Frequency squares are a generalization of Latin squares. A frequency square of order n with frequency vector l=(l1,...,ls) is an n x n matrix with entries from {1,2,...,s} such that the number i appears li times in every row and column. Using the computer I enumerated frequency squares of order n<=8 and the corresponding isotopism classes (squares inequivalent under rearrangements of rows, columns and entries). The work is described in the paper Frequency squares of orders 7 and 8, Utilitas Mathematica 72 (2007), 89-95. Through this page lists of isotopism class representatives and the programs used for the computations can be accessed.
| n | l | No. classes | n | l | No. classes |
| 6 | (2,2,2) | 46 | 8 | (5,3) | 51 |
| (2,2,1,1) | 106 | (5,2,1) | 624 | ||
| (2,1,1,1,1) | 56 | (5,1,1,1) | 370 | ||
| 7 | (5,2) | 4 | (4,4) | 156 | |
| (5,1,1) | 4 | (4,3,1) | 19 041 | ||
| (4,3) | 16 | (4,2,2) | 112 043 | ||
| (4,2,1) | 92 | (4,2,1,1) | 347 263 | ||
| (4,1,1,1) | 56 | (4,1,1,1,1) | 93 561 | ||
| (3,3,1) | 226 | (3,3,2) | 766 361 | ||
| (3,2,2) | 1 939 | (3,3,1,1) | 1 211 710 | ||
| (3,2,1,1) | 5 300 | (3,2,2,1) | 27 865 024 | ||
| (3,1,1,1,1) | 1 398 | (3,2,1,1,1) | 29 632 348 | ||
| (2,2,2,1) | 15 269 | (3,1,1,1,1,1) | 4 735 238 | ||
| (2,2,1,1,1) | 22 813 | (2,2,2,2) | 26 983 466 | ||
| (2,1,1,1,1,1) | 6 941 | (2,2,2,1,1) | 171 710 120 | ||
| (1,1,1,1,1,1,1) | 564 | (2,2,1,1,1,1) | 137 000 435 | ||
| 8 | (6,2) | 7 | (2,1,1,1,1,1,1) | 29 163 047 | |
| (6,1,1) | 7 | (1,1,1,1,1,1,1,1) | 1 676 267 |
Large lists are gnu-zipped and the largest ones are not on-line due to limited disk space, but I can supply them on DVD upon request.
Vedran Krcadinac, 1.3.2011.