RFE September 2026: almost trivalent maps

[Sean A. Irvine]

Hi,

The six sequences A002006, A002007, A002008, A002009, A002010, A002012 all have the name "Almost trivalent maps" and are lacking in detail. They are likely all related to A002005 for which we have considerably more information and detail. Please determine what these sequences are and improve the entries accordingly.

Last month's question was resolved very quickly, thanks to Aitzaz Imtiaz, Michael S. Branicky, David Consiglio, Jr., James C. McMahon and others.

I track these requests for enhancement here (a few others still remain open):

https://oeis.org/wiki/User:Sean_A._Irvine/Requests_for_Enhancements#Requests_for_Enhancements

Sean.

[Elijah Beregovsky]

Dear Sean!

I happened to notice these sequences a couple days ago and here’s what I’ve been able to find so far:

A map is an embedding of a (multi)graph into a compact orientable surface (for most of the sequences in question the surface is likely a sphere) such that the connected components of the complement (faces) are homeomorphic to an open disk, considered up to continuous deformation.

A map is labelled by giving each half-edge a label. A map is rooted by distinguishing one of the half-edges. Then its incident vertex is called the root vertex, and the face to the left of it is called the root face.

A rooted map is called almost trivalent or almost cubic if all of its non-root vertices have degree 3. Equivalently, if we take a dual map, it’s almost cubic (also called a near-triangulation) if all of the non-root faces are triangles. A near-triangulation is of type (n,m) if it has n interior (=non-root) faces and the exterior (=root) face is an m-gon. According to Mathai and Rathie the original paper by Mullin, Nemeth and Schellenberg (which I could not locate) contained the tables for the number of maps for n,m=1,2,...14, and so is likely the source of the OEIS sequences.

Mathai and Rathie give several formulas for the number of maps t(n,m), some of which correspond to existing OEIS sequences:

t(0,2r)     = Catalan numbers

t(1,2r-1)  = C(2n,n-1)

t(2,2r)     = 2(r+1)C(2n,n-1) = A002011

t(3,2r-1) = 2/3 * (r+1)(r+2)C(2n,n-1) = A002012

t(4,2r)    = 2/3 * (r+1)(r^2+7r+16)C(2n,n-1) which is not on the OEIS, but it gives, e.g. t(4,6)=1840=A002008(3), so probably related.

t(5,2r-1)  = 2/15 * (r+1)(r^3+11r^2+50r+58)C(2n,n-1), also not on the OEIS, but gives, e.g. t(5,5)=2672=A002007(3) so also likely related.

Gonna search some more and maybe edit all of these if I figure it out before y’all do.

Best wishes,

Elijah

[Elijah Beregovsky]

Looking at the sequences more closely I notice that:

A002006(n) is equal to t(2n,4),

A002007(n) is equal to t(2n+1,5),

A002008(n) is equal to t(2n,6),

A002009(n) is equal to t(2n+1,7),

A002010(n) is equal to t(2n,8) for the first three terms. 

I conjecture these are actually the definitions. Could someone calculate some more terms, please, just to be sure?

Elijah

[Brendan]

My library has the 1970 proceedings these sequences appeared in and I'll have a copy early next week.  Whoever wants a copy, send me mail.

Brendan.

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[Brendan]

To my surprise, the paper of Mullin et al arrived on the weekend.  I sent a copy to Charles,

does anyone else want a copy?

Brendan.

[Sean A. Irvine]

I've annotated the paper with the corresponding A-numbers (as a thin justification for the OEIS storing a copy) and made it available in the relevant sequences.

https://oeis.org/A002005/a002005.pdf

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