A222188
Table read by antidiagonals: number of toroidal m X n binary arrays, allowing rotation and/or reflection of the rows and/or the columns.
Original entry on oeis.org
2, 3, 3, 4, 7, 4, 6, 13, 13, 6, 8, 34, 36, 34, 8, 13, 78, 158, 158, 78, 13, 18, 237, 708, 1459, 708, 237, 18, 30, 687, 4236, 14676, 14676, 4236, 687, 30, 46, 2299, 26412, 184854, 340880, 184854, 26412, 2299, 46
Offset: 1
Array begins:
2, 3, 4, 6, 8, 13, 18, 30, ...
3, 7, 13, 34, 78, 237, 687, 2299, ...
4, 13, 36, 158, 708, 4236, 26412, 180070, ...
6, 34, 158, 1459, 14676, 184854, 2445918, 33888844, ...
8, 78, 708, 14676, 340880, 8999762, 245619576, 6873769668, ...
...
- G. C. Greubel, Table of n, a(n) for the first 100 antidiagonals, flattened
- S. N. Ethier, Counting toroidal binary arrays, arXiv:1301.2352 [math.CO], 2013 and J. Int. Seq. 16 (2013) #13.4.7.
- S. N. Ethier and Jiyeon Lee, Counting toroidal binary arrays, II, arXiv:1502.03792v1 [math.CO], Feb 12, 2015 and J. Int. Seq. 18 (2015) # 15.8.3.
- S. N. Ethier and Jiyeon Lee, Parrondo games with two-dimensional spatial dependence, arXiv:1510.06947 [math.PR], 2015.
- Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023. See p. 3.
-
b1[m_, n_] := Sum[EulerPhi[c]*EulerPhi[d]*2^(m*n/LCM[c, d]), {c, Divisors[ m]}, {d, Divisors[n]}]/(4*m*n); b2a[m_, n_] := If[OddQ[m], 2^((m+1)*n/2) /(4*n), (2^(m*n/2) + 2^((m+2)*n/2))/(8*n)]; b2b[m_, n_] := DivisorSum[n, If[# >= 2, EulerPhi[#]*2^((m*n)/#), 0]&]/(4*n); b2c[m_, n_] := If[OddQ[ m], Sum[If [OddQ[n/GCD[j, n]], 2^((m+1)*GCD[j, n]/2) - 2^(m*GCD[j, n]), 0], {j, 1, n-1}]/(4*n), Sum[If[OddQ[n/GCD[j, n]], 2^(m*GCD[j, n]/2) + 2^((m+2)*GCD[j, n]/2) - 2^(m*GCD[j, n]+1), 0], {j, 1, n-1}]/(8*n)]; b2[m_, n_] := b2a[m, n] + b2b[m, n] + b2c[m, n]; b3[m_, n_] := b2[n, m]; b4oo[m_, n_] := 2^((m*n-3)/2); b4eo[m_, n_] := 3*2^(m*n/2 - 3); b4ee[m_, n_] := 7*2^(m*n/2-4); a[m_, n_] := Module[{b}, If [OddQ[m], If [OddQ[n], b = b4oo[m, n], b = b4eo[m, n]], If[OddQ[n], b = b4eo[m, n], b = b4ee[m, n]]]; b += b1[m, n] + b2[m, n] + b3[m, n]; Return[b]]; Table[a[m - n+1, n], {m, 1, 10}, {n, 1, m}] // Flatten (* Jean-François Alcover, Dec 05 2015, adapted from Michel Marcus's PARI script *)
-
odd(n) = n%2;
b1(m,n) = sumdiv(m, c, sumdiv(n, d, eulerphi(c)*eulerphi(d)*2^(m*n/lcm(c,d))))/(4*m*n);
b2a(m,n) = if (odd(m), 2^((m+1)*n/2)/(4*n), (2^(m*n/2)+2^((m+2)*n/2))/(8*n));
b2b(m,n) = sumdiv(n, d, if (d>=2, eulerphi(d)*2^((m*n)/d), 0))/(4*n);
b2c(m,n) = if (odd(m), sum(j=1, n-1, if (odd(n/gcd(j,n)), 2^((m+1)*gcd(j,n)/2)-2^(m*gcd(j,n))))/(4*n), sum(j=1, n-1, if (odd(n/gcd(j,n)), 2^(m*gcd(j,n)/2)+2^((m+2)*gcd(j,n)/2)-2^(m*gcd(j,n)+1)))/(8*n));
b2(m,n) = b2a(m,n) + b2b(m,n) + b2c(m,n);
b3(m,n) = b2(n,m);
b4oo(m,n) = 2^((m*n - 3)/2);
b4eo(m,n) = 3*2^(m*n/2 - 3);
b4ee(m,n) = 7*2^(m*n/2 - 4);
a(m,n) = {if (odd(m), if (odd(n), b = b4oo(m,n), b = b4eo(m,n)), if (odd(n), b = b4eo(m,n), b = b4ee(m,n))); b += b1(m,n) + b2(m,n) + b3(m,n); return (b);}
\\ Michel Marcus, Feb 13 2013
A255016
Number of toroidal n X n binary arrays, allowing rotation and/or reflection of rows and/or columns as well as matrix transposition.
