A036368 -id:A036368 - cp's OEIS frontend

A355049 Number of chiral pairs of orthoplex n-ominoes with cell centers determining n-3 space.

8, 76, 440, 2019, 8147, 30367, 107061, 361655, 1181761, 3762817, 11733393, 35957132, 108591703, 323914688, 955984083, 2795513143, 8108894051, 23354358683, 66838785954, 190211189706, 538567451991, 1517943035326

Offset: 7

Robert A. Russell, Jun 16 2022

Orthoplex polyominoes are connected sets of cells of regular tilings with Schläfli symbols {}, {4}, {3,4}, {3,3,4}, {3,3,3,4}, etc. These are tilings of regular orthoplexes projected on their circumspheres. Orthoplex polyominoes are equivalent to multidimensional polyominoes that do not extend more than two units along any axis, i.e., fit within a 2^d cube. Each member of a chiral pair is a reflection but not a rotation of the other.

			a(7)=8 because there are 8 pairs of chiral heptominoes in 2^4 space. See trunks 1, 6, 8, 12, 13, 19, 27, and 28 in the linked Trunk Generating Functions.

Robert A. Russell, Table of n, a(n) for n = 7..100
Robert A. Russell, Trunk Generating Functions

Cf. A355047 (oriented), A355048 (unoriented), A355050 (achiral) A355051 (asymmetric), A045648 (rooted chiral).

Other dimensions: A036368 (n-2), A045649 (n-1), A355054 (multidimensional).

sc[n_,k_] := sc[n,k] = c[n+1-k,1] + If[n<2k, 0, sc[n-k,k](-1)^k];
c[1,1] := 1; c[n_,1] := c[n,1] = Sum[c[i,1] sc[n-1,i]i, {i,1,n-1}]/(n-1);
c[n_,k_] := c[n, k] = Sum[c[i, 1] c[n-i, k-1], {i,1,n-1}];
nmax = 30; K[x_] := Sum[c[i,1] x^i, {i,0,nmax}]
Drop[CoefficientList[Series[(14 K[x]^6 + 3 K[x]^7 + 6 K[x]^4 K[-x^2] + 6 K[x]^5 K[-x^2] - 18 K[x]^2 K[-x^2]^2 + 3 K[x]^3 K[-x^2]^2 - 10 K[-x^2]^3 - 6 K[x] K[-x^2]^3 + 4 K[x^3]^2 - 6 K[x] K[-x^2] K[-x^4] + 4 K[-x^6]) / 24 + K[x]^3 (38 K[x]^4 + 9 K[x]^5 + 4 K[x]^2 K[-x^2] + 10 K[x]^3 K[-x^2] - 2 K[-x^2]^2 + K[x] K[-x^2]^2) / (8(1-K[x])) + K[x]^6 (5 K[x] + 16 K[x]^2 + 6 K[x]^3 + K[-x^2] + 2 K[x] K[-x^2]) / (2(1-K[x])^2) - K[-x^2]^2 (K[x]^4 + 2 K[x] K[-x^2] + 4 K[x]^2 K[-x^2] + 2 K[-x^2]^2 + 5 K[x] K[-x^2]^2 + K[-x^4] + K[x] K[-x^4]) / (4(1-K[-x^2])) + K[x]^7 (2 + 42 K[x] + 51 K[x]^2 + 24 K[x]^3 + 3 K[-x^2]) / (12(1-K[x])^3) + (K[x] K[x^3]^2) / (3(1-K[x^3])) - K[x]^2 K[-x^2]^2 (2 K[x] + 5 K[x]^3 + 2 K[-x^2] + K[x] K[-x^2]) / (4(1-K[x]) (1-K[-x^2])) - K[-x^2]^4 (8 + K[x] + 8 K[x] K[-x^2]) / (8(1-K[-x^2])^2) + K[x]^9 (17 + 8 K[x]) / (8(1-K[x])^4) - K[x]^5 (1 + 4 K[x]) K[-x^2]^2 / (4(1-K[x])^2 (1-K[-x^2])) + (K[x] K[-x^4]^2) / (4(1-K[-x^4])) + (3 K[x]^10) / (8(1-K[x])^5) - ((K[x]^6 K[-x^2]^2) / (4(1-K[x])^3 (1-K[-x^2]))) - (((1 + K[x]) K[-x^2]^5) / (4(1-K[-x^2])^3)) + ((1 + K[x]) K[-x^2] K[-x^4]^2) / (4(1-K[-x^2]) (1-K[-x^4])) - ((K[x]^2 K[-x^2]^4) / (8(1-K[x]) (1-K[-x^2])^2)), {x,0,nmax}], x], 7]

a(n) = A355047(n) - A355048(n) = (A355047(n) - A355050(n)) / 2 = A355048(n) - A355050(n).

