The first few most recent pending statistics:
Identifier
St000000:
Graphs
⟶ ℤ
Values
([],1)
=>
0
([],2)
=>
0
([(0,1)],2)
=>
1
([],3)
=>
0
([(1,2)],3)
=>
1
([(0,2),(1,2)],3)
=>
1
([(0,1),(0,2),(1,2)],3)
=>
2
([],4)
=>
0
([(2,3)],4)
=>
1
([(1,3),(2,3)],4)
=>
1
([(0,3),(1,3),(2,3)],4)
=>
1
([(0,3),(1,2)],4)
=>
2
([(0,3),(1,2),(2,3)],4)
=>
2
([(1,2),(1,3),(2,3)],4)
=>
2
([(0,3),(1,2),(1,3),(2,3)],4)
=>
2
([(0,2),(0,3),(1,2),(1,3)],4)
=>
1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=>
2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=>
3
([],5)
=>
0
([(3,4)],5)
=>
1
([(2,4),(3,4)],5)
=>
1
([(1,4),(2,4),(3,4)],5)
=>
1
([(0,4),(1,4),(2,4),(3,4)],5)
=>
1
([(1,4),(2,3)],5)
=>
2
([(1,4),(2,3),(3,4)],5)
=>
2
([(0,1),(2,4),(3,4)],5)
=>
2
([(2,3),(2,4),(3,4)],5)
=>
2
([(0,4),(1,4),(2,3),(3,4)],5)
=>
2
([(1,4),(2,3),(2,4),(3,4)],5)
=>
2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=>
2
([(1,3),(1,4),(2,3),(2,4)],5)
=>
1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=>
2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=>
2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=>
2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=>
2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=>
1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=>
2
([(0,4),(1,3),(2,3),(2,4)],5)
=>
2
([(0,1),(2,3),(2,4),(3,4)],5)
=>
3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=>
3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=>
3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=>
3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=>
2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=>
3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=>
3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=>
3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=>
3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=>
3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=>
2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=>
2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=>
3
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=>
4
([],6)
=>
0
([(4,5)],6)
=>
1
([(3,5),(4,5)],6)
=>
1
([(2,5),(3,5),(4,5)],6)
=>
1
([(1,5),(2,5),(3,5),(4,5)],6)
=>
1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=>
1
([(2,5),(3,4)],6)
=>
2
([(2,5),(3,4),(4,5)],6)
=>
2
([(1,2),(3,5),(4,5)],6)
=>
2
([(3,4),(3,5),(4,5)],6)
=>
2
([(1,5),(2,5),(3,4),(4,5)],6)
=>
2
([(0,1),(2,5),(3,5),(4,5)],6)
=>
2
([(2,5),(3,4),(3,5),(4,5)],6)
=>
2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=>
2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=>
2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=>
2
([(2,4),(2,5),(3,4),(3,5)],6)
=>
1
([(0,5),(1,5),(2,4),(3,4)],6)
=>
2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=>
2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=>
2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=>
2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=>
2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=>
2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=>
2
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=>
1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=>
2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=>
2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=>
1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
2
([(0,5),(1,4),(2,3)],6)
=>
3
([(1,5),(2,4),(3,4),(3,5)],6)
=>
2
([(0,1),(2,5),(3,4),(4,5)],6)
=>
3
([(1,2),(3,4),(3,5),(4,5)],6)
=>
3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=>
3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=>
3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=>
3
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=>
3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=>
3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=>
3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=>
3
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=>
3
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=>
2
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=>
3
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=>
3
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=>
3
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=>
3
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
3
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=>
3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=>
3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=>
3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=>
2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=>
3
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=>
2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=>
3
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=>
3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
3
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=>
3
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=>
3
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=>
3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=>
3
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
3
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
3
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=>
3
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
3
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
3
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=>
3
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=>
2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
3
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=>
3
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
2
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=>
3
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=>
3
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
3
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
3
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=>
2
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
3
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=>
2
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=>
2
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=>
2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=>
2
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=>
2
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=>
3
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=>
2
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=>
3
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=>
3
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=>
3
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=>
3
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=>
3
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
3
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
3
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
3
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
3
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=>
4
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=>
3
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=>
4
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
