Number of convex polyhedra whose faces are regular polygons and whose largest face is an $n$-gon

Question

I'm trying to count up the number of convex polyhedra whose faces are regular polygons and whose largest face is an $n$-gon. (I.e. either a uniform polyhedron or a Johnson solid.) If I've done my counting correct, this does not appear in the On-Line Encyclopedia of Integer Sequences, and I'd like to add it.

Are my lists complete and correct?

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For $n = 3$, I've counted $8$:

Tetrahedron, octahedron, icosahedron, $J_{12}$, $J_{13}$, $J_{17}$, $J_{51}$, $J_{84}$.

For $n=4$, I've counted $30$:

Cube, cuboctahedron, rhombicuboctahedron, snub cube, triangular prism, square antiprism, $J_{1}$, $J_{7}$, $J_{8}$, $J_{10}$, $J_{14}$, $J_{15}$, $J_{16}$, $J_{26}$, $J_{27}$, $J_{28}$, $J_{29}$, $J_{35}$, $J_{36}$, $J_{37}$, $J_{44}$, $J_{45}$, $J_{49}$, $J_{50}$, $J_{85}$, $J_{86}$, $J_{87}$, $J_{88}$, $J_{89}$, $J_{90}$.

For $n = 5$, I've counted $37$:

Dodecahedron, icosidodecahedron, rhombicosidodecahedron, snub dodecahedron, pentagonal prism, pentagonal antiprism, $J_{2}$, $J_{9}$, $J_{11}$, $J_{30}$, $J_{31}$, $J_{38}$, $J_{39}$, $J_{46}$, $J_{52}$, $J_{53}$, $J_{62}$, $J_{63}$, $J_{64}$, $J_{91}$, $J_{32}$, $J_{33}$, $J_{40}$, $J_{41}$, $J_{47}$, $J_{61}$, $J_{59}$, $J_{60}$, $J_{58}$, $J_{34}$, $J_{42}$, $J_{43}$, $J_{48}$, $J_{72}$, $J_{73}$, $J_{74}$, $J_{75}$.

For $n = 6$, I've counted $14$:

Truncated tetrahedron, truncated octahedron, truncated icosahedron, hexagonal prism, hexagonal antiprism, $J_{3}$, $J_{18}$, $J_{22}$, $J_{54}$, $J_{55}$, $J_{56}$, $J_{57}$, $J_{65}$, $J_{92}$.

For $n = 7$, I've counted $2$:

Heptagonal prism, heptagonal antiprism.

For $n = 8$, I've counted $9$:

Truncated cube, truncated cuboctahedron, octagonal prism, octagonal antiprism, $J_{4}$, $J_{19}$, $J_{23}$, $J_{66}$, $J_{67}$.

For $n = 9$, I've counted $2$:

Enneagonal prism, enneagonal antiprism.

For $n = 10$, I've counted $22$:

Truncated dodecahedron, truncated icosidodecahedron, decagonal prism, decagonal antiprism, $J_{5}$, $J_{6}$, $J_{20}$, $J_{21}$, $J_{24}$, $J_{25}$, $J_{68}$, $J_{69}$, $J_{70}$, $J_{71}$, $J_{76}$, $J_{77}$, $J_{78}$, $J_{79}$, $J_{80}$, $J_{81}$, $J_{82}$, $J_{83}$.

For $n > 10$, I've counted $2$:

$n$-gonal prism, $n$-gonal antiprism.

Answer 1

It turns out your list is complete. Having recognized the five Platonic and 13 Archimedean solids by the names you list, I turn to the Johnson solids, which are defined as all other strictly convex (dihedral angles all <180°) polyhedra consisting of regular faces besides prisms and antiprisms. Norman Johnson identified 92 solids, all of which are in your list, in 1966; the numbering we use today is from his identification. That the 92 solids so identified is the complete list of Johnson solids was proved by Victor Zalgaller in 1969.

I also checked your count of the Johnson solids for each individual value of $n$ in your lists. These too are correct.

Answer 2

After Oscar posted his confirmation, I triple-checked with the following Mathematica program and was able to have the computer confirm it too.

MaxFace[l_] := Max[Length /@ l];
a[n_] := Count[
  Join[
    MaxFace /@ PolyhedronData["Platonic", "FaceIndices"],
    MaxFace /@ PolyhedronData["Archimedean", "FaceIndices"],
    MaxFace /@ PolyhedronData["Johnson", "FaceIndices"],
    Range[4, n], (*Prisms, including triangular prism, excluding cube*)
    Range[4, n]  (*Antiprisms, excluding octahedron*)
  ],
  n
]