***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * This information is distributed in the hope that it will be useful, * * but WITHOUT ANY WARRANTY; without even the implied warranty of * * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * ***************************************************************************** ----------------------------------------------------------------------------- Statistic identifier: St001757 ----------------------------------------------------------------------------- Collection: Graphs ----------------------------------------------------------------------------- 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. ----------------------------------------------------------------------------- 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 toric_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 toggle_labelling(G, pi, n, 1) def toric_promotion_labelling_orbits(G): G = G.canonical_label().copy(immutable=True) return toric_promotion_labelling_orbits_aux(G) @cached_function def toric_promotion_labelling_orbits_aux(G): n = G.num_verts() return orbit_decomposition(Permutations(n), lambda pi: toric_promotion_labelling(G, pi)) def statistic(G): return len(toric_promotion_labelling_orbits(G)) ----------------------------------------------------------------------------- Statistic 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 ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 90 ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 120 ([],6) => 144 ([(4,5)],6) => 144 ([(3,5),(4,5)],6) => 144 ([(2,5),(3,5),(4,5)],6) => 96 ([(1,5),(2,5),(3,5),(4,5)],6) => 48 ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 144 ([(2,5),(3,4)],6) => 144 ([(2,5),(3,4),(4,5)],6) => 96 ([(1,2),(3,5),(4,5)],6) => 144 ([(3,4),(3,5),(4,5)],6) => 144 ([(1,5),(2,5),(3,4),(4,5)],6) => 48 ([(0,1),(2,5),(3,5),(4,5)],6) => 96 ([(2,5),(3,4),(3,5),(4,5)],6) => 96 ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 144 ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 48 ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 144 ([(2,4),(2,5),(3,4),(3,5)],6) => 80 ([(0,5),(1,5),(2,4),(3,4)],6) => 144 ([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 48 ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 144 ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 80 ([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 48 ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 144 ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 96 ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 48 ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 144 ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 96 ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 48 ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 96 ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 96 ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 48 ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 96 ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 96 ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 240 ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 240 ([(0,5),(1,4),(2,3)],6) => 144 ([(1,5),(2,4),(3,4),(3,5)],6) => 48 ([(0,1),(2,5),(3,4),(4,5)],6) => 96 ([(1,2),(3,4),(3,5),(4,5)],6) => 144 ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 144 ([(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) => 144 ([(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) => 144 ([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 48 ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 96 ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 40 ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 72 ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 48 ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => 144 ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 48 ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 96 ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 48 ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 72 ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 144 ([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => 80 ([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => 144 ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => 96 ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => 144 ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => 144 ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 80 ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 144 ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => 96 ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => 96 ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 144 ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 96 ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 96 ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 96 ([(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) => 144 ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 120 ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6) => 96 ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 96 ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 72 ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 168 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 144 ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 96 ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 48 ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 144 ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 96 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 240 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 240 ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 96 ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 48 ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 96 ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 48 ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 144 ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 96 ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6) => 156 ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => 48 ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 96 ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 96 ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 48 ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 144 ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 72 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 144 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 168 ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => 216 ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 96 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 216 ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 216 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 192 ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 144 ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 48 ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 96 ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 144 ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 144 ([(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) => 168 ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 240 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 360 ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 360 ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 240 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 336 ([(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) => 360 ([(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) => 96 ([(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) => 96 ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 96 ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 144 ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 144 ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6) => 144 ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 144 ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 96 ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 168 ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 192 ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 192 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 288 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 288 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6) => 144 ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 144 ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 192 ([(0,3),(0,4),(0,5),(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),(3,4)],6) => 240 ([(0,1),(0,3),(0,5),(1,2),(1,4),(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),(3,4),(3,5),(4,5)],6) => 192 ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 288 ([(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) => 360 ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 240 ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 240 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 240 ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 144 ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 48 ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 144 ([(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) => 288 ([(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) => 432 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 312 ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 288 ([(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) => 384 ([(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) => 456 ([(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) => 504 ([(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) => 576 ([(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 ----------------------------------------------------------------------------- Created: Dec 14, 2021 at 15:56 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Dec 14, 2021 at 15:56 by Martin Rubey