A287239 Number of inequivalent n X n matrices over an alphabet of size 6 under action of dihedral group of the square D_4, with one-sixth each of 1s, 2s, 3s, 4s, 5s and 6s (ordered occurrences rounded up/down if n^2 != 0 mod 6).
1, 1, 1, 5688, 504508320, 2029169127793680, 333772217080092664473600, 1966297518276227170017585421188600, 474436367892839446541884570454351985506872320, 4529567636413022031420100639004131328550592354551163392000, 1664947024157601976065851576560401128416782438266187161307818265349050000
Offset: 0
Keywords
Examples
For n = 3 the a(3) = 5688 solutions are colorings of 3 X 3 matrices in 6 colors inequivalent under the action of D_4 with exactly 2 occurrences of each color (coefficient of x1^2 x2^2 x3^2 x4^2 x5^2 x6^2).
Links
- María Merino, Table of n, a(n) for n = 0..34
- M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).
Formula
G.f.: g(x1,x2,x3,x4,x5,x6)=1/8*(y1^(n^2)+2*y1^n*y2^((n^2-n)/2)+3*y2^(n^2/2)+2*y4^(n^2/4)) if n even and 1/8*(y1^(n^2)+4*y1^n*y2^((n^2-n)/2)+y1*y2^((n^2-1)/2)+2*y1*y4^((n^2-1)/4)) if n odd, where coefficient correspond to y1=Sum_{i=1..6} x_i, y2=Sum_{i=1..6} x_i^2, y4=Sum_{i=1..6} x_i^4 and occurrences of numbers are ceiling(n^2/6) for the first k numbers and floor(n^2/6) for the last (6-k) numbers, if n^2 = k mod 6.
Comments