A317059 a(n) is the number of time-dependent assembly trees satisfying the edge gluing rule for a complete graph on n vertices.

1, 1, 3, 21, 255, 4815, 130095, 4763115, 226955925, 13646570175, 1010560060125, 90363456777825, 9599238270346725, 1194935000536101825, 172283712268118826375, 28481473075454845070625, 5351643310498951112521875, 1134140509146174954631081875, 269235074280949277622074328375

Offset: 1

Author

Nick Mayers, Robert Short, Aria Dougherty, Jul 20 2018

Keywords

• easy
• nonn

Comments

A time-dependent assembly tree for a connected graph G=(V, E) on n vertices is a rooted tree, each node of which is label a subset U of V and a nonnegative integer i such that:

1) each internal node has at least two children,

2) there are leaves labeled (v, 0) for each vertex v in V,

3) the label on the root is (V, m) for 1 <= m <= n-1,

4) for each node (U, i) with i

5) if (U, i) and (U', i') are adjacent nodes with U a subset of U', then i

6) for each 0 <= i <= m, there exists a node (U, i) with U a subset of V.

A time-dependent assembly tree is said to satisfy the edge gluing rule if each internal vertex v of G has exactly two children and if U_1 and U_2 are the labels of the children of internal vertex v, then there is an edge (v_1,v_2) in the edge set of G such that v_1 is in U_1 and v_2 is in U_2.

a(n) is also the number of labeled histories possible for n leaves if simultaneous bifurcations are allowed. a(n) is also the number of single-elimination sports tournament schedules possible for n teams if matches involve pairs of teams, arbitrarily many arenas are available, and labeled teams have been specified, but the bracket of matches has not been specified. - Noah A Rosenberg, Feb 20 2025

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..261
• M. Bona and A. Vince, The Number of Ways to Assemble a Graph, arXiv preprint arXiv:1204.3842 [math.CO], 2012.
• Emily H. Dickey and Noah A. Rosenberg, Labelled histories with multifurcation and simultaneity, Phil. Trans. R. Soc. B 380 (2025), 20230307.
• A. Dougherty, N. Mayers, and R. Short, How to Build a Graph in n Days: Some Variants on Graph Assembly, arXiv preprint arXiv:1807.08079 [math.CO], 2018.

Crossrefs

Cf. A006472, A047781, A317057, A317058, A317060.

Programs

Mathematica

Nest[Function[{a, n}, Append[a, Sum[(n!/((2^j) j! (n - 2 j)!)) a[[n - j]], {j, Floor[n/2]}]]][#, Length@ # + 1] &, {1, 1}, 17] (* Michael De Vlieger, Jul 26 2018 *)

PARI

lista(nn) = my(v = vector(nn)); for (n=1, nn, if (n<=2, v[n] = 1, v[n] = sum(j=1, n\2, (n!/((2^j)*j!*(n-2*j)!))*v[n-j]))); v; \\ Michel Marcus, Aug 08 2018

Sage

@cached_function
def TimeDepenEdgeComp(n):
    if n==1:
        return 1
    elif n==2:
        return 1
    else:
        return sum((factorial(n)/((2^j)*factorial(j)*factorial(n-2*j)))*TimeDepenEdgeComp(n-j) for j in range(1, n//2+1))
print(",".join(str(TimeDepenEdgeComp(i)) for i in range(1, 20)))

Formula

a(n) = Sum_{j=1..floor(n/2)}(n!/((2^j)j!(n-2j)!))*a(n-j), a(1)=a(2)=1.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.