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Identifier
Values
=>
Cc0020;cc-rep
([],1)=>0 ([],2)=>1 ([(0,1)],2)=>1 ([],3)=>0 ([(1,2)],3)=>1 ([(0,2),(1,2)],3)=>2 ([(0,1),(0,2),(1,2)],3)=>3 ([],4)=>0 ([(2,3)],4)=>0 ([(1,3),(2,3)],4)=>1 ([(0,3),(1,3),(2,3)],4)=>3 ([(0,3),(1,2)],4)=>1 ([(0,3),(1,2),(2,3)],4)=>3 ([(1,2),(1,3),(2,3)],4)=>3 ([(0,3),(1,2),(1,3),(2,3)],4)=>6 ([(0,2),(0,3),(1,2),(1,3)],4)=>6 ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>10 ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>15 ([],5)=>0 ([(3,4)],5)=>0 ([(2,4),(3,4)],5)=>0 ([(1,4),(2,4),(3,4)],5)=>1 ([(0,4),(1,4),(2,4),(3,4)],5)=>4 ([(1,4),(2,3)],5)=>0 ([(1,4),(2,3),(3,4)],5)=>1 ([(0,1),(2,4),(3,4)],5)=>1 ([(2,3),(2,4),(3,4)],5)=>0 ([(0,4),(1,4),(2,3),(3,4)],5)=>4 ([(1,4),(2,3),(2,4),(3,4)],5)=>3 ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)=>9 ([(1,3),(1,4),(2,3),(2,4)],5)=>4 ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)=>10 ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>8 ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)=>9 ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>18 ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)=>20 ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>32 ([(0,4),(1,3),(2,3),(2,4)],5)=>4 ([(0,1),(2,3),(2,4),(3,4)],5)=>3 ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)=>9 ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)=>18 ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)=>10 ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)=>19 ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)=>32 ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)=>18 ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>16 ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>31 ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>51 ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)=>33 ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)=>52 ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>77 ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>110 ([],6)=>0 ([(4,5)],6)=>0 ([(3,5),(4,5)],6)=>0 ([(2,5),(3,5),(4,5)],6)=>0 ([(1,5),(2,5),(3,5),(4,5)],6)=>1 ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>5 ([(2,5),(3,4)],6)=>0 ([(2,5),(3,4),(4,5)],6)=>0 ([(1,2),(3,5),(4,5)],6)=>0 ([(3,4),(3,5),(4,5)],6)=>0 ([(1,5),(2,5),(3,4),(4,5)],6)=>1 ([(0,1),(2,5),(3,5),(4,5)],6)=>1 ([(2,5),(3,4),(3,5),(4,5)],6)=>0 ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>5 ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>3 ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>12 ([(2,4),(2,5),(3,4),(3,5)],6)=>0 ([(0,5),(1,5),(2,4),(3,4)],6)=>1 ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)=>4 ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)=>5 ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>0 ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>3 ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)=>5 ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)=>14 ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>8 ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>12 ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>26 ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>12 ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)=>14 ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>32 ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>20 ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>26 ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>52 ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>64 ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>96 ([(0,5),(1,4),(2,3)],6)=>0 ([(1,5),(2,4),(3,4),(3,5)],6)=>1 ([(0,1),(2,5),(3,4),(4,5)],6)=>1 ([(1,2),(3,4),(3,5),(4,5)],6)=>0 ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)=>5 ([(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)=>12 ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>9 ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>27 ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)=>5 ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)=>14 ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>11 ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)=>15 ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>8 ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)=>12 ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>30 ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>26 ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)=>21 ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)=>53 ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)=>5 ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)=>4 ([(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)=>14 ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)=>12 ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)=>12 ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>8 ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)=>12 ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)=>30 ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)=>26 ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>27 ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>54 ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>0 ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>26 ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>16 ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>47 ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)=>34 ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)=>62 ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>52 ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>53 ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>58 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)=>98 ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>57 ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>40 ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>47 ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>91 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>108 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>160 ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)=>32 ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>24 ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>57 ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>45 ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>97 ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)=>15 ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)=>33 ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)=>30 ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>31 ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>26 ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)=>30 ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>58 ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>53 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)=>59 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>99 ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)=>65 ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)=>57 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>109 ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>103 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>168 ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>97 ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>75 ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>91 ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>152 ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)=>104 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>99 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>161 ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>254 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)=>117 ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>180 ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>174 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>270 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>387 ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)=>9 ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)=>26 ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)=>27 ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>16 ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>54 ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>47 ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>93 ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)=>59 ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)=>105 ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>91 ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>99 ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>98 ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>162 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>176 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>256 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6)=>63 ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>59 ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>105 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>170 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)=>111 ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>176 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>169 ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>264 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>389 ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>168 ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>161 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>255 ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>152 ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>125 ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>235 ([(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)=>378 ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>561 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>272 ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>262 ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>271 ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>398 ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>408 ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>572 ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>792 ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>1080
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Description
The number of two-component spanning forests of a graph.
A spanning subgraph is a subgraph which contains all vertices of the ambient graph. A forest is a graph which contains no cycles, and has any number of connected components. A two-component spanning forest is a spanning subgraph which contains no cycles and has two connected components.
References
[1] Kassel, A., Kenyon, R., Wu, W. Random two-component spanning forests MathSciNet:3414453 zbMATH:1334.82011 DOI:10.1214/14-AIHP625 arXiv:1203.4858
[2] Number of forests with two connected components in the complete graph K_n. OEIS:A083483
[3] Number a(n) of forests with two components in the complete bipartite graph K_n,n. OEIS:A100070
Code
def statistic(g):
    n = len(g.vertices())
    L = g.kirchhoff_matrix()
    # reduced Laplacian matrix
    Lred = L[:n-1,:n-1]
    a = 0
    for i in range(n-1):
        skip_i = [j for j in range(n-1) if j != i]
        # add number of forests rooted at i and n-1
        a += Lred[skip_i,skip_i].det()
    for (i,j,_) in g.edges():
        if i != n-1 and j != n-1 and i != j:
            skip_ij = [k for k in range(n-1) if k != i and k != j]
            # subtract number of forests rooted at e+, e-, and n-1
            a -= Lred[skip_ij, skip_ij].det()
    return a
Created
Jul 26, 2022 at 11:37 by Harry Richman
Updated
Jul 26, 2022 at 11:37 by Harry Richman