Shortest self-avoiding walks on Z4 for which self-trapping can occur. Hugo Pfoertner, http://www.pfoertner.org/ December 2, 2002 If the first step is chosen as (0,0,0,0) -> (1,0,0,0) the number of self-avoiding walks of length 15 is 505_919_033_077. 1728 walks are trapped after 15 steps. With the requirements: 1) First step changing one of the coordinates positive 2) Before a step in a co-ordinate direction i=2,3,4 is made, it has to be preceded by a step in direction i-1, the following 36 walks remain. The entries in the first column preceding the path coordinates indicate, which points form the "cage" for the final trap location (which is always 0,1,0,0 given the above conditions) No. Count Walk # 1/Probability 1 292 165470834281 678223072849 Cage Walk Coords -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 2 0 0 2 0 2 0 0 -1 2 0 0 -1 -1 1 0 0 -1 1 1 0 3 0 1 1 0 0 1 1 1 4 0 1 0 1 0 1 -1 1 -3 0 1 -1 0 0 1 -1 -1 -4 0 1 0 -1 0 1 0 0 2 308 166315840744 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 2 0 0 2 0 2 0 0 0 2 1 0 3 0 1 1 0 -1 1 1 0 -1 -1 1 0 0 -1 1 0 1 4 0 1 0 1 0 1 -1 1 -3 0 1 -1 0 0 1 -1 -1 -4 0 1 0 -1 0 1 0 0 3 313 166340581149 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 2 0 0 2 0 2 0 0 0 2 1 0 3 0 1 1 0 0 1 1 1 4 0 1 0 1 -1 1 0 1 -1 -1 1 0 0 -1 1 -1 0 -3 0 1 -1 0 0 1 -1 -1 -4 0 1 0 -1 0 1 0 0 4 314 166340585036 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 2 0 0 2 0 2 0 0 0 2 1 0 3 0 1 1 0 0 1 1 1 4 0 1 0 1 -1 1 0 1 -1 -1 1 0 0 -1 1 0 -1 -4 0 1 0 -1 0 1 -1 -1 -3 0 1 -1 0 0 1 0 0 5 315 166340924913 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 2 0 0 2 0 2 0 0 0 2 1 0 3 0 1 1 0 0 1 1 1 4 0 1 0 1 0 1 -1 1 -3 0 1 -1 0 -1 1 -1 0 -1 -1 1 0 0 -1 1 0 -1 -4 0 1 0 -1 0 1 0 0 6 316 166340935500 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 2 0 0 2 0 2 0 0 0 2 1 0 3 0 1 1 0 0 1 1 1 4 0 1 0 1 0 1 -1 1 -3 0 1 -1 0 0 1 -1 -1 -4 0 1 0 -1 -1 1 0 -1 -1 -1 1 0 0 0 1 0 0 7 398 186702770644 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 -1 1 1 0 -1 -1 1 0 0 -1 2 0 0 2 0 2 0 0 0 2 0 1 4 0 1 0 1 0 1 -1 1 -3 0 1 -1 0 0 1 -1 -1 -4 0 1 0 -1 0 1 0 0 8 400 186707570169 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 -1 1 1 0 -1 -1 1 0 0 -1 1 -1 0 -3 0 1 -1 0 0 1 -1 1 4 0 1 0 1 0 2 0 1 2 0 2 0 0 0 2 0 -1 -4 0 1 0 -1 0 1 0 0 9 405 186716738854 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 -1 1 1 0 -1 -1 1 0 0 -1 1 0 1 4 0 1 0 1 0 2 0 1 2 0 2 0 0 0 2 -1 0 -3 0 1 -1 0 0 1 -1 -1 -4 0 1 0 -1 0 1 0 