4
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=>
4
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
4
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
4
([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=>
3
([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
4
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
3
([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
4
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=>
4
([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=>
3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=>
4
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=>
3
([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=>
3
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
4
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
4
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
4
([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=>
4
([(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)
=>
4
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=>
3
Description
The biclique partition number of a graph.
The biclique partition number of a graph is the minimum number of pairwise edge disjoint complete bipartite subgraphs so that each edge belongs to exactly one of them. A theorem of Graham and Pollak [1] asserts that the complete graph $K_n$ has biclique partition number $n - 1$.
The biclique partition number of a graph is the minimum number of pairwise edge disjoint complete bipartite subgraphs so that each edge belongs to exactly one of them. A theorem of Graham and Pollak [1] asserts that the complete graph $K_n$ has biclique partition number $n - 1$.
References
[1] MR0289210
WARNING - could not verify link HTTP Error 404: Not Found
[2] WARNING - could not verify link HTTP Error 404: Not Found
WARNING - could not verify link HTTP Error 404: Not Found
[2] WARNING - could not verify link HTTP Error 404: Not Found
Code
@cached_function
def statistic(G, certificate=False, verbose=False):
"""
sage: [statistic(graphs.CompleteGraph(n)) for n in range(1,6)]
[0, 1, 2, 3, 4]
sage: [statistic(graphs.PathGraph(n)) for n in range(1,8)]
[0, 1, 1, 2, 2, 3, 3]
sage: [statistic(graphs.CycleGraph(n)) for n in range(1,8)]
[0, 1, 2, 1, 3, 3, 4]
sage: n=5; all(statistic(G) <= n-G.independent_set(value_only=True) for G in graphs(n))
True
"""
E = G.edges(labels=False)
if not E:
return 0
for c in range(1, len(E)+1):
for p in SetPartitions(E, c):
if verbose: print("P", p)
for b in p:
V = [u for e in b for u in e]
H = G.subgraph(edges=b, vertices=V)
if verbose: print("V", H.vertices(), "E", H.edges(labels=False))
is_bipartite, bipartition = H.is_bipartite(certificate=True)
if is_bipartite:
if verbose: print(bipartition)
k = list(bipartition.values()).count(0)
l = H.num_verts() - k
B = graphs.CompleteBipartiteGraph(k, l)
if not H.is_isomorphic(B):
if verbose: print("not complete")
break
else:
if verbose: print("not bipartite")
break
else:
if certificate:
return c, p
return c
Created
Jun 28, 2022 at 12:09 by Martin Rubey
Updated
Jun 28, 2022 at 12:09 by Martin Rubey
Identifier
St000324:
Permutations ⟶ ℤ
Values
Modified entries:
[1,2,3]
=>
3654
[1,3,2]
=>
5436
[3,1,2]
=>
2436
[3,2,1]
=>
3624
[1,2,3,4]
=>
72162
[1,4,3,2]
=>
16272
[2,1,3,4]
=>
72108
[2,1,4,3]
=>
10872
[2,3,4,1]
=>
72108
[2,4,3,1]
=>
10872
[3,1,2,4]
=>
72108
[3,2,1,4]
=>
3672
[3,4,1,2]
=>
7236
[3,4,2,1]
=>
10872
[4,1,2,3]
=>
48108
[4,3,2,1]
=>
10848
[1,2,3,4,5]
=>
144486
[1,2,3,5,4]
=>
216324
[1,2,4,3,5]
=>
216324
[1,2,4,5,3]
=>
216324
[1,2,5,3,4]
=>
216324
[1,2,5,4,3]
=>
324216
[1,3,2,4,5]
=>
216324
[1,3,2,5,4]
=>
108216
[1,3,4,2,5]
=>
216108
[1,3,4,5,2]
=>
216324
[1,3,5,2,4]
=>
216108
[1,3,5,4,2]
=>
324216
[1,4,2,3,5]
=>
216324
[1,4,2,5,3]
=>
108216
[1,4,3,2,5]
=>
324216
[1,4,3,5,2]
=>
324216
[1,4,5,2,3]
=>
216108
[1,4,5,3,2]
=>
324216
[1,5,2,3,4]
=>
216324
[1,5,2,4,3]
=>
108216
[1,5,3,2,4]
=>
324216
[1,5,3,4,2]
=>
324216
[1,5,4,2,3]
=>
324216
[1,5,4,3,2]
=>
486144
[2,1,3,4,5]
=>
144324
[2,1,5,4,3]
=>
324144
[2,3,1,4,5]
=>
72108
[2,3,1,5,4]
=>
10872
[2,3,4,1,5]
=>
72108
[2,3,4,5,1]
=>
144324
[2,3,5,1,4]
=>
72108
[2,4,1,3,5]
=>
72108
[2,4,1,5,3]
=>
10872
[2,4,3,1,5]
=>
10872
[2,4,5,1,3]
=>
72108
[2,5,1,3,4]
=>
72108
[2,5,1,4,3]
=>
10872
[2,5,3,1,4]
=>
10872
[2,5,4,1,3]
=>
10872
[2,5,4,3,1]
=>
324144
[3,1,2,4,5]
=>
144324
[3,1,4,2,5]
=>
72216
[3,1,4,5,2]
=>
144216
[3,1,5,2,4]
=>
72216
[3,1,5,4,2]
=>
216144
[3,2,1,4,5]
=>
72216
[3,2,1,5,4]
=>
108144
[3,2,4,5,1]
=>
144216
[3,2,5,4,1]
=>
216144
[3,4,1,2,5]
=>
72108
[3,4,2,1,5]
=>
10872
[3,4,2,5,1]
=>
21672
[3,4,5,1,2]
=>
144108
[3,5,1,2,4]
=>
72108
[3,5,2,1,4]
=>
10872
[3,5,2,4,1]
=>
21672
[3,5,4,1,2]
=>
21672
[3,5,4,2,1]
=>
324144
[4,1,2,3,5]
=>
144324
[4,1,2,5,3]
=>
144216
[4,1,3,2,5]
=>
72216
[4,1,3,5,2]
=>
144216
[4,1,5,2,3]
=>
144216
[4,1,5,3,2]
=>
216144
[4,2,1,3,5]
=>
72216
[4,2,1,5,3]
=>
72144
[4,2,3,5,1]
=>
144216
[4,2,5,1,3]
=>
14472
[4,2,5,3,1]
=>
216144
[4,3,1,2,5]
=>
72216
[4,3,1,5,2]
=>
72144
[4,3,2,1,5]
=>
108144
[4,3,2,5,1]
=>
216144
[4,3,5,1,2]
=>
14472
[4,3,5,2,1]
=>
216144
[4,5,1,2,3]
=>
144108
[4,5,2,1,3]
=>
21672
[4,5,2,3,1]
=>
21672
[4,5,3,1,2]
=>
21672
[4,5,3,2,1]
=>
324144
[5,1,2,3,4]
=>
96324
[5,1,2,4,3]
=>
144216
[5,1,3,2,4]
=>
144216
[5,1,3,4,2]
=>
144216
[5,1,4,2,3]
=>
144216
[5,1,4,3,2]
=>
216144
[5,2,1,3,4]
=>
144216
[5,2,1,4,3]
=>
72144
[5,2,3,1,4]
=>
14472
[5,2,3,4,1]
=>
144216
[5,2,4,1,3]
=>
14472
[5,2,4,3,1]
=>
216144
[5,3,1,2,4]
=>
144216
[5,3,1,4,2]
=>
72144
[5,3,2,1,4]
=>
216144
[5,3,2,4,1]
=>
216144
[5,3,4,1,2]
=>
14472
[5,3,4,2,1]
=>
216144
[5,4,1,2,3]
=>
144216
[5,4,1,3,2]
=>
72144
[5,4,2,1,3]
=>
216144
[5,4,2,3,1]
=>
216144
[5,4,3,1,2]
=>
216144
[5,4,3,2,1]
=>
32496
[1,2,3,4,5,6]
=>
2881458
[1,2,3,4,6,5]
=>
432972
[1,2,3,5,4,6]
=>
432972
[1,2,3,5,6,4]
=>
432972
[1,2,3,6,4,5]
=>
432972
[1,2,4,3,5,6]
=>
432972
[1,2,4,3,6,5]
=>
216648
[1,2,4,5,3,6]
=>
432324
[1,2,4,5,6,3]
=>
432972
[1,2,4,6,3,5]
=>
432324
[1,2,5,3,4,6]
=>
432972
[1,2,5,3,6,4]
=>
216648
[1,2,5,6,3,4]
=>
432324
[1,2,6,3,4,5]
=>
432972
[1,2,6,3,5,4]
=>
216648
[1,2,6,5,4,3]
=>
972432
[1,3,2,4,5,6]
=>
432972
[1,3,2,4,6,5]
=>
216648
[1,3,2,5,4,6]
=>
216648
[1,3,2,5,6,4]
=>
216648
[1,3,2,6,4,5]
=>
216648
[1,3,2,6,5,4]
=>
324432
[1,3,4,2,5,6]
=>
432324
[1,3,4,5,2,6]
=>
432324
[1,3,4,5,6,2]
=>
432972
[1,3,4,6,2,5]
=>
432324
[1,3,5,2,4,6]
=>
432324
[1,3,5,4,2,6]
=>
648216
[1,3,5,6,2,4]
=>
432324
[1,3,6,2,4,5]
=>
432324
[1,3,6,4,2,5]
=>
648216
[1,3,6,5,2,4]
=>
648216
[1,3,6,5,4,2]
=>
972432
[1,4,2,3,5,6]
=>
432972
[1,4,2,3,6,5]
=>
216648
[1,4,2,5,3,6]
=>
216648
[1,4,2,5,6,3]
=>
216648
[1,4,2,6,3,5]
=>
216648
[1,4,2,6,5,3]
=>
324432
[1,4,3,2,6,5]
=>
324432
[1,4,3,5,2,6]
=>
648216
[1,4,3,6,2,5]
=>
648216
[1,4,3,6,5,2]
=>
324432
[1,4,5,2,3,6]
=>
432324
[1,4,5,3,2,6]
=>
648216
[1,4,5,3,6,2]
=>
648216
[1,4,5,6,2,3]
=>
432324
[1,4,6,2,3,5]
=>
432324
[1,4,6,3,2,5]
=>
648216
[1,4,6,3,5,2]
=>
648216
[1,4,6,5,2,3]
=>
648216
[1,4,6,5,3,2]
=>
972432
[1,5,2,3,4,6]
=>
432972
[1,5,2,3,6,4]
=>
216648
[1,5,2,4,3,6]
=>
216648
[1,5,2,4,6,3]
=>
216648
[1,5,2,6,3,4]
