A174579 Number of spanning trees in the n-triangular grid graph.

1, 3, 54, 5292, 2723220, 7242690816, 98719805835000, 6861326937782575104, 2423821818614367091537296, 4342290918217084382837760000000, 39389085041906366256386454778172877408, 1807026244113880332171608161401397806958116864

Offset: 0

Author

Alois P. Heinz, Nov 29 2010

Keywords

• nonn

Comments

The n-triangular grid graph has n+1 rows with k vertices in row k. Each vertex is connected to the neighbors in the same row and up to two vertices in each of the neighboring rows. The Graph has A000217(n+1) vertices and 3*A000217(n) edges altogether.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..50
• Eric Weisstein's World of Mathematics, Spanning Tree
• Eric Weisstein's World of Mathematics, Triangular Grid Graph
• Wikipedia, Kirchhoff's theorem

Crossrefs

Cf. A000217, A112676, A266513.

Programs

Maple

with(LinearAlgebra):
tr:= n-> n*(n+1)/2:
a:= proc(n) local h, i, M;
      if n=0 then 1 else
        M:= Matrix(tr(n+1), shape=symmetric);
        for h in [seq(seq([[i, i+j], [i, i+j+1], [i+j, i+j+1]][],
                           i=tr(j-1)+1 .. tr(j-1)+j), j=1..n)]
        do M[h[]]:= -1 od;
        for i to tr(n+1) do M[i, i]:= -add(M[i, j], j=1..tr(n+1)) od;
        Determinant(DeleteColumn(DeleteRow(M, 1), 1))
      fi
    end:
seq(a(n), n=0..12);

Mathematica

tr[n_] := n*(n + 1)/2;
a[0] = 1; a[n_] := Module[{T, M}, T = Table[Table[{{i, i+j}, {i, i+j+1}, {i + j, i+j+1}}, {i, tr[j-1]+1, tr[j-1] + j}], {j, 1, n}] // Flatten[#, 2]&; M = Array[0&, {tr[n+1], tr[n+1]}]; Do[{i, j} = h; M[[i, j]] = -1, {h, T}]; M = M + Transpose[M]; For[i = 1, i <= tr[n+1], i++, M[[i, i]] = -Sum[M[[i, j]], {j, 1, tr[n+1]}]]; Det[Rest /@ Rest[M]]];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Jun 02 2018, from Maple *)

Python

# Using graphillion
from graphillion import GraphSet
def make_n_triangular_grid_graph(n):
    s = 1
    grids = []
    for i in range(n + 1, 1, -1):
        for j in range(i - 1):
            a, b, c = s + j, s + j + 1, s + i + j
            grids.extend([(a, b), (a, c), (b, c)])
        s += i
    return grids
def A174579(n):
    if n == 0: return 1
    universe = make_n_triangular_grid_graph(n)
    GraphSet.set_universe(universe)
    spanning_trees = GraphSet.trees(is_spanning=True)
    return spanning_trees.len()
print([A174579(n) for n in range(8)])  # Seiichi Manyama, Nov 30 2020

Extensions

Indexing changed by Alois P. Heinz, Jun 14 2017