A047770 -id:A047770 - cp's OEIS frontend

A047769 Number of chiral pairs of dissectable polyhedra with n tetrahedral cells and symmetry of type M.

0, 0, 0, 1, 0, 5, 0, 26, 0, 133, 0, 708, 0, 3860, 0, 21604, 0, 123266, 0, 715216, 0, 4206956, 0, 25032840, 0, 150413322, 0, 911379384, 0, 5562367173, 0, 34164355715, 0, 211015212580, 0, 1309815397995, 0, 8166460799097, 0, 51120054233490

Offset: 1

N. J. A. Sloane

nonn

One of 17 different symmetry types comprising A007173 and A027610 and one of 7 for A371350. Also the number of tetrahedral clusters or polyominoes of the regular tiling with Schläfli symbol {3,3,oo}, both having type M chiral symmetry and n tetrahedral cells. The axis of symmetry is the altitude of a tetrahedral face (21); the order of the symmetry group is 2. An achiral polyomino is identical to its reflection. - Robert A. Russell, Mar 22 2024

L. W. Beineke and R. E. Pippert, Enumerating dissectable polyhedra by their automorphism groups, Canad. J. Math., 26 (1974), 50-67.
Robert A. Russell, Mathematica Graphics3D program for A047769 example

Cf. A047770.

Cf. A007173 (oriented), A027610 (unoriented), A371350 (chiral), A001764 (rooted), A047764 (type Q), A047765 (type P), A047766 (type O).

Table[If[OddQ[n],0,(Binomial[3n/2,n/2]/(n+1)-If[0==Mod[n,4], 2Binomial[3n/4,n/4]/(n+2),4Binomial[(3n-2)/4,(n-2)/4]/(n+2),0]- If[2==Mod[n,6],3Binomial[3(n-2)/6,(n-2)/6]/(n+1)- If[2==Mod[n,12],6Binomial[3(n-2)/12,(n-2)/12],12Binomial[(3n-12)/12,
(n-8)/12]]/(n+4),0])/2],{n,60}] (* Robert A. Russell, Mar 22 2024 *)

If n=2m then (1/2)*(A001764(m) - 2*A047766(n) - A047765(n) - A047764(n)), otherwise 0.

G.f.: (G(z^2) - G(z^4) - z^2 * (G(z^4)^2 + G(z^6) - G(z^12) - z^6*G(z^12)^2)) / 2, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. - Robert A. Russell, Mar 22 2024

cp's OEIS Frontend

A047769 Number of chiral pairs of dissectable polyhedra with n tetrahedral cells and symmetry of type M.

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