=>
216648
[1,5,2,6,4,3]
=>
324432
[1,5,3,2,6,4]
=>
324432
[1,5,3,4,2,6]
=>
648216
[1,5,3,6,2,4]
=>
648216
[1,5,3,6,4,2]
=>
324432
[1,5,4,2,6,3]
=>
324432
[1,5,4,3,2,6]
=>
972432
[1,5,4,3,6,2]
=>
972432
[1,5,4,6,2,3]
=>
648216
[1,5,4,6,3,2]
=>
972432
[1,5,6,2,3,4]
=>
432324
[1,5,6,3,2,4]
=>
648216
[1,5,6,3,4,2]
=>
648216
[1,5,6,4,2,3]
=>
648216
[1,5,6,4,3,2]
=>
972432
[1,6,2,3,4,5]
=>
432972
[1,6,2,3,5,4]
=>
216648
[1,6,2,4,3,5]
=>
216648
[1,6,2,4,5,3]
=>
216648
[1,6,2,5,3,4]
=>
216648
[1,6,2,5,4,3]
=>
324432
[1,6,3,2,5,4]
=>
324432
[1,6,3,4,2,5]
=>
648216
[1,6,3,5,2,4]
=>
648216
[1,6,3,5,4,2]
=>
324432
[1,6,4,2,5,3]
=>
324432
[1,6,4,3,2,5]
=>
972432
[1,6,4,3,5,2]
=>
972432
[1,6,4,5,2,3]
=>
648216
[1,6,4,5,3,2]
=>
972432
[1,6,5,2,4,3]
=>
324432
[1,6,5,3,2,4]
=>
972432
[1,6,5,3,4,2]
=>
972432
[1,6,5,4,2,3]
=>
972432
[1,6,5,4,3,2]
=>
1458288
[2,1,3,4,5,6]
=>
288972
[2,1,3,4,6,5]
=>
432648
[2,1,3,5,4,6]
=>
432648
[2,1,3,5,6,4]
=>
432648
[2,1,3,6,4,5]
=>
432648
[2,1,3,6,5,4]
=>
648432
[2,1,4,3,5,6]
=>
432648
[2,1,4,3,6,5]
=>
216432
[2,1,4,5,3,6]
=>
432216
[2,1,4,5,6,3]
=>
432648
[2,1,4,6,3,5]
=>
432216
[2,1,4,6,5,3]
=>
648432
[2,1,5,3,4,6]
=>
432648
[2,1,5,3,6,4]
=>
216432
[2,1,5,4,3,6]
=>
648432
[2,1,5,4,6,3]
=>
648432
[2,1,5,6,3,4]
=>
432216
[2,1,5,6,4,3]
=>
648432
[2,1,6,3,4,5]
=>
432648
[2,1,6,3,5,4]
=>
216432
[2,1,6,4,3,5]
=>
648432
[2,1,6,4,5,3]
=>
648432
[2,1,6,5,3,4]
=>
648432
[2,1,6,5,4,3]
=>
972288
[2,3,1,4,5,6]
=>
144324
[2,3,1,6,5,4]
=>
324144
[2,3,4,1,5,6]
=>
144108
[2,3,4,5,1,6]
=>
144324
[2,3,4,5,6,1]
=>
288972
[2,3,4,6,1,5]
=>
144324
[2,3,4,6,5,1]
=>
432648
[2,3,5,1,4,6]
=>
144108
[2,3,5,4,6,1]
=>
432648
[2,3,5,6,1,4]
=>
144324
[2,3,5,6,4,1]
=>
432648
[2,3,6,1,4,5]
=>
144108
[2,3,6,4,5,1]
=>
432648
[2,3,6,5,4,1]
=>
648432
[2,4,1,3,5,6]
=>
144324
[2,4,1,6,5,3]
=>
324144
[2,4,3,1,6,5]
=>
108144
[2,4,3,5,6,1]
=>
432648
[2,4,3,6,5,1]
=>
216432
[2,4,5,1,3,6]
=>
144108
[2,4,5,3,6,1]
=>
432216
[2,4,5,6,1,3]
=>
144324
[2,4,5,6,3,1]
=>
432648
[2,4,6,1,3,5]
=>
144108
[2,4,6,3,5,1]
=>
432216
[2,4,6,5,3,1]
=>
648432
[2,5,1,3,4,6]
=>
144324
[2,5,1,6,4,3]
=>
324144
[2,5,3,1,6,4]
=>
108144
[2,5,3,4,6,1]
=>
432648
[2,5,3,6,4,1]
=>
216432
[2,5,4,1,6,3]
=>
108144
[2,5,4,3,1,6]
=>
324144
[2,5,4,3,6,1]
=>
648432
[2,5,4,6,3,1]
=>
648432
[2,5,6,1,3,4]
=>
144108
[2,5,6,3,4,1]
=>
432216
[2,5,6,4,3,1]
=>
648432
[2,6,1,3,4,5]
=>
144324
[2,6,1,5,4,3]
=>
324144
[2,6,3,1,5,4]
=>
108144
[2,6,3,4,5,1]
=>
432648
[2,6,3,5,4,1]
=>
216432
[2,6,4,1,5,3]
=>
108144
[2,6,4,3,1,5]
=>
324144
[2,6,4,3,5,1]
=>
648432
[2,6,4,5,3,1]
=>
648432
[2,6,5,1,4,3]
=>
108144
[2,6,5,3,1,4]
=>
324144
[2,6,5,3,4,1]
=>
648432
[2,6,5,4,1,3]
=>
324144
[2,6,5,4,3,1]
=>
972288
[3,1,2,4,5,6]
=>
288972
[3,1,2,4,6,5]
=>
432648
[3,1,2,5,4,6]
=>
432648
[3,1,2,5,6,4]
=>
432648
[3,1,2,6,4,5]
=>
432648
[3,1,2,6,5,4]
=>
648432
[3,1,4,2,5,6]
=>
144648
[3,1,4,2,6,5]
=>
216432
[3,1,4,5,2,6]
=>
144216
[3,1,4,5,6,2]
=>
288648
[3,1,4,6,2,5]
=>
144216
[3,1,5,2,4,6]
=>
144648
[3,1,5,2,6,4]
=>
216432
[3,1,5,4,2,6]
=>
216432
[3,1,5,6,2,4]
=>
144216
[3,1,6,2,4,5]
=>
144648
[3,1,6,2,5,4]
=>
216432
[3,1,6,4,2,5]
=>
216432
[3,1,6,5,2,4]
=>
216432
[3,1,6,5,4,2]
=>
648288
[3,2,1,4,5,6]
=>
144648
[3,2,1,4,6,5]
=>
216432
[3,2,1,5,4,6]
=>
216432
[3,2,1,5,6,4]
=>
216432
[3,2,1,6,4,5]
=>
216432
[3,2,1,6,5,4]
=>
324288
[3,2,4,1,5,6]
=>
144216
[3,2,4,1,6,5]
=>
216144
[3,2,4,5,1,6]
=>
144216
[3,2,4,5,6,1]
=>
288648
[3,2,4,6,1,5]
=>
144216
[3,2,5,1,4,6]
=>
144216
[3,2,5,1,6,4]
=>
216144
[3,2,5,4,1,6]
=>
216144
[3,2,5,6,1,4]
=>
144216
[3,2,6,1,4,5]
=>
144216
[3,2,6,1,5,4]
=>
216144
[3,2,6,4,1,5]
=>
216144
[3,2,6,5,1,4]
=>
216144
[3,2,6,5,4,1]
=>
648288
[3,4,1,2,5,6]
=>
144324
[3,4,1,5,2,6]
=>
72216
[3,4,1,5,6,2]
=>
144216
[3,4,1,6,2,5]
=>
72216
[3,4,1,6,5,2]
=>
216144
[3,4,2,1,6,5]
=>
108144
[3,4,2,5,1,6]
=>
21672
[3,4,2,5,6,1]
=>
432216
[3,4,2,6,1,5]
=>
21672
[3,4,2,6,5,1]
=>
216144
[3,4,5,1,2,6]
=>
144108
[3,4,5,1,6,2]
=>
144216
[3,4,5,2,6,1]
=>
432216
[3,4,5,6,1,2]
=>
288324
[3,4,5,6,2,1]
=>
432648
[3,4,6,1,2,5]
=>
144108
[3,4,6,1,5,2]
=>
144216
[3,4,6,2,5,1]
=>
432216
[3,4,6,5,1,2]
=>
432216
[3,4,6,5,2,1]
=>
648432
[3,5,1,2,4,6]
=>
144324
[3,5,1,4,2,6]
=>
72216
[3,5,1,4,6,2]
=>
144216
[3,5,1,6,2,4]
=>
72216
[3,5,1,6,4,2]
=>
216144
[3,5,2,1,6,4]
=>
108144
[3,5,2,4,1,6]
=>
21672
[3,5,2,4,6,1]
=>
432216
[3,5,2,6,1,4]
=>
21672
[3,5,2,6,4,1]
=>
216144
[3,5,4,1,6,2]
=>
216144
[3,5,4,2,1,6]
=>
324144
[3,5,4,2,6,1]
=>
648144
[3,5,4,6,1,2]
=>
432216
[3,5,4,6,2,1]
=>
648432
[3,5,6,1,2,4]
=>
144108
[3,5,6,1,4,2]
=>
144216
[3,5,6,2,4,1]
=>
432216
[3,5,6,4,1,2]
=>
432216
[3,5,6,4,2,1]
=>
648432
[3,6,1,2,4,5]
=>
144324
[3,6,1,4,2,5]
=>
72216
[3,6,1,4,5,2]
=>
144216
[3,6,1,5,2,4]
=>
72216
[3,6,1,5,4,2]
=>
216144
[3,6,2,1,5,4]
=>
108144
[3,6,2,4,1,5]
=>
21672
[3,6,2,4,5,1]
=>
432216
[3,6,2,5,1,4]
=>
21672
[3,6,2,5,4,1]
=>
216144
[3,6,4,1,5,2]
=>
216144
[3,6,4,2,1,5]
=>
324144
[3,6,4,2,5,1]
=>
648144
[3,6,4,5,1,2]
=>
432216
[3,6,4,5,2,1]
=>
648432
[3,6,5,1,4,2]
=>
216144
[3,6,5,2,1,4]
=>
324144
[3,6,5,2,4,1]
=>
648144
[3,6,5,4,1,2]
=>
648144
[3,6,5,4,2,1]
=>
972288
[4,1,2,3,5,6]
=>
288972
[4,1,2,3,6,5]
=>
432648
[4,1,2,5,3,6]
=>
144648
[4,1,2,5,6,3]
=>
288648
[4,1,2,6,3,5]
=>
144648
[4,1,3,2,5,6]
=>
144648
[4,1,3,2,6,5]
=>
216432
[4,1,3,5,2,6]
=>
144216
[4,1,3,5,6,2]
=>
288648
[4,1,3,6,2,5]
=>
144216
[4,1,5,2,3,6]
=>
144648
[4,1,5,2,6,3]
=>
144432
[4,1,5,3,2,6]
=>
216432
[4,1,5,6,2,3]
=>
288216
[4,1,6,2,3,5]
=>
144648
[4,1,6,2,5,3]
=>
144432
[4,1,6,3,2,5]
=>
216432
[4,1,6,5,3,2]
=>
648288
[4,2,1,3,5,6]
=>
144648
[4,2,1,3,6,5]
=>
216432
[4,2,1,5,3,6]
=>
72432
[4,2,1,5,6,3]
=>
144432
[4,2,1,6,3,5]
=>
72432
[4,2,1,6,5,3]
=>
216288
[4,2,3,1,5,6]
=>
144216
[4,2,3,1,6,5]
=>
216144
[4,2,3,5,1,6]
=>
144216
[4,2,3,5,6,1]
=>
288648
[4,2,3,6,1,5]
=>
144216
[4,2,5,1,3,6]
=>
144216
[4,2,5,3,1,6]
=>
216144
[4,2,5,6,1,3]
=>
288216
[4,2,6,1,3,5]
=>
144216
[4,2,6,3,1,5]
=>
216144
[4,2,6,5,1,3]
=>
432144
[4,2,6,5,3,1]
=>
648288
[4,3,1,2,5,6]
=>
144648
[4,3,1,2,6,5]
=>
216432
[4,3,1,5,2,6]
=>
72432
[4,3,1,5,6,2]
=>
144432
[4,3,1,6,2,5]
=>
72432
[4,3,1,6,5,2]
=>
216288
[4,3,2,1,5,6]
=>
216432
[4,3,2,1,6,5]
=>
108288
[4,3,2,5,1,6]
=>
216144
[4,3,2,6,1,5]
=>
216144
[4,3,2,6,5,1]
=>
216288
[4,3,5,1,2,6]
=>
144216
[4,3,5,2,1,6]
=>
216144
[4,3,5,2,6,1]
=>
432144
[4,3,5,6,1,2]
=>
288216
[4,3,6,1,2,5]
=>
144216
[4,3,6,2,1,5]
=>
216144
[4,3,6,2,5,1]
=>
432144
[4,3,6,5,1,2]
=>
432144
[4,3,6,5,2,1]
=>
648288
[4,5,1,2,3,6]
=>
144324
[4,5,1,2,6,3]
=>
144216
[4,5,1,3,2,6]
=>
72216
[4,5,1,3,6,2]
=>
144216
[4,5,1,6,2,3]
=>
144216
[4,5,1,6,3,2]
=>
216144
[4,5,2,1,6,3]
=>
216144
[4,5,2,3,1,6]
=>
21672
[4,5,2,3,6,1]
=>
432216
[4,5,2,6,1,3]
=>
43272
[4,5,2,6,3,1]
=>
216144
[4,5,3,1,6,2]
=>
216144
[4,5,3,2,1,6]
=>
324144
[4,5,3,2,6,1]
=>
648144
[4,5,3,6,1,2]
=>
43272
[4,5,3,6,2,1]
=>
648144
[4,5,6,1,2,3]
=>
288108
[4,5,6,1,3,2]
=>
144216
[4,5,6,2,1,3]
=>
432216
[4,5,6,2,3,1]
=>
432216
[4,5,6,3,1,2]
=>
432216
[4,5,6,3,2,1]
=>
648432
[4,6,1,2,3,5]
=>
144324
[4,6,1,2,5,3]
=>
144216
[4,6,1,3,2,5]
=>
72216
[4,6,1,3,5,2]
=>
144216
[4,6,1,5,2,3]
=>
144216
[4,6,1,5,3,2]
=>
