The Generating Functions for 2D Self-Avoiding Walks of (Globally) Bounded Width The generating function (in the variable s) enumerating 2D self-avoiding walks of width <= 1 equals 6 4 3 (s - s - 3 s + 3 s + 2) s - ------------------------------ 2 2 2 (s - 1) (s + s - 1) (s + 1) The generating function (in the variable s) enumerating 2D self-avoiding walks of width <= 2 equals 22 10 4 6 5 2 3 11 - (-2 - 4 s - 50 s + 54 s + 5 s - 5 s - 6 s + 3 s + 12 s + 20 s 24 23 27 26 25 12 13 14 - 14 s - 36 s + 6 s + 8 s - 4 s - 130 s - 151 s + 65 s 15 9 8 7 16 17 21 20 + 201 s + 8 s - s + 7 s + 53 s - 194 s + 15 s + 89 s 18 19 / 2 2 2 - 147 s + 30 s ) s / ((s + s - 1) (s - 1) (s + 1) / 6 5 4 3 2 2 4 2 2 (2 s + 2 s + s + 2 s + s - 1) (2 s - 1) (s + 1) (s + s + 1)) The generating function (in the variable s) enumerating 2D self-avoiding walks of width <= 3 equals 25 27 22 29 28 - s (2 + 152989 s - 309729 s - 149260 s + 395177 s + 2316 s 34 47 46 77 10 73 + 701500 s - 837845 s - 1174335 s + 3568 s + 238 s - 183952 s 78 76 75 74 72 70 - 5994 s + 69216 s + 74206 s - 16815 s - 232836 s + 389563 s 69 71 4 6 5 2 3 + 402764 s + 72767 s + 84 s - 192 s + 92 s - 20 s - 9 s 42 41 79 81 80 82 - 842711 s - 771734 s - 6460 s - 13648 s - 5828 s - 6216 s 85 83 84 86 24 11 + 2320 s + 9016 s + 8144 s - 3840 s + 226705 s - 2961 s 43 48 23 26 33 + 393660 s + 609707 s + 21480 s - 163568 s + 38327 s 50 30 32 12 31 + 123711 s + 246079 s - 578908 s - 2868 s - 332737 s 36 67 37 35 39 - 475244 s - 771741 s - 825772 s + 461609 s + 833984 s 40 51 38 52 92 91 + 436031 s - 1339604 s + 80176 s - 825716 s - 192 s - 288 s 93 45 13 14 88 89 90 + 64 s + 319071 s + 5533 s + 11894 s - 176 s + 880 s + 832 s 87 15 9 8 7 44 - 2680 s - 2136 s + 1241 s + 238 s - 428 s + 1110256 s 16 17 21 20 18 19 - 29898 s - 25638 s - 111637 s + 26580 s + 33753 s + 83859 s 65 57 62 64 63 + 458864 s - 670566 s - 946610 s + 1103759 s + 381952 s 68 61 59 60 58 - 178078 s - 1120419 s + 1291390 s - 37210 s + 817406 s 56 55 49 53 54 - 1194116 s - 106777 s + 1375093 s + 657623 s + 1326159 s 66 / 3 4 6 7 8 18 14 10 - 456598 s ) / ((-1 + 2 s + s - s - s + s ) (-2 s + 8 s + 2 s / 4 12 2 8 16 6 20 2 19 18 + 1 + 3 s - s - 4 s - 4 s - 5 s + 2 s + 4 s ) (2 s + 2 s 17 16 15 14 13 12 11 10 9 - 7 s - 6 s + 5 s + 8 s + 7 s - 17 s - 8 s + 3 s + 10 s 8 7 6 5 4 3 2 + 3 s - 8 s + 2 s - s + 6 s - 3 s - 2 s + 1) (s - 1) (s + s + 1) 3 4 5 6 2 2 4 2 (-1 + s + 2 s + s + 2 s + 2 s ) (2 s - 1) (s + 1) )