Values
=>
Cc0020;cc-rep
([],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
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>40
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)=>72
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>136
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>116
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)=>60
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)=>152
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)=>100
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)=>144
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)=>96
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)=>124
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)=>112
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)=>108
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>144
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)=>112
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)=>144
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)=>144
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>120
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>152
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>120
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>144
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>48
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>168
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)=>156
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)=>196
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>168
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>148
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>196
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)=>216
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>168
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>72
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>136
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>176
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>272
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>288
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)=>160
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>64
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>172
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>84
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>200
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)=>140
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)=>180
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)=>144
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>156
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>128
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)=>136
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>196
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>160
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)=>184
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>208
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)=>224
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)=>176
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>256
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>240
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>272
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>192
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>96
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>184
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>216
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)=>224
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>192
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>240
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>312
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)=>336
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>352
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>296
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>384
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>408
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)=>144
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)=>136
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)=>144
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>144
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>168
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>168
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>192
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)=>188
([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)=>232
([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>192
([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>212
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>232
([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>256
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>312
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>336
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6)=>200
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>184
([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>240
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>280
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)=>264
([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>308
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>264
([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>344
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>408
([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>288
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>264
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>312
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>216
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>120
([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>240
([(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)=>360
([(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)=>480
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>368
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>336
([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>384
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>432
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>480
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>528
([(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)=>600
([(0,1),(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)=>720
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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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