Original entry on oeis.org
1, 2, 6, 26, 805, 172112, 239123150, 1436120190288, 36028817512382026, 3731252531904348833632, 1584563250300891724601560272, 2746338834266358751489231123956672, 19358285762613388352671214587818634041520
Offset: 0
- Michael De Vlieger, Table of n, a(n) for n = 0..57
- S. N. Ethier and Jiyeon Lee, Counting toroidal binary arrays, II, arXiv:1502.03792 [math.CO], Feb 12, 2015 and J. Int. Seq. 18 (2015) # 15.8.3.
- S. N. Ethier and Jiyeon Lee, Parrondo games with two-dimensional spatial dependence, arXiv preprint arXiv:1510.06947 [math.PR], 2015.
- Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023. See p. 3.
- Wikipedia, Pólya enumeration theorem.
Cf.
A184271 (number of m X n binary arrays allowing rotation of rows/columns),
A179043 (main diagonal of
A184271),
A222188 (number of m X n binary arrays allowing rotation/reflection of rows/columns),
A209251 (main diagonal of
A222188),
A255015 (number of n X n binary arrays allowing rotation of rows/columns as well as matrix transposition).
-
a[n_] := (8 n^2)^(-1) Sum[If[Mod[n, c] == 0, Sum[If[Mod[n, d] == 0, EulerPhi[c] EulerPhi[d] 2^(n^2/ LCM[c, d]), 0], {d, 1, n}], 0], {c, 1, n}] + (4 n)^(-1) Sum[If[Mod[n, d] == 0, EulerPhi[d] 2^(n^2/d), 0], {d, 1, n}] + If[Mod[n, 2] == 1, (4 n)^(-1) Sum[If[Mod[n, d] == 0 && Mod[d, 2] == 1, EulerPhi[d] (2^(n (n + 1)/(2 d)) - 2^(n^2/d)), 0], {d, 1, n}],(8 n)^(-1) Sum[If[Mod[n, d] == 0 && Mod[d, 2] == 1, EulerPhi[d] (2^(n^2/(2 d)) + 2^(n (n + 2)/(2 d)) - 2 2^(n^2/d)), 0], {d, 1, n}]] + (1/2) If[Mod[n, 2] == 1, 2^((n^2 - 3)/2), 7 2^(n^2/2 - 4)] + (4 n)^(-1) Sum[If[Mod[n, d] == 0, EulerPhi[d] 2^(n (n + d - 2 IntegerPart[d/2])/(2 d)), 0], {d, 1, n}] + If[Mod[n, 2] == 1, 2^((n^2 - 5)/4), 5 2^(n^2/4 - 3)];
A255015
Number of toroidal n X n binary arrays, allowing rotation of rows and/or columns as well as matrix transposition.
Original entry on oeis.org
1, 2, 6, 44, 2209, 674384, 954623404, 5744406453840, 144115192471496836, 14925010120653819583840, 6338253001142965335834871200, 10985355337065423791175013899922368, 77433143050453552587418968170813573149024
Offset: 0
- Michael De Vlieger, Table of n, a(n) for n = 0..57
- S. N. Ethier and Jiyeon Lee, Counting toroidal binary arrays, II, arXiv:1502.03792v1 [math.CO], Feb 12, 2015 and J. Int. Seq. 18 (2015) # 15.8.3.
- Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023. See p. 3.
- Wikipedia, Polya enumeration theorem.
Cf.
A184271 (number of m X n binary arrays allowing rotation of rows/columns),
A179043 (main diagonal of
A184271),
A222188 (number of m X n binary arrays allowing rotation/reflection of rows/columns),
A209251 (main diagonal of
A222188),
A255016 (number of n X n binary arrays allowing rotation/reflection of rows/columns as well as matrix transposition).
-
a[n_] := (2 n^2)^(-1) Sum[If[Mod[n, c] == 0, Sum[If[Mod[n, d] == 0, EulerPhi[c] EulerPhi[d] 2^(n^2/ LCM[c, d]), 0], {d, 1, n}], 0], {c, 1, n}] + (2 n)^(-1) Sum[If[Mod[n, d] == 0, EulerPhi[d] 2^(n (n + d - 2 IntegerPart[d/2])/(2 d)), 0], {d, 1, n}];
Showing 1-3 of 3 results.