G.f.: (14 C(x)^6 + 3 C(x)^7 + 6 C(x)^4 C(-x^2) + 6 C(x)^5 C(-x^2) - 18 C(x)^2 C(-x^2)^2 + 3 C(x)^3 C(-x^2)^2 - 10 C(-x^2)^3 - 6 C(x) C(-x^2)^3 + 4 C(x^3)^2 - 6 C(x) C(-x^2) C(-x^4) + 4 C(-x^6)) / 24 + C(x)^3 (38 C(x)^4 + 9 C(x)^5 + 4 C(x)^2 C(-x^2) + 10 C(x)^3 C(-x^2) - 2 C(-x^2)^2 + C(x) C(-x^2)^2) / (8(1-C(x))) + C(x)^6 (5 C(x) + 16 C(x)^2 + 6 C(x)^3 + C(-x^2) + 2 C(x) C(-x^2)) / (2(1-C(x))^2) - C(-x^2)^2 (C(x)^4 + 2 C(x) C(-x^2) + 4 C(x)^2 C(-x^2) + 2 C(-x^2)^2 + 5 C(x) C(-x^2)^2 + C(-x^4) + C(x) C(-x^4)) / (4(1-C(-x^2))) + C(x)^7 (2 + 42 C(x) + 51 C(x)^2 + 24 C(x)^3 + 3 C(-x^2)) / (12(1-C(x))^3) + (C(x) C(x^3)^2) / (3(1-C(x^3))) - C(x)^2 C(-x^2)^2 (2 C(x) + 5 C(x)^3 + 2 C(-x^2) + C(x) C(-x^2)) / (4(1-C(x)) (1-C(-x^2))) - C(-x^2)^4 (8 + C(x) + 8 C(x) C(-x^2)) / (8(1-C(-x^2))^2) + C(x)^9 (17 + 8 C(x)) / (8(1-C(x))^4) - C(x)^5 (1 + 4 C(x)) C(-x^2)^2 / (4(1-C(x))^2 (1-C(-x^2))) + (C(x) C(-x^4)^2) / (4(1-C(-x^4))) + (3 C(x)^10) / (8(1-C(x))^5) - ((C(x)^6 C(-x^2)^2) / (4(1-C(x))^3 (1-C(-x^2)))) - (((1 + C(x)) C(-x^2)^5) / (4(1-C(-x^2))^3)) + ((1 + C(x)) C(-x^2) C(-x^4)^2) / (4(1-C(-x^2)) (1-C(-x^4))) - ((C(x)^2 C(-x^2)^4) / (8(1-C(x)) (1-C(-x^2))^2)) where C(x) is the generating function for chiral n-ominoes in n-1 space, one cell labeled in A045648.

A355998 Number of fixed orthoplex n-ominoes with cell centers determining (n-2)-space.

Original entry on oeis.org

1, 48, 1728, 62720, 2457600, 105815808, 5017600000, 261227298816, 14860167413760, 918839084134400, 61439672177393700, 4421589120000000000, 340976534987475000000, 28064307240230900000000, 2456376885785930000000000

Offset: 4

Robert A. Russell, Jul 22 2022

nonn

Orthoplex polyominoes are connected sets of cells of regular tilings with Schläfli symbols {}, {4}, {3,4}, {3,3,4}, {3,3,3,4}, etc. These are tilings of regular orthoplexes projected on their circumspheres. Orthoplex polyominoes are equivalent to multidimensional polyominoes that do not extend more than two units along any axis, i.e., fit within a 2^d cube. Two fixed polyominoes are identical only if one is a translation of the other.

			For A(4)=1, all 4 squares of the 2^2 space are used.

Cf. A171860 (multidimensional), A036367 (unoriented), A036368 (chiral), A036369 (asymmetric).

Diagonal 2 of A355997.

Table[2^(n-3) n^(n-5) (n-2) (n-3)^2, {n,4,30}]

a(n) = 2^(n-3) * n^(n-5) * (n-2) * (n-3)^2.

a(n) ~ A171860(n) / 2.

cp's OEIS Frontend

A355049 Number of chiral pairs of orthoplex n-ominoes with cell centers determining n-3 space.

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