(i), pi.index(j)):
return pi
sigma = [j if e == i else i if e == j else e for e in pi]
return Permutation(sigma)
def promotion_labelling(G, pi):
n = G.num_verts()
assert set(G.vertices()) == set(range(n))
for i in range(1, n):
pi = toggle_labelling(G, pi, i, i+1)
return pi
def promotion_labelling_orbits(G):
G = G.canonical_label().copy(immutable=True)
return promotion_labelling_orbits_aux(G)
@cached_function
def promotion_labelling_orbits_aux(G):
n = G.num_verts()
return orbit_decomposition(Permutations(n),
lambda pi: promotion_labelling(G, pi))
def statistic(G):
return len(promotion_labelling_orbits(G))
Created
Dec 14, 2021 at 16:03 by Martin Rubey
Updated
Dec 14, 2021 at 16:03 by Martin Rubey
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