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 $n - 1$.
References
[1] MR0289210
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
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.
diff Desc.
The shape of the tree associated to a permutation.

A permutation can be mapped to a rooted tree
as 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].
T
he statistic is given by 2^w 3^h, where w is the width [[St000325]] and h is the height [[St000308]] of this tree.
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.
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
Identifier
St000000: Lattices ⟶ ℤ (values match St001615The number of join prime elements of a lattice., St001617The dimension of the space of valuations of a lattice., St001622The number of join-irreducible elements of a lattice.)
Values
([(0,2),(2,1)],3) => 2
([(0,1),(0,2),(1,3),(2,3)],4)