216144
[4,6,2,1,5,3]
=>
216144
[4,6,2,3,1,5]
=>
21672
[4,6,2,3,5,1]
=>
432216
[4,6,2,5,1,3]
=>
43272
[4,6,2,5,3,1]
=>
216144
[4,6,3,1,5,2]
=>
216144
[4,6,3,2,1,5]
=>
324144
[4,6,3,2,5,1]
=>
648144
[4,6,3,5,1,2]
=>
43272
[4,6,3,5,2,1]
=>
648144
[4,6,5,1,2,3]
=>
432216
[4,6,5,1,3,2]
=>
216144
[4,6,5,2,1,3]
=>
648144
[4,6,5,2,3,1]
=>
648144
[4,6,5,3,1,2]
=>
648144
[4,6,5,3,2,1]
=>
972288
[5,1,2,3,4,6]
=>
288972
[5,1,2,3,6,4]
=>
288648
[5,1,2,4,3,6]
=>
144648
[5,1,2,4,6,3]
=>
288648
[5,1,2,6,3,4]
=>
288648
[5,1,3,2,4,6]
=>
144648
[5,1,3,2,6,4]
=>
144432
[5,1,3,4,2,6]
=>
144216
[5,1,3,4,6,2]
=>
288648
[5,1,3,6,2,4]
=>
288216
[5,1,4,2,3,6]
=>
144648
[5,1,4,2,6,3]
=>
144432
[5,1,4,3,2,6]
=>
216432
[5,1,4,6,2,3]
=>
288216
[5,1,6,2,3,4]
=>
288648
[5,1,6,2,4,3]
=>
144432
[5,1,6,4,3,2]
=>
648288
[5,2,1,3,4,6]
=>
144648
[5,2,1,3,6,4]
=>
144432
[5,2,1,4,3,6]
=>
72432
[5,2,1,4,6,3]
=>
144432
[5,2,1,6,3,4]
=>
144432
[5,2,1,6,4,3]
=>
216288
[5,2,3,1,4,6]
=>
144216
[5,2,3,4,1,6]
=>
144216
[5,2,3,4,6,1]
=>
288648
[5,2,3,6,1,4]
=>
288216
[5,2,4,1,3,6]
=>
144216
[5,2,4,3,1,6]
=>
216144
[5,2,4,6,1,3]
=>
288216
[5,2,6,1,3,4]
=>
288216
[5,2,6,3,1,4]
=>
432144
[5,2,6,4,1,3]
=>
432144
[5,2,6,4,3,1]
=>
648288
[5,3,1,2,4,6]
=>
144648
[5,3,1,2,6,4]
=>
144432
[5,3,1,4,2,6]
=>
72432
[5,3,1,4,6,2]
=>
144432
[5,3,1,6,2,4]
=>
144432
[5,3,1,6,4,2]
=>
216288
[5,3,2,1,4,6]
=>
216432
[5,3,2,1,6,4]
=>
216288
[5,3,2,4,1,6]
=>
216144
[5,3,2,6,1,4]
=>
432144
[5,3,2,6,4,1]
=>
216288
[5,3,4,1,2,6]
=>
144216
[5,3,4,2,1,6]
=>
216144
[5,3,4,2,6,1]
=>
432144
[5,3,4,6,1,2]
=>
288216
[5,3,6,1,2,4]
=>
288216
[5,3,6,2,1,4]
=>
432144
[5,3,6,2,4,1]
=>
432144
[5,3,6,4,1,2]
=>
432144
[5,3,6,4,2,1]
=>
648288
[5,4,1,2,3,6]
=>
144648
[5,4,1,2,6,3]
=>
144432
[5,4,1,3,2,6]
=>
72432
[5,4,1,3,6,2]
=>
144432
[5,4,1,6,2,3]
=>
144432
[5,4,1,6,3,2]
=>
216288
[5,4,2,1,3,6]
=>
216432
[5,4,2,1,6,3]
=>
216288
[5,4,2,3,1,6]
=>
216144
[5,4,2,6,1,3]
=>
432144
[5,4,2,6,3,1]
=>
216288
[5,4,3,1,2,6]
=>
216432
[5,4,3,1,6,2]
=>
216288
[5,4,3,2,1,6]
=>
324288
[5,4,3,2,6,1]
=>
648288
[5,4,3,6,1,2]
=>
432144
[5,4,3,6,2,1]
=>
648288
[5,4,6,1,2,3]
=>
288216
[5,4,6,2,1,3]
=>
432144
[5,4,6,2,3,1]
=>
432144
[5,4,6,3,1,2]
=>
432144
[5,4,6,3,2,1]
=>
648288
[5,6,1,2,3,4]
=>
288324
[5,6,1,2,4,3]
=>
144216
[5,6,1,3,2,4]
=>
144216
[5,6,1,3,4,2]
=>
144216
[5,6,1,4,2,3]
=>
144216
[5,6,1,4,3,2]
=>
216144
[5,6,2,1,3,4]
=>
432216
[5,6,2,1,4,3]
=>
216144
[5,6,2,3,1,4]
=>
43272
[5,6,2,3,4,1]
=>
432216
[5,6,2,4,1,3]
=>
43272
[5,6,2,4,3,1]
=>
216144
[5,6,3,1,2,4]
=>
432216
[5,6,3,1,4,2]
=>
216144
[5,6,3,2,1,4]
=>
648144
[5,6,3,2,4,1]
=>
648144
[5,6,3,4,1,2]
=>
43272
[5,6,3,4,2,1]
=>
648144
[5,6,4,1,2,3]
=>
432216
[5,6,4,1,3,2]
=>
216144
[5,6,4,2,1,3]
=>
648144
[5,6,4,2,3,1]
=>
648144
[5,6,4,3,1,2]
=>
648144
[5,6,4,3,2,1]
=>
972288
[6,1,2,3,4,5]
=>
192972
[6,1,2,3,5,4]
=>
288648
[6,1,2,4,3,5]
=>
288648
[6,1,2,4,5,3]
=>
288648
[6,1,2,5,3,4]
=>
288648
[6,1,3,2,4,5]
=>
288648
[6,1,3,2,5,4]
=>
144432
[6,1,3,4,2,5]
=>
288216
[6,1,3,4,5,2]
=>
288648
[6,1,3,5,2,4]
=>
288216
[6,1,4,2,3,5]
=>
288648
[6,1,4,2,5,3]
=>
144432
[6,1,4,5,2,3]
=>
288216
[6,1,5,2,3,4]
=>
288648
[6,1,5,2,4,3]
=>
144432
[6,1,5,4,3,2]
=>
648288
[6,2,1,3,4,5]
=>
288648
[6,2,1,3,5,4]
=>
144432
[6,2,1,4,3,5]
=>
144432
[6,2,1,4,5,3]
=>
144432
[6,2,1,5,3,4]
=>
144432
[6,2,1,5,4,3]
=>
216288
[6,2,3,1,4,5]
=>
288216
[6,2,3,4,1,5]
=>
288216
[6,2,3,4,5,1]
=>
288648
[6,2,3,5,1,4]
=>
288216
[6,2,4,1,3,5]
=>
288216
[6,2,4,3,1,5]
=>
432144
[6,2,4,5,1,3]
=>
288216
[6,2,5,1,3,4]
=>
288216
[6,2,5,3,1,4]
=>
432144
[6,2,5,4,1,3]
=>
432144
[6,2,5,4,3,1]
=>
648288
[6,3,1,2,4,5]
=>
288648
[6,3,1,2,5,4]
=>
144432
[6,3,1,4,2,5]
=>
144432
[6,3,1,4,5,2]
=>
144432
[6,3,1,5,2,4]
=>
144432
[6,3,1,5,4,2]
=>
216288
[6,3,2,1,5,4]
=>
216288
[6,3,2,4,1,5]
=>
432144
[6,3,2,5,1,4]
=>
432144
[6,3,2,5,4,1]
=>
216288
[6,3,4,1,2,5]
=>
288216
[6,3,4,2,1,5]
=>
432144
[6,3,4,2,5,1]
=>
432144
[6,3,4,5,1,2]
=>
288216
[6,3,5,1,2,4]
=>
288216
[6,3,5,2,1,4]
=>
432144
[6,3,5,2,4,1]
=>
432144
[6,3,5,4,1,2]
=>
432144
[6,3,5,4,2,1]
=>
648288
[6,4,1,2,3,5]
=>
288648
[6,4,1,2,5,3]
=>
144432
[6,4,1,3,2,5]
=>
144432
[6,4,1,3,5,2]
=>
144432
[6,4,1,5,2,3]
=>
144432
[6,4,1,5,3,2]
=>
216288
[6,4,2,1,5,3]
=>
216288
[6,4,2,3,1,5]
=>
432144
[6,4,2,5,1,3]
=>
432144
[6,4,2,5,3,1]
=>
216288
[6,4,3,1,5,2]
=>
216288
[6,4,3,2,1,5]
=>
648288
[6,4,3,2,5,1]
=>
648288
[6,4,3,5,1,2]
=>
432144
[6,4,3,5,2,1]
=>
648288
[6,4,5,1,2,3]
=>
288216
[6,4,5,2,1,3]
=>
432144
[6,4,5,2,3,1]
=>
432144
[6,4,5,3,1,2]
=>
432144
[6,4,5,3,2,1]
=>
648288
[6,5,1,2,3,4]
=>
288648
[6,5,1,2,4,3]
=>
144432
[6,5,1,3,2,4]
=>
144432
[6,5,1,3,4,2]
=>
144432
[6,5,1,4,2,3]
=>
144432
[6,5,1,4,3,2]
=>
216288
[6,5,2,1,4,3]
=>
216288
[6,5,2,3,1,4]
=>
432144
[6,5,2,4,1,3]
=>
432144
[6,5,2,4,3,1]
=>
216288
[6,5,3,1,4,2]
=>
216288
[6,5,3,2,1,4]
=>
648288
[6,5,3,2,4,1]
=>
648288
[6,5,3,4,1,2]
=>
432144
[6,5,3,4,2,1]
=>
648288
[6,5,4,1,3,2]
=>
216288
[6,5,4,2,1,3]
=>
648288
[6,5,4,2,3,1]
=>
648288
[6,5,4,3,1,2]
=>
648288
[6,5,4,3,2,1]
=>
972192
Description
The shape of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices {0,1,2,…,n} and root 0 in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by $2^w 3^h$, where $w$ is the width St000325The width of the tree associated to a permutation. and $h$ is the height St000308The height of the tree associated to a permutation. of this tree.
A permutation can be mapped to a rooted tree with vertices {0,1,2,…,n} and root 0 in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by $2^w 3^h$, where $w$ is the width St000325The width of the tree associated to a permutation. and $h$ is the height St000308The height of the tree associated to a permutation. of this tree.
diff Desc.
The shape of the tree associated to a permutation.
A permutation can be mapped to a rooted treeas described on the [[http://www.findstat.org/Permutations#Other_maps|permutations page]], and twith vertices {0,1,2,…,n} and root 0 in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by 2^w 3^h, where w is the width [[St000325]] and h is the height [[St000308]] of this tree.
A permutation can be mapped to a rooted tree
The statistic is given by 2^w 3^h, where w is the width [[St000325]] and h is the height [[St000308]] of this tree.
References
Code
def statistic(pi):
if pi == []: return (0, 0)
h, w, i, branch, next = 0, 0, 0, [0], pi[0]
while true:
while next < branch[len(branch)-1]:
branch.pop()
current = 0
w += 1
while next > current:
i += 1
h = max(h, len(branch))
if i == len(pi): return 2^w*3^h
branch.append(next)
current, next = next, pi[i]
diff Code
def statistic(pi): if pi == []: return (0, 0) h, w, i, branch, next = 0, 0,len(pi)0, [0], pi[0] while true: while next < branch[len(branch)-1]: branch.pop() current = 0 w += 1 while next > current: i-+= 1 h = max(h, len(branch)) if i ==0len(pi): return 2^w*3^h branch.append(next) current, next = next, pi[i]
Created
Dec 11, 2015 at 11:05 by Peter Luschny
Updated
Feb 25, 2022 at 16:54 by Nadia Lafreniere
Identifier
St000000:
Dyck paths
⟶ ℤ
(values match
St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra.)