0 10 406 186716742726 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 -1 1 1 0 -1 -1 1 0 0 -1 1 0 1 4 0 1 0 1 0 2 0 1 2 0 2 0 0 0 2 0 -1 -4 0 1 0 -1 0 1 -1 -1 -3 0 1 -1 0 0 1 0 0 11 407 186716838179 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 -1 1 1 0 -1 -1 1 0 0 -1 1 0 1 4 0 1 0 1 0 1 -1 1 -3 0 1 -1 0 0 2 -1 0 2 0 2 0 0 0 2 0 -1 -4 0 1 0 -1 0 1 0 0 12 408 186716843446 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 -1 1 1 0 -1 -1 1 0 0 -1 1 0 1 4 0 1 0 1 0 1 -1 1 -3 0 1 -1 0 0 1 -1 -1 -4 0 1 0 -1 0 2 0 -1 2 0 2 0 0 0 1 0 0 13 410 187098518904 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 0 2 1 0 2 0 2 0 0 -1 2 0 0 -1 -1 1 0 0 -1 1 0 1 4 0 1 0 1 0 1 -1 1 -3 0 1 -1 0 0 1 -1 -1 -4 0 1 0 -1 0 1 0 0 14 412 187115409215 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 0 2 1 0 2 0 2 0 0 0 2 -1 0 -3 0 1 -1 0 0 1 -1 1 4 0 1 0 1 -1 1 0 1 -1 -1 1 0 0 -1 1 0 -1 -4 0 1 0 -1 0 1 0 0 15 417 187124382930 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 0 2 1 0 2 0 2 0 0 0 2 0 1 4 0 1 0 1 -1 1 0 1 -1 -1 1 0 0 -1 1 -1 0 -3 0 1 -1 0 0 1 -1 -1 -4 0 1 0 -1 0 1 0 0 16 418 187124386848 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 0 2 1 0 2 0 2 0 0 0 2 0 1 4 0 1 0 1 -1 1 0 1 -1 -1 1 0 0 -1 1 0 -1 -4 0 1 0 -1 0 1 -1 -1 -3 0 1 -1 0 0 1 0 0 17 419 187124644581 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 0 2 1 0 2 0 2 0 0 0 2 0 1 4 0 1 0 1 0 1 -1 1 -3 0 1 -1 0 -1 1 -1 0 -1 -1 1 0 0 -1 1 0 -1 -4 0 1 0 -1 0 1 0 0 18 420 187124655213 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 0 2 1 0 2 0 2 0 0 0 2 0 1 4 0 1 0 1 0 1 -1 1 -3 0 1 -1 0 0 1 -1 -1 -4 0 1 0 -1 -1 1 0 -1 -1 -1 1 0 0 0 1 0 0 19 439 187954827978 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 0 1 1 1 4 0 1 0 1 -1 1 0 1 -1 -1 1 0 0 -1 2 0 0 2 0 2 0 0 0 2 -1 0 -3 0 1 -1 0 0 1 -1 -1 -4 0 1 0 -1 0 1 0 0 20 440 187954831852 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 0 1 1 1 4 0 1 0 1 -1 1 0 1 -1 -1 1 0 0 -1 2 0 0 2 0 2 0 0 0 2 0 -1 -4 0 1 0 -1 0 1 -1 -1 -3 0 1 -1 0 0 1 0 0 21 441 187954928177 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 0 1 1 1 4 0 1 0 1 -1 1 0 1 -1 -1 1 0 0 -1 1 -1 0 -3 0 1 -1 0 0 2 -1 0 2 0 2 0 0 0 2 0 -1 -4 0 1 0 -1 0 1 0 0 22 442 187954933434 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 0 1 1 1 4 0 1 0 1 -1 1 0 1 -1 -1 1 0 0 -1 1 -1 0 -3 0 1 -1 0 0 1 -1 -1 -4 0 1 0 -1 0 2 0 -1 2 0 2 0 0 0 1 0 0 23 443 187955114150 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 0 1 1 1 4 0 1 0 1 -1 1 0 1 -1 -1 1 0 0 -1 1 0 -1 -4 0 1 0 -1 0 2 0 -1 2 0 2 0 0 0 2 -1 0 -3 0 1 -1 0 0 1 0 0 24 444 187955116299 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 0 1 1 1 4 0 1 0 1 -1 1 0 1 -1 -1 1 0 0 -1 1 0 -1 -4 0 1 0 -1 0 1 -1 -1 -3 0 1 -1 0 0 2 -1 0 2 0 2 0 0 0 1 0 0 25 445 187967014975 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 0 1 1 1 4 0 1 0 1 0 2 0 1 2 0 2 0 0 -1 2 0 0 -1 -1 1 0 0 -1 1 -1 0 -3 0 1 -1 0 0 1 -1 -1 -4 0 1 0 -1 0 1 0 0 26 446 187967018892 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 0 1 1 1 4 0 1 0 1 0 2 0 1 2 0 2 0 0 -1 2 0 0 -1 -1 1 0 0 -1 1 0 -1 -4 0 1 0 -1 0 1 -1 -1 -3 0 1 -1 0 0 1 0 0 27 447 187967355886 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 0 1 1 1 4 0 1 0 1 0 2 0 1 2 0 2 0 0 0 2 -1 0 -3 0 1 -1 0 -1 1 -1 0 -1 -1 1 0 0 -1 1 0 -1 -4 0 1 0 -1 0 1 0 0 28 448 187967364720 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 0 1 1 1 4 0 1 0 1 0 2 0 1 2 0 2 0 0 0 2 -1 0 -3 0 1 -1 0 0 1 -1 -1 -4 0 1 0 -1 -1 1 0 -1 -1 -1 1 0 0 0 1 0 0 29 449 187967539900 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 0 1 1 1 4 0 1 0 1 0 2 0 1 2 0 2 0 0 0 2 0 -1 -4 0 1 0 -1 -1 1 0 -1 -1 -1 1 0 0 -1 1 -1 0 -3 0 1 -1 0 0 1 0 0 30 450 187967545583 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 0 1 1 1 4 0 1 0 1 0 2 0 1 2 0 2 0 0 0 2 0 -1 -4 0 1 0 -1 0 1 -1 -1 -3 0 1 -1 0 -1 1 -1 0 -1 -1 1 0 0 0 1 0 0 31 451 187971839187 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 0 1 1 1 4 0 1 0 1 0 1 -1 1 -3 0 1 -1 0 -1 1 -1 0 -1 -1 1 0 0 -1 2 0 0 2 0 2 0 0 0 2 0 -1 -4 0 1 0 -1 0 1 0 0 32 452 187971843137 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 0 1 1 1 4 0 1 0 1 0 1 -1 1 -3 0 1 -1 0 -1 1 -1 0 -1 -1 1 0 0 -1 1 0 -1 -4 0 1 0 -1 0 2 0 -1 2 0 2 0 0 0 1 0 0 33 453 187972104233 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 0 1 1 1 4 0 1 0 1 0 1 -1 1 -3 0 1 -1 0 0 2 -1 0 2 0 2 0 0 -1 2 0 0 -1 -1 1 0 0 -1 1 0 -1 -4 0 1 0 -1 0 1 0 0 34 454 187972113004 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 0 1 1 1 4 0 1 0 1 0 1 -1 1 -3 0 1 -1 0 0 2 -1 0 2 0 2 0 0 0 2 0 -1 -4 0 1 0 -1 -1 1 0 -1 -1 -1 1 0 0 0 1 0 0 35 455 187972366833 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 0 1 1 1 4 0 1 0 1 0 1 -1 1 -3 0 1 -1 0 0 1 -1 -1 -4 0 1 0 -1 -1 1 0 -1 -1 -1 1 0 0 -1 2 0 0 2 0 2 0 0 0 1 0 0 36 456 187972372477 678223072849 -2 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 3 0 1 1 0 0 1 1 1 4 0 1 0 1 0 1 -1 1 -3 0 1 -1 0 0 1 -1 -1 -4 0 1 0 -1 0 2 0 -1 2 0 2 0 0 -1 2 0 0 -1 -1 1 0 0 0 1 0 0