A339045 -id:A339045 - cp's OEIS frontend

A338999 Number of connected multigraphs with n edges and rooted at two indistinguishable vertices whose removal leaves a connected graph.

1, 1, 3, 11, 43, 180, 804, 3763, 18331, 92330, 478795, 2547885, 13880832, 77284220, 439146427, 2543931619, 15010717722, 90154755356, 550817917537, 3421683388385, 21601986281226, 138548772267326, 902439162209914, 5967669851051612, 40053432076016812

Offset: 1

Rainer Rosenthal, Nov 18 2020

nonn

This sequence counts the CDE-descendants of a single edge A-Z.
[C]onnect: different nodes {P,Q} != {A,Z} may form a new edge P-Q.
[D]issect: any edge P-Q may be dissected into P-M-Q with a new node M.
[E]xtend:  any node P not in {A,Z} may form a new edge P-Q with a new node Q.
These basic operations were motivated by A338487, which seemed to count the CDE-descendants of K_4 with edge A-Z removed.

			The a(3) = 3 CDE-descendants of A-Z with 3 edges are
.
         A          A          A
        ( )        /          /
         o        o - o      o - o
         |           /        \
         Z          Z          Z
.
        DCC        DD         DE
.

Technology Review's Puzzle Corner, How many different resistances can be obtained by combining 10 one ohm resistors? Oct 3, 2003.

Joel Karnofsky, Solution of problem from Technology Review's Puzzle Corner Oct 3, 2003, Feb 23, 2004.

Cf. A180414, A337517, A338487, A339038, A339045, A339065.

\\ See A339065 for G.
InvEulerT(v)={my(p=log(1+x*Ser(v))); dirdiv(vector(#v,n,polcoef(p,n)), vector(#v,n,1/n))}
seq(n)={my(A=O(x*x^n), g=G(2*n, x+A,[]), gr=G(2*n, x+A,[1])/g, u=InvEulerT(Vec(-1+G(2*n, x+A,[1,1])/(g*gr^2))), t=InvEulerT(Vec(-1+G(2*n, x+A,[2])/(g*subst(gr,x,x^2)))), v=vector(n)); for(n=1, #v, v[n]=(u[n]+t[n]-if(n%2==0,u[n/2]-v[n/2]))/2); v} \\ Andrew Howroyd, Nov 20 2020

a(7)-a(25) from Andrew Howroyd, Nov 20 2020

A339205 Number of 2-connected multigraphs with n edges rooted at two noninterchangeable vertices and have no decomposition into parallel components rooted at the two distinguished vertices.

Original entry on oeis.org

1, 0, 0, 0, 1, 6, 39, 221, 1237

Offset: 1

Rainer Rosenthal, Nov 27 2020

Variant of A339123, treating the distinguished points as not interchangeable.

Graphs that are 2-connected also have no decomposition into series components, so the graphs enumerated by this sequence are the minimal subset of oriented networks which when combined in series and parallel produce all possible networks with a source and a sink and in which every edge lies on a path between the source and the sink.

			a(6) = 6, Figures 5 and 6 are A/Z-mirrored images of 1 and 3:
.
      A          A          A          A          A          A
    // \        / \        d \        / \        / \        / \
   //   \      /___\      /   \      /   \      /   \      /   \
   o-----o    o --- o    o-----o    o--o--o    o-----o    o-----o
    \   /      \   /      \   /      \   /     \\   /      \   /
     \ /        \ /        \ /        \ /       \\ /        q /
      Z          Z          Z          Z          Z          Z
.
    Fig. 1     Fig. 2     Fig. 3     Fig. 4     Fig. 5    Fig. 6

Cf. A338999, A339045, A339123.

cp's OEIS Frontend

A338999 Number of connected multigraphs with n edges and rooted at two indistinguishable vertices whose removal leaves a connected graph.

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