Values
[1,0]
=>
1
[1,0,1,0]
=>
1
[1,1,0,0]
=>
1
[1,0,1,0,1,0]
=>
2
[1,0,1,1,0,0]
=>
1
[1,1,0,0,1,0]
=>
1
[1,1,0,1,0,0]
=>
1
[1,1,1,0,0,0]
=>
1
[1,0,1,0,1,0,1,0]
=>
2
[1,0,1,0,1,1,0,0]
=>
2
[1,0,1,1,0,0,1,0]
=>
1
[1,0,1,1,0,1,0,0]
=>
2
[1,0,1,1,1,0,0,0]
=>
1
[1,1,0,0,1,0,1,0]
=>
2
[1,1,0,0,1,1,0,0]
=>
1
[1,1,0,1,0,0,1,0]
=>
2
[1,1,0,1,0,1,0,0]
=>
2
[1,1,0,1,1,0,0,0]
=>
1
[1,1,1,0,0,0,1,0]
=>
1
[1,1,1,0,0,1,0,0]
=>
1
[1,1,1,0,1,0,0,0]
=>
1
[1,1,1,1,0,0,0,0]
=>
1
[1,0,1,0,1,0,1,0,1,0]
=>
3
[1,0,1,0,1,0,1,1,0,0]
=>
2
[1,0,1,0,1,1,0,0,1,0]
=>
2
[1,0,1,0,1,1,0,1,0,0]
=>
2
[1,0,1,0,1,1,1,0,0,0]
=>
2
[1,0,1,1,0,0,1,0,1,0]
=>
2
[1,0,1,1,0,0,1,1,0,0]
=>
1
[1,0,1,1,0,1,0,0,1,0]
=>
2
[1,0,1,1,0,1,0,1,0,0]
=>
2
[1,0,1,1,0,1,1,0,0,0]
=>
2
[1,0,1,1,1,0,0,0,1,0]
=>
1
[1,0,1,1,1,0,0,1,0,0]
=>
1
[1,0,1,1,1,0,1,0,0,0]
=>
2
[1,0,1,1,1,1,0,0,0,0]
=>
1
[1,1,0,0,1,0,1,0,1,0]
=>
2
[1,1,0,0,1,0,1,1,0,0]
=>
2
[1,1,0,0,1,1,0,0,1,0]
=>
1
[1,1,0,0,1,1,0,1,0,0]
=>
2
[1,1,0,0,1,1,1,0,0,0]
=>
1
[1,1,0,1,0,0,1,0,1,0]
=>
2
[1,1,0,1,0,0,1,1,0,0]
=>
2
[1,1,0,1,0,1,0,0,1,0]
=>
2
[1,1,0,1,0,1,0,1,0,0]
=>
2
[1,1,0,1,0,1,1,0,0,0]
=>
2
[1,1,0,1,1,0,0,0,1,0]
=>
1
[1,1,0,1,1,0,0,1,0,0]
=>
2
[1,1,0,1,1,0,1,0,0,0]
=>
2
[1,1,0,1,1,1,0,0,0,0]
=>
1
[1,1,1,0,0,0,1,0,1,0]
=>
2
[1,1,1,0,0,0,1,1,0,0]
=>
1
[1,1,1,0,0,1,0,0,1,0]
=>
2
[1,1,1,0,0,1,0,1,0,0]
=>
2
[1,1,1,0,0,1,1,0,0,0]
=>
1
[1,1,1,0,1,0,0,0,1,0]
=>
2
[1,1,1,0,1,0,0,1,0,0]
=>
2
[1,1,1,0,1,0,1,0,0,0]
=>
2
[1,1,1,0,1,1,0,0,0,0]
=>
1
[1,1,1,1,0,0,0,0,1,0]
=>
1
[1,1,1,1,0,0,0,1,0,0]
=>
1
[1,1,1,1,0,0,1,0,0,0]
=>
1
[1,1,1,1,0,1,0,0,0,0]
=>
1
[1,1,1,1,1,0,0,0,0,0]
=>
1
[1,0,1,0,1,0,1,0,1,0,1,0]
=>
3
[1,0,1,0,1,0,1,0,1,1,0,0]
=>
3
[1,0,1,0,1,0,1,1,0,0,1,0]
=>
2
[1,0,1,0,1,0,1,1,0,1,0,0]
=>
3
[1,0,1,0,1,0,1,1,1,0,0,0]
=>
2
[1,0,1,0,1,1,0,0,1,0,1,0]
=>
2
[1,0,1,0,1,1,0,0,1,1,0,0]
=>
2
[1,0,1,0,1,1,0,1,0,0,1,0]
=>
3
[1,0,1,0,1,1,0,1,0,1,0,0]
=>
3
[1,0,1,0,1,1,0,1,1,0,0,0]
=>
2
[1,0,1,0,1,1,1,0,0,0,1,0]
=>
2
[1,0,1,0,1,1,1,0,0,1,0,0]
=>
2
[1,0,1,0,1,1,1,0,1,0,0,0]
=>
2
[1,0,1,0,1,1,1,1,0,0,0,0]
=>
2
[1,0,1,1,0,0,1,0,1,0,1,0]
=>
2
[1,0,1,1,0,0,1,0,1,1,0,0]
=>
2
[1,0,1,1,0,0,1,1,0,0,1,0]
=>
1
[1,0,1,1,0,0,1,1,0,1,0,0]
=>
2
[1,0,1,1,0,0,1,1,1,0,0,0]
=>
1
[1,0,1,1,0,1,0,0,1,0,1,0]
=>
3
[1,0,1,1,0,1,0,0,1,1,0,0]
=>
2
[1,0,1,1,0,1,0,1,0,0,1,0]
=>
3
[1,0,1,1,0,1,0,1,0,1,0,0]
=>
2
[1,0,1,1,0,1,0,1,1,0,0,0]
=>
2
[1,0,1,1,0,1,1,0,0,0,1,0]
=>
2
[1,0,1,1,0,1,1,0,0,1,0,0]
=>
2
[1,0,1,1,0,1,1,0,1,0,0,0]
=>
2
[1,0,1,1,0,1,1,1,0,0,0,0]
=>
2
[1,0,1,1,1,0,0,0,1,0,1,0]
=>
2
[1,0,1,1,1,0,0,0,1,1,0,0]
=>
1
[1,0,1,1,1,0,0,1,0,0,1,0]
=>
2
[1,0,1,1,1,0,0,1,0,1,0,0]
=>
2
[1,0,1,1,1,0,0,1,1,0,0,0]
=>
1
[1,0,1,1,1,0,1,0,0,0,1,0]
=>
2
[1,0,1,1,1,0,1,0,0,1,0,0]
=>
2
[1,0,1,1,1,0,1,0,1,0,0,0]
=>
2
[1,0,1,1,1,0,1,1,0,0,0,0]
=>
2
[1,0,1,1,1,1,0,0,0,0,1,0]
=>
1
[1,0,1,1,1,1,0,0,0,1,0,0]
=>
1
[1,0,1,1,1,1,0,0,1,0,0,0]
=>
1
[1,0,1,1,1,1,0,1,0,0,0,0]
=>
2
[1,0,1,1,1,1,1,0,0,0,0,0]
=>
1
[1,1,0,0,1,0,1,0,1,0,1,0]
=>
3
[1,1,0,0,1,0,1,0,1,1,0,0]
=>
2
[1,1,0,0,1,0,1,1,0,0,1,0]
=>
2
[1,1,0,0,1,0,1,1,0,1,0,0]
=>
2
[1,1,0,0,1,0,1,1,1,0,0,0]
=>
2
[1,1,0,0,1,1,0,0,1,0,1,0]
=>
2
[1,1,0,0,1,1,0,0,1,1,0,0]
=>
1
[1,1,0,0,1,1,0,1,0,0,1,0]
=>
2
[1,1,0,0,1,1,0,1,0,1,0,0]
=>
2
[1,1,0,0,1,1,0,1,1,0,0,0]
=>
2
[1,1,0,0,1,1,1,0,0,0,1,0]
=>
1
[1,1,0,0,1,1,1,0,0,1,0,0]
=>
1
[1,1,0,0,1,1,1,0,1,0,0,0]
=>
2
[1,1,0,0,1,1,1,1,0,0,0,0]
=>
1
[1,1,0,1,0,0,1,0,1,0,1,0]
=>
3
[1,1,0,1,0,0,1,0,1,1,0,0]
=>
2
[1,1,0,1,0,0,1,1,0,0,1,0]
=>
2
[1,1,0,1,0,0,1,1,0,1,0,0]
=>
2
[1,1,0,1,0,0,1,1,1,0,0,0]
=>
2
[1,1,0,1,0,1,0,0,1,0,1,0]
=>
3
[1,1,0,1,0,1,0,0,1,1,0,0]
=>
2
[1,1,0,1,0,1,0,1,0,0,1,0]
=>
2
[1,1,0,1,0,1,0,1,0,1,0,0]
=>
2
[1,1,0,1,0,1,0,1,1,0,0,0]
=>
2
[1,1,0,1,0,1,1,0,0,0,1,0]
=>
2
[1,1,0,1,0,1,1,0,0,1,0,0]
=>
2
[1,1,0,1,0,1,1,0,1,0,0,0]
=>
2
[1,1,0,1,0,1,1,1,0,0,0,0]
=>
2
[1,1,0,1,1,0,0,0,1,0,1,0]
=>
2
[1,1,0,1,1,0,0,0,1,1,0,0]
=>
1
[1,1,0,1,1,0,0,1,0,0,1,0]
=>
2
[1,1,0,1,1,0,0,1,0,1,0,0]
=>
2
[1,1,0,1,1,0,0,1,1,0,0,0]
=>
2
[1,1,0,1,1,0,1,0,0,0,1,0]
=>
2
[1,1,0,1,1,0,1,0,0,1,0,0]
=>
2
[1,1,0,1,1,0,1,0,1,0,0,0]
=>
2
[1,1,0,1,1,0,1,1,0,0,0,0]
=>
2
[1,1,0,1,1,1,0,0,0,0,1,0]
=>
1
[1,1,0,1,1,1,0,0,0,1,0,0]
=>
1
[1,1,0,1,1,1,0,0,1,0,0,0]
=>
2
[1,1,0,1,1,1,0,1,0,0,0,0]
=>
2
[1,1,0,1,1,1,1,0,0,0,0,0]
=>
1
[1,1,1,0,0,0,1,0,1,0,1,0]
=>
2
[1,1,1,0,0,0,1,0,1,1,0,0]
=>
2
[1,1,1,0,0,0,1,1,0,0,1,0]
=>
1
[1,1,1,0,0,0,1,1,0,1,0,0]
=>
2
[1,1,1,0,0,0,1,1,1,0,0,0]
=>
1
[1,1,1,0,0,1,0,0,1,0,1,0]
=>
2
[1,1,1,0,0,1,0,0,1,1,0,0]
=>
2
[1,1,1,0,0,1,0,1,0,0,1,0]
=>
2
[1,1,1,0,0,1,0,1,0,1,0,0]
=>
2
[1,1,1,0,0,1,0,1,1,0,0,0]
=>
2
[1,1,1,0,0,1,1,0,0,0,1,0]
=>
1
[1,1,1,0,0,1,1,0,0,1,0,0]
=>
2
[1,1,1,0,0,1,1,0,1,0,0,0]
=>
2
[1,1,1,0,0,1,1,1,0,0,0,0]
=>
1
[1,1,1,0,1,0,0,0,1,0,1,0]
=>
2
[1,1,1,0,1,0,0,0,1,1,0,0]
=>
2
[1,1,1,0,1,0,0,1,0,0,1,0]
=>
2
[1,1,1,0,1,0,0,1,0,1,0,0]
=>
2
[1,1,1,0,1,0,0,1,1,0,0,0]
=>
2
[1,1,1,0,1,0,1,0,0,0,1,0]
=>
2
[1,1,1,0,1,0,1,0,0,1,0,0]
=>
2
[1,1,1,0,1,0,1,0,1,0,0,0]
=>
2
[1,1,1,0,1,0,1,1,0,0,0,0]
=>
2
[1,1,1,0,1,1,0,0,0,0,1,0]
=>
1
[1,1,1,0,1,1,0,0,0,1,0,0]
=>
2
[1,1,1,0,1,1,0,0,1,0,0,0]
=>
2
[1,1,1,0,1,1,0,1,0,0,0,0]
=>
2
[1,1,1,0,1,1,1,0,0,0,0,0]
=>
1
[1,1,1,1,0,0,0,0,1,0,1,0]
=>
2
[1,1,1,1,0,0,0,0,1,1,0,0]
=>
1
[1,1,1,1,0,0,0,1,0,0,1,0]
=>
2
[1,1,1,1,0,0,0,1,0,1,0,0]
=>
2
[1,1,1,1,0,0,0,1,1,0,0,0]
=>
1
[1,1,1,1,0,0,1,0,0,0,1,0]
=>
2
[1,1,1,1,0,0,1,0,0,1,0,0]
=>
2
[1,1,1,1,0,0,1,0,1,0,0,0]
=>
2
[1,1,1,1,0,0,1,1,0,0,0,0]
=>
1
[1,1,1,1,0,1,0,0,0,0,1,0]
=>
2
[1,1,1,1,0,1,0,0,0,1,0,0]
=>
2
[1,1,1,1,0,1,0,0,1,0,0,0]
=>
2
[1,1,1,1,0,1,0,1,0,0,0,0]
=>
2
[1,1,1,1,0,1,1,0,0,0,0,0]
=>
1
[1,1,1,1,1,0,0,0,0,0,1,0]
=>
1
[1,1,1,1,1,0,0,0,0,1,0,0]
=>
1
[1,1,1,1,1,0,0,0,1,0,0,0]
=>
1
[1,1,1,1,1,0,0,1,0,0,0,0]
=>
1
[1,1,1,1,1,0,1,0,0,0,0,0]
=>
1
[1,1,1,1,1,1,0,0,0,0,0,0]
=>
1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=>
4
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=>
3
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=>
3
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=>
3
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=>
3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=>
2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=>
2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=>
3
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=>
3
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=>
3
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=>
2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=>
2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=>
3
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=>
2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=>
2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=>
2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=>
2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=>
2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=>
2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=>
3
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=>
3
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=>
3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=>
3
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=>
3
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=>
2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=>
3
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=>
3
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=>
2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=>
2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=>
2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=>
2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=>
2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=>
2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=>
3
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=>
3
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=>
3
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=>
2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=>
2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=>
2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=>
2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=>
2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=>
2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=>
3
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=>
2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=>
2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=>
2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=>
2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=>
2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=>
1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=>
2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=>
2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=>
2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=>
1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=>
1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=>
2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=>
1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=>
3
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=>
3
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=>
2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=>
3
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=>
2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=>
3
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=>
3
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=>
2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=>
3
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=>
2
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=>
2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=>
3
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=>
2
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=>
2
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=>
2
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=>
2
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=>
3
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=>
3
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=>
2
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=>
3
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=>
3
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=>
2
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=>
2
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=>
2
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=>
2
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=>
2
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=>
2
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=>
2
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=>
2
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=>
2
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=>
1
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=>
2
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=>
1
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=>
2
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=>
2
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=>
2
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=>
2
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=>
2
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=>
1
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=>
2
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=>
2
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=>
1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=>
3
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=>
2
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=>
3
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=>
2
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=>
2
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=>
3
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=>
2
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=>
2
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=>
2
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=>
2
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=>
2
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=>
2
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=>
2
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=>
2
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=>
2
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=>
1
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=>
2
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=>
2
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=>
1
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=>
2
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=>
2
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=>
2
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=>
1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=>
2
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=>
2
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=>
2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=>
2
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=>
2
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=>
1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=>
1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=>
1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=>
1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=>
2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=>
1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=>
3
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=>
3
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=>
2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=>
3
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=>
2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=>
2
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=>
2
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=>
3
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=>
3
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=>
2
[1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=>
2
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=>
2
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=>
2
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=>
2
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=>
2
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=>
2
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=>
1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=>
2
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=>
1
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=>
3
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=>
2
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=>
3
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=>
2
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=>
2
[1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=>
2
[1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=>
2
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=>
2
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=>
2
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=>
2
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=>
1
[1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=>
2
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=>
2
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=>
1
[1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=>
2
[1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=>
2
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=>
2
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=>
2
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=>
1
[1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=>
