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A376747
Number of non-isomorphic colorings of a toroidal n X n grid using exactly two swappable colors.
4
0, 4, 31, 2107, 671103, 954459519, 5744387279871, 144115188277194943, 14925010118699132241919, 6338253001141180784480847871, 10985355337065420437221545952731135, 77433143050453552574875182200691073835007, 2213872302702432822841084717014014514981767643135, 256208234097415541381052629523530965709132732687965552639
OFFSET
1,2
REFERENCES
F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.
FORMULA
a(n) = (1/(n^2*2!))*(Sum_{sigma in S_2} Sum_{d|n} Sum_{f|n} phi(d) phi(f) [[forall j_l(sigma) > 0 : l|lcm(d,f) ]] P(gcd(d,f)*(n/d)*(n/f), sigma)) where P(F, sigma) = F! [z^F] Product_{l=1..2} (exp(lz)-1)^j_l(sigma). The notation j_l(sigma) is from the Harary text and gives the number of cycles of length l in the permutation sigma. [[.]] is an Iverson bracket.
CROSSREFS
Main diagonal of A294791.
Sequence in context: A005841 A005828 A084764 * A061789 A103909 A309906
KEYWORD
nonn
AUTHOR
Marko Riedel, Oct 03 2024
STATUS
approved