1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=>
1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=>
2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=>
1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=>
3
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=>
3
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=>
2
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=>
3
[1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=>
2
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=>
2
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=>
2
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=>
3
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=>
3
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=>
2
[1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=>
2
[1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=>
2
[1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=>
2
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=>
2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=>
3
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=>
3
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=>
2
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=>
3
[1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=>
2
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=>
3
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=>
2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=>
3
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=>
3
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=>
2
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=>
2
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=>
2
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=>
2
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=>
2
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=>
2
[1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=>
2
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=>
3
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=>
3
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=>
2
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=>
2
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=>
2
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=>
2
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=>
2
[1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=>
2
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=>
2
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=>
2
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=>
2
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=>
2
[1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=>
2
[1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=>
2
[1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=>
1
[1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=>
2
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=>
1
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=>
3
[1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=>
2
[1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=>
3
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=>
2
[1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=>
2
[1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=>
2
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=>
2
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=>
2
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=>
2
[1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=>
3
[1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=>
2
[1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=>
3
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=>
2
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=>
2
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=>
2
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=>
2
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=>
2
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=>
2
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=>
2
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=>
2
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=>
2
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=>
2
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=>
2
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=>
2
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=>
1
[1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=>
2
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=>
2
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=>
1
[1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=>
2
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=>
2
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=>
2
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=>
2
[1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=>
2
[1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=>
2
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=>
2
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=>
2
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=>
2
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=>
1
[1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=>
1
[1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=>
1
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=>
2
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=>
2
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=>
1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=>
3
[1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=>
2
[1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=>
2
[1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=>
2
[1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=>
2
[1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=>
2
[1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=>
1
[1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=>
2
[1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=>
2
[1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=>
2
[1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=>
1
[1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=>
1
[1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=>
2
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=>
1
[1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=>
3
[1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=>
2
[1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=>
2
[1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=>
2
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=>
2
[1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=>
3
[1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=>
2
[1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=>
2
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=>
2
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=>
2
[1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=>
2
[1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=>
2
[1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=>
2
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=>
2
[1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=>
2
[1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=>
1
[1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=>
2
[1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=>
2
[1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=>
2
[1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=>
2
[1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=>
2
[1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=>
2
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=>
2
[1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=>
1
[1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=>
1
[1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=>
2
[1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=>
2
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=>
1
[1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=>
3
[1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=>
2
[1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=>
2
[1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=>
2
[1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=>
2
[1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=>
3
[1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=>
2
[1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=>
2
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=>
2
[1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=>
2
[1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=>
2
[1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=>
2
[1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=>
2
[1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=>
2
[1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=>
3
[1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=>
2
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=>
2
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=>
2
[1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=>
2
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=>
2
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=>
2
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=>
2
[1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=>
2
[1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=>
2
[1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=>
2
[1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=>
2
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=>
2
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=>
2
[1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=>
2
[1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=>
1
[1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=>
2
[1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=>
2
[1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=>
2
[1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=>
2
[1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=>
2
[1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=>
2
[1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=>
2
[1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=>
2
[1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=>
2
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=>
2
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=>
2
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=>
2
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=>
1
[1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=>
1
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=>
2
[1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=>
2
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=>
2
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=>
1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=>
2
[1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=>
2
[1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=>
1
[1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=>
2
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=>
1
[1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=>
2
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=>
2
[1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=>
2
[1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=>
2
[1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=>
2
[1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=>
1
[1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=>
2
[1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=>
2
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=>
1
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=>
2
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=>
2
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=>
2
[1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=>
2
[1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=>
2
[1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=>
2
[1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=>
2
[1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=>
2
[1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=>
2
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=>
1
[1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=>
2
[1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=>
2
[1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=>
2
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=>
1
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=>
2
[1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=>
2
[1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=>
2
[1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=>
2
[1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=>
2
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=>
2
[1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=>
2
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=>
2
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=>
2
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=>
2
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=>
2
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=>
2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=>
2
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=>
2
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=>
1
[1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=>
2
[1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=>
2
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=>
2
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=>
2
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=>
1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=>
2
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=>
1
[1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=>
2
[1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=>
2
[1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=>
1
[1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=>
2
[1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=>
2
[1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=>
2
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=>
1
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=>
2
[1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=>
2
[1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=>
2
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=>
2
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=>
1
[1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=>
2
[1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=>
2
[1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=>
2
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=>
2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=>
2
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=>
1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=>
1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=>
1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=>
1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=>
1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=>
1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=>
1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=>
1
Description
Return g/2 when g is even and (g+1)/2 when g is odd where g is the global dimension of the Nakayama algebra corresponding to the Dyck path.
Created
Dec 15, 2020 at 15:27 by Rene Marczinzik
Updated
Dec 15, 2020 at 15:27 by Rene Marczinzik
Identifier
St000000:
Dyck paths
⟶ ℤ
(values match
St000674The number of hills of a Dyck path.)
Values
[1,0,1,0]
=>
2
[1,0,1,0,1,0]
=>
3
[1,1,0,0,1,0]
=>
1
[1,0,1,0,1,0,1,0]
=>
4
[1,0,1,1,0,0,1,0]
=>
2
[1,1,0,0,1,0,1,0]
=>
2
[1,1,0,1,0,0,1,0]
=>
1
[1,1,1,0,0,0,1,0]
=>
1
[1,0,1,0,1,0,1,0,1,0]
=>
5
[1,0,1,0,1,1,0,0,1,0]
=>
3
[1,0,1,1,0,0,1,0,1,0]
=>
3
[1,0,1,1,0,1,0,0,1,0]
=>
2
[1,0,1,1,1,0,0,0,1,0]
=>
2
[1,1,0,0,1,0,1,0,1,0]
=>
3
[1,1,0,0,1,1,0,0,1,0]
=>
1
[1,1,0,1,0,0,1,0,1,0]
=>
2
[1,1,0,1,0,1,0,0,1,0]
=>
1
[1,1,0,1,1,0,0,0,1,0]
=>
1
[1,1,1,0,0,0,1,0,1,0]
=>
2
[1,1,1,0,0,1,0,0,1,0]
=>
1
[1,1,1,0,1,0,0,0,1,0]
=>
1
[1,1,1,1,0,0,0,0,1,0]
=>
1
[1,0,1,0,1,0,1,0,1,0,1,0]
=>
6
[1,0,1,0,1,0,1,1,0,0,1,0]
=>
4
[1,0,1,0,1,1,0,0,1,0,1,0]
=>
4
[1,0,1,0,1,1,0,1,0,0,1,0]
=>
3
[1,0,1,0,1,1,1,0,0,0,1,0]
=>
3
[1,0,1,1,0,0,1,0,1,0,1,0]
=>
4
[1,0,1,1,0,0,1,1,0,0,1,0]
=>
2
[1,0,1,1,0,1,0,0,1,0,1,0]
=>
3
[1,0,1,1,0,1,0,1,0,0,1,0]
=>
2
[1,0,1,1,0,1,1,0,0,0,1,0]
=>
2
[1,0,1,1,1,0,0,0,1,0,1,0]
=>
3
[1,0,1,1,1,0,0,1,0,0,1,0]
=>
2
[1,0,1,1,1,0,1,0,0,0,1,0]
=>
2
[1,0,1,1,1,1,0,0,0,0,1,0]
=>
2
[1,1,0,0,1,0,1,0,1,0,1,0]
=>
4
[1,1,0,0,1,0,1,1,0,0,1,0]
=>
2
[1,1,0,0,1,1,0,0,1,0,1,0]
=>
2
[1,1,0,0,1,1,0,1,0,0,1,0]
=>
1
[1,1,0,0,1,1,1,0,0,0,1,0]
=>
1
[1,1,0,1,0,0,1,0,1,0,1,0]
=>
3
[1,1,0,1,0,0,1,1,0,0,1,0]
=>
1
[1,1,0,1,0,1,0,0,1,0,1,0]
=>
2
[1,1,0,1,0,1,0,1,0,0,1,0]
=>
1
[1,1,0,1,0,1,1,0,0,0,1,0]
=>
1
[1,1,0,1,1,0,0,0,1,0,1,0]
=>
2
[1,1,0,1,1,0,0,1,0,0,1,0]
=>
1
[1,1,0,1,1,0,1,0,0,0,1,0]
=>
1
[1,1,0,1,1,1,0,0,0,0,1,0]
=>
1
[1,1,1,0,0,0,1,0,1,0,1,0]
=>
3
[1,1,1,0,0,0,1,1,0,0,1,0]
=>
1
[1,1,1,0,0,1,0,0,1,0,1,0]
=>
2
[1,1,1,0,0,1,0,1,0,0,1,0]
=>
1
[1,1,1,0,0,1,1,0,0,0,1,0]
=>
1
[1,1,1,0,1,0,0,0,1,0,1,0]
=>
2
[1,1,1,0,1,0,0,1,0,0,1,0]
=>
1
[1,1,1,0,1,0,1,0,0,0,1,0]
=>
1
[1,1,1,0,1,1,0,0,0,0,1,0]
=>
1
[1,1,1,1,0,0,0,0,1,0,1,0]
=>
2
[1,1,1,1,0,0,0,1,0,0,1,0]
=>
1
[1,1,1,1,0,0,1,0,0,0,1,0]
=>
1
[1,1,1,1,0,1,0,0,0,0,1,0]
=>
1
[1,1,1,1,1,0,0,0,0,0,1,0]
=>
1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=>
7
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=>
5
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=>
5
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=>
4
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=>
4
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=>
5
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=>
3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=>
4
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=>
3
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=>
3
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=>
4
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=>
3
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=>
3
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=>
3
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=>
5
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=>
3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=>
3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=>
2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=>
2
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=>
4
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=>
2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=>
3
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=>
2
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=>
2
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=>
3
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=>
2
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=>
2
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=>
2
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=>
4
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=>
2
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=>
3
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=>
2
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=>
2
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=>
3
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=>
2
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=>
2
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=>
2
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=>
3
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=>
2
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=>
2
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=>
2
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=>
2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=>
5
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=>
3
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=>
3
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=>
2
[1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=>
2
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=>
3
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=>
1
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=>
2
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=>
1
[1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=>
1
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=>
2
[1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=>
1
[1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=>
1
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=>
1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=>
4
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=>
2
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=>
2
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=>
1
[1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=>
1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=>
3
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=>
1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=>
2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=>
1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=>
1
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=>
2
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=>
1
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=>
1
[1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=>
1
[1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=>
3
[1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=>
1
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=>
2
[1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=>
1
[1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=>
1
[1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=>
2
[1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=>
1
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=>
1
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=>
1
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=>
2
[1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=>
1
[1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=>
1
[1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=>
1
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=>
1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=>
4
[1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=>
2
[1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=>
2
[1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=>
1
[1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=>
1
[1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=>
3
[1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=>
1
[1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=>
2
[1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=>
1
[1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=>
1
[1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=>
2
[1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=>
1
[1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=>
1
[1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=>
1
[1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=>
3
[1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=>
1
[1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=>
2
[1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=>
1
[1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=>
1
[1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=>
2
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=>
1
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=>
1
[1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=>
1
[1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=>
2
[1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=>
1
[1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=>
1
[1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=>
1
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=>
1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=>
3
[1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=>
1
[1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=>
2
[1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=>
1
[1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=>
1
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=>
2
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=>
1
[1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=>
1
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=>
1
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=>
2
[1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=>
1
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=>
1
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=>
1
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=>
1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=>
2
[1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=>
1
[1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=>
1
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=>
1
[1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=>
1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=>
1
Description
The number of fixed points of Ringel's bijection for cyclic Nakayama algebras with Magnitude one.
We use Christian Stump's bijection to Dyck paths.
We use Christian Stump's bijection to Dyck paths.
References
[1] Ringel, C. M. The finitistic dimension of a Nakayama algebra arXiv:2008.10044
Code
sage: def to_dyck(seq): ....: n = len(seq) ....: assert( seq.count(n) == 1 ) ....: i = seq.index(n) ....: seq = seq[i+1:] + seq[:i] ....: a = i = 0 ....: low_seq = [0] ....: high_seq = [] ....: for i,a in enumerate(seq): ....: if a < n: ....: low_seq.append(n-a) ....: else: ....: high_seq.append(a-n-1) ....: Ds = DyckWords() ....: high_seq = high_seq[::-1] ....: return DyckWord(list(Ds.from_area_sequence(high_seq)) + list(Ds.from_area_sequence(low_seq).reverse()))
Created
Dec 04, 2020 at 00:38 by Rene Marczinzik
Updated
Dec 04, 2020 at 00:38 by Rene Marczinzik
Identifier
St000000:
Posets
⟶ ℤ
Values
([(0,2),(2,1)],3)
=>
3
([(0,1),(0,2),(1,3),(2,3)],4)
=>
4
([(0,3),(2,1),(3,2)],4)
=>
4
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=>
1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=>
5
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=>
4
([(0,4),(2,3),(3,1),(4,2)],5)
=>
5
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=>
5
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=>
1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=>
2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=>
2
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=>
4
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=>
2
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=>
2
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=>
1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=>
6
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=>
5
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=>
4
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=>
6
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=>
6
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=>
6
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=>
6
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=>
5
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=>
1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=>
2
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=>
2
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=>
1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=>
4
([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=>
3
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=>
3
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)
=>
1
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=>
1
([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7)
=>
4
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=>
4
([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=>
2
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=>
6
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=>
2
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=>
7
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7)
=>
2
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=>
2
([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=>
7
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=>
1
([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=>
2
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
=>
2
([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=>
1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=>
2
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=>
1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=>
2
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=>
2
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=>
3
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=>
7
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=>
5
([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7)
=>
3
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=>
7
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=>
5
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=>
6
([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=>
6
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=>
7
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=>
3
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=>
6
([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
=>
6
([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=>
4
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=>
2
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=>
2
([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=>
6
([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=>
3
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=>
3
([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
=>
4
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=>
7
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=>
2
([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=>
6
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=>
5
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=>
3
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=>
7
([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=>
7
([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=>
5
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Code
DeclareOperation("2gorensteininjectivemodules",[IsList]);
InstallMethod(2gorensteininjectivemodules, "for a representation of a quiver", [IsList],0,function(LIST)
local A,L,LL,M,B,n,T,D,injA,W,simA,S,P,projA,R,RegA,CoRegA;
A:=LIST[1];
injA:=IndecInjectiveModules(A);
W:=Filtered(injA,x->IsProjectiveModule(x)=true or InjDimensionOfModule(Source(ProjectiveCover(NthSyzygy(x,1))),33)<=1);
return(Size(W));
end);
Created
Oct 03, 2020 at 20:59 by Rene Marczinzik
Updated
Oct 03, 2020 at 20:59 by Rene Marczinzik
Identifier
St000000:
Posets
⟶ ℤ
Values
([(0,2),(2,1)],3)
=>
2
([(0,1),(0,2),(1,3),(2,3)],4)
=>
4
([(0,3),(2,1),(3,2)],4)
=>
3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=>
9
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=>
5
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=>
6
([(0,4),(2,3),(3,1),(4,2)],5)
=>
4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=>
5
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=>
16
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=>
10
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=>
10
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=>
8
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=>
10
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=>
10
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=>
12
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=>
7
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=>
7
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=>
8
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=>
6
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=>
6
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=>
5
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=>
6
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=>
7
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=>
25
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=>
17
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=>
17
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=>
18
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=>
11
([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=>
11
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=>
11
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)
=>
15
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=>
17
([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7)
=>
10
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=>
11
([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=>
12
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=>
9
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=>
16
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=>
8
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7)
=>
12
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=>
17
([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=>
7
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=>
20
([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=>
13
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
=>
13
([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=>
14
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=>
12
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=>
15
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=>
13
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=>
13
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=>
11
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=>
8
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=>
9
([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7)
=>
11
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=>
7
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=>
9
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=>
9
([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=>
9
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=>
8
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=>
11
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=>
8
([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
=>
9
([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=>
10
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=>
13
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=>
12
([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=>
8
([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=>
11
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=>
11
([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
=>
10
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=>
6
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=>
13
([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=>
8
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=>
9
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=>
11
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=>
7
([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=>
7
([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=>
9
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Code
DeclareOperation("daP1top",[IsList]);
InstallMethod(daP1top, "for a representation of a quiver", [IsList],0,function(LIST)
local A,L,LL,M,B,n,T,D,injA,W,simA,S,P,projA,R,RegA,CoRegA;
A:=LIST[1];
CoRegA:=DirectSumOfQPAModules(IndecInjectiveModules(A));
T:=TopOfModule(NthSyzygy(CoRegA,1));
return(Dimension(T));
end);
Created
Oct 03, 2020 at 20:51 by Rene Marczinzik
Updated
Oct 03, 2020 at 20:51 by Rene Marczinzik
Identifier
St000000:
Lattices
⟶ ℤ
Values
([(0,2),(2,1)],3)
=>
1
([(0,1),(0,2),(1,3),(2,3)],4)
=>
2
([(0,3),(2,1),(3,2)],4)
=>
1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=>
2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=>
2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=>
1
([(0,4),(2,3),(3,1),(4,2)],5)
=>
1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=>
2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=>
2
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=>
2
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=>
2
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=>
1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=>
1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=>
1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=>
2
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=>
1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=>
2
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=>
2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=>
2
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=>
2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=>
2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=>
1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=>
2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=>
2
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=>
2
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=>
2
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=>
2
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=>
1
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=>
1
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=>
1
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=>
1
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=>
2
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=>
1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=>
2
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=>
1
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=>
1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=>
1
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=>
2
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=>
2
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=>
1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=>
2
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=>
2
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=>
2
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=>
2
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=>
2
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=>
2
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=>
2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=>
2
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=>
2
([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=>
2
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=>
2
([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=>
2
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)
=>
2
([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=>
2
([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=>
2
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=>
2
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=>
2
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7)
=>
2
([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=>
2
([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=>
1
([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=>
2
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=>
1
([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7)
=>
2
([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7)
=>
2
([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
=>
2
([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=>
2
([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=>
1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
=>
2
([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=>
2
([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
=>
2
([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=>
1
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=>
2
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=>
1
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=>
2
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=>
2
([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=>
1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Code
DeclareOperation("projdimsimpleminimum",[IsList]);
InstallMethod(projdimsimpleminimum, "for a representation of a quiver", [IsList],0,function(LIST)
local A,L,LL,M,B,n,T,D,injA,W,simA,S,P,projA;
A:=LIST[1];
projA:=IndecProjectiveModules(A);P:=Filtered(projA,x->IsInjectiveModule(x)=true)[1];
S:=TopOfModule(P);
return(ProjDimensionOfModule(S,33));
end);
Created
Oct 03, 2020 at 20:23 by Rene Marczinzik
Updated
Oct 03, 2020 at 20:23 by Rene Marczinzik
Identifier
St000000:
Lattices
⟶ ℤ
Values
([(0,2),(2,1)],3)
=>
0
([(0,1),(0,2),(1,3),(2,3)],4)
=>
1
([(0,3),(2,1),(3,2)],4)
=>
0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=>
4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=>
2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=>
1
([(0,4),(2,3),(3,1),(4,2)],5)
=>
0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=>
1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=>
5
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=>
5
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=>
4
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=>
4
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=>
2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=>
1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=>
3
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=>
1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=>
4
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=>
1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=>
4
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=>
3
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=>
2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=>
0
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=>
2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=>
6
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=>
6
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=>
6
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=>
5
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=>
5
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=>
5
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=>
4
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=>
4
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=>
5
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=>
4
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=>
4
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=>
1
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=>
3
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=>
2
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=>
6
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=>
5
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=>
4
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=>
6
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=>
5
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=>
2
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=>
5
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=>
4
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=>
5
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=>
5
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=>
4
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=>
1
([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=>
4
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=>
5
([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=>
5
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)
=>
6
([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=>
6
([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=>
3
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=>
2
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=>
3
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7)
=>
5
([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=>
5
([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=>
3
([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=>
4
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=>
2
([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7)
=>
4
([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7)
=>
4
([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
=>
3
([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=>
4
([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=>
1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
=>
5
([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=>
2
([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
=>
4
([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=>
2
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=>
3
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=>
0
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=>
3
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=>
3
([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=>
1
Description
Number of indecomposable injective modules with projective dimension 2.
Code
DeclareOperation("indinjprojdim2",[IsList]);
InstallMethod(indinjprojdim2, "for a representation of a quiver", [IsList],0,function(LIST)
local A,L,LL,M,B,n,T,D,injA,W,simA;
A:=LIST[1];
injA:=IndecInjectiveModules(A);
W:=Filtered(injA,x->ProjDimensionOfModule(x,33)=2);
return(Size(W));
end);
Created
Oct 03, 2020 at 20:07 by Rene Marczinzik
Updated
Oct 03, 2020 at 20:07 by Rene Marczinzik
Identifier
St000000:
Lattices
⟶ ℤ
Values
([(0,2),(2,1)],3)
=>
0
([(0,1),(0,2),(1,3),(2,3)],4)
=>
1
([(0,3),(2,1),(3,2)],4)
=>
0
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=>
1
([(0,4),(2,3),(3,1),(4,2)],5)
=>
0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=>
1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=>
1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=>
1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=>
1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=>
2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=>
0
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=>
1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=>
2
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=>
2
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=>
1
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=>
2
([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=>
1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=>
0
([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=>
1
([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=>
2
([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=>
2
([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=>
1
([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=>
2
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=>
3
([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=>
2
([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=>
3
([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=>
2
([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=>
1
([(0,5),(1,7),(2,7),(3,4),(4,6),(5,3),(6,1),(6,2)],8)
=>
1
([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=>
1
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=>
0
([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=>
2
([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=>
0
([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=>
1
([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=>
0
([(0,4),(0,5),(2,8),(3,8),(4,7),(5,7),(6,2),(6,3),(7,6),(8,1)],9)
=>
2
([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=>
2
([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
=>
2
([(0,6),(2,8),(3,8),(4,7),(5,1),(6,4),(7,2),(7,3),(8,5)],9)
=>
1
([(0,2),(0,3),(2,8),(3,8),(4,6),(5,4),(6,1),(7,5),(8,7)],9)
=>
1
([(0,4),(0,5),(2,7),(3,7),(4,8),(5,8),(6,1),(7,6),(8,2),(8,3)],9)
=>
2
([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=>
3
([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=>
3
([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=>
3
([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=>
2
([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=>
3
([(0,7),(2,8),(3,8),(4,5),(5,1),(6,4),(7,2),(7,3),(8,6)],9)
=>
1
([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=>
4
([(0,6),(1,8),(2,8),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(8,5)],9)
=>
2
([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=>
2
([(0,5),(1,8),(2,8),(3,7),(4,7),(5,6),(6,1),(6,2),(8,3),(8,4)],9)
=>
2
([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=>
3
([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=>
3
([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=>
0
([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=>
2
([(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4),(7,8)],9)
=>
2
([(0,6),(1,8),(2,8),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],9)
=>
1
([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=>
0
([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
=>
1
([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
=>
1
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Code
DeclareOperation("2regularsimples",[IsList]);
InstallMethod(2regularsimples, "for a representation of a quiver", [IsList],0,function(LIST)
local A,L,LL,M,B,n,T,D,injA,W,simA,RegA;
A:=LIST[1];
simA:=SimpleModules(A);
RegA:=DirectSumOfQPAModules(IndecProjectiveModules(A));
W:=Filtered(simA,x->ProjDimensionOfModule(x,33)=2 and gradeofmodule([A,x])=2 and Size(ExtOverAlgebra(NthSyzygy(x,1),RegA)[2])=1);
return(Size(W));
end);
Created
Oct 03, 2020 at 18:31 by Rene Marczinzik
Updated
Oct 03, 2020 at 18:31 by Rene Marczinzik
Identifier
St000000:
Lattices
⟶ ℤ
Values
([(0,2),(2,1)],3)
=>
3
([(0,1),(0,2),(1,3),(2,3)],4)
=>
3
([(0,3),(2,1),(3,2)],4)
=>
4
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=>
4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=>
4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=>
4
([(0,4),(2,3),(3,1),(4,2)],5)
=>
5
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=>
4
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=>
5
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=>
5
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=>
4
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=>
5
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=>
5
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=>
5
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=>
5
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=>
5
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=>
5
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=>
5
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=>
5
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=>
5
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=>
4
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=>
6
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=>
5
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=>
6
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=>
6
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=>
5
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=>
5
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=>
6
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=>
6
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=>
6
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=>
6
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=>
5
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=>
5
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=>
5
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=>
6
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=>
6
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=>
5
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=>
6
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=>
5
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=>
6
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=>
6
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=>
6
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=>
5
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=>
6
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=>
6
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=>
6
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=>
6
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=>
5
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=>
6
([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=>
6
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=>
5
([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=>
6
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)
=>
6
([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=>
5
([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=>
5
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=>
5
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=>
5
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7)
=>
5
([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=>
5
([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=>
6
([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=>
5
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=>
6
([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7)
=>
5
([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7)
=>
4
([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
=>
5
([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=>
6
([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=>
6
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
=>
6
([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=>
6
([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
=>
6
([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=>
6
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=>
5
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=>
7
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=>
6
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=>
6
([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=>
6
Description
The number of simple modules with projective dimension at most 1.
Created
Oct 03, 2020 at 18:15 by Rene Marczinzik
Updated
Oct 03, 2020 at 18:15 by Rene Marczinzik