#This is an Umbral Scheme for walks in the 2D #square-lattice that avoid #retracing and rectangular subwalks 2 [F[1]() = -t - F[1]() t - 2 F[2](1, 1) t - 2 F[3](1, 1, 1) t, F[2](x[1], x[2]) 4 3 t F[1]() t = - --------------------------- - ---------------------------- 2 2 2 2 (1 - t x[1]) (1 - t x[2]) (-1 + t x[1]) (t x[2] - 1) 3 F[2](t x[1], t x[2]) t x[1] + ---------------------------- - F[2](t x[2], x[1]) t 2 (t x[2] - 1) (-1 + t x[1]) 3 F[2](1, 1) t + 2 ------------------------------------------ 2 2 (-1 + t x[1]) (-1 + t x[1]) (t x[2] - 1) 4 F[2](1, t x[1]) t x[1] - 2 ------------------------------------------ 2 2 (-1 + t x[1]) (-1 + t x[1]) (t x[2] - 1) 2 F[2](1, t x[2]) t - --------------------------- 2 (t x[2] - 1) (-1 + t x[1]) 4 F[3](1, 1, t x[1]) t x[1] - 2 ------------------------------------------ 2 2 (-1 + t x[1]) (-1 + t x[1]) (t x[2] - 1) 3 F[3](1, 1, 1) t + 2 ------------------------------------------ 2 2 (-1 + t x[1]) (-1 + t x[1]) (t x[2] - 1) 3 2 F[3](t x[1], t x[2], x[1]) t x[1] F[3](1, t x[2], x[1]) t - ----------------------------------- + ----------------------- -1 + t x[1] -1 + t x[1] 2 F[3](1, 1, t x[2]) t - F[3](t x[2], x[1], x[2]) t x[2] - --------------------------- 2 (t x[2] - 1) (-1 + t x[1]) 3 F[3](1, t x[1], t x[2]) t x[1] + -------------------------------, F[3](x[1], x[2], x[3]) = 2 (t x[2] - 1) (-1 + t x[1]) 6 t - ----------------------------------------- 2 2 2 (1 - t x[1]) (1 - t x[2]) (1 - t x[3]) 5 F[1]() t + ------------------------------------------ 2 2 2 (-1 + t x[1]) (t x[3] - 1) (t x[2] - 1) 5 F[2](t x[1], t x[2]) t x[1] - ----------------------------------------- 2 2 (t x[2] - 1) (-1 + t x[1]) (t x[3] - 1) 2 2 F[2](t x[2], x[1]) t (-t x[1] - 1 + t x[3]) + --------------------------------------------- 2 (t x[3] - 1) %1 5 F[2](1, 1) t - 2 -------------------------------------------------------- 2 2 2 (t x[2] - 1) (t x[3] - 1) (-1 + t x[1]) (-1 + t x[1]) 6 F[2](1, t x[1]) t x[1] + 2 -------------------------------------------------------- 2 2 2 (t x[2] - 1) (t x[3] - 1) (-1 + t x[1]) (-1 + t x[1]) 4 F[2](1, t x[2]) t F[2](t x[3], x[2]) t + ----------------------------------------- - -------------------- 2 2 %1 (t x[2] - 1) (-1 + t x[1]) (t x[3] - 1) 2 F[2](x[1], x[2]) t F[2](t x[2], t x[3]) t + ------------------ + ----------------------- %1 2 (t x[3] - 1) %1 6 F[3](1, 1, t x[1]) t x[1] + 2 -------------------------------------------------------- 2 2 2 (t x[2] - 1) (t x[3] - 1) (-1 + t x[1]) (-1 + t x[1]) 5 F[3](1, 1, 1) t - 2 -------------------------------------------------------- 2 2 2 (t x[2] - 1) (t x[3] - 1) (-1 + t x[1]) (-1 + t x[1]) 5 2 2 F[3](t x[1], t x[2], x[1]) t x[1] F[3](1, t x[2], x[1]) t + ----------------------------------- - ------------------------ 2 (-1 + t x[1]) %1 (t x[3] - 1) (-1 + t x[1]) 2 2 F[3](t x[2], t x[3], x[2]) t x[2] F[3](t x[2], x[1], x[2]) t x[2] - ---------------------------------- + -------------------------------- %1 %1 2 4 F[3](1, t x[2], t x[3]) t F[3](1, 1, t x[2]) t + -------------------------- + ----------------------------------------- 2 2 2 (t x[3] - 1) %1 (t x[2] - 1) (-1 + t x[1]) (t x[3] - 1) F[3](x[1], x[2], x[3]) t x[3] F[3](t x[3], x[2], x[3]) t x[3] + ----------------------------- - ------------------------------- %1 %1 5 F[3](1, t x[1], t x[2]) t x[1] - -----------------------------------------], [4, 8, 8] 2 2 (t x[2] - 1) (-1 + t x[1]) (t x[3] - 1) %1 := t x[3] - x[1] [1, 4] [1, 4, 12] [1, 4, 12, 36] #This are the first 30 terms in the enumerating sequence #notice that it starts out the same as self-avoiding walks [1, 4, 12, 36, 100, 284, 780, 2172, 5980, 16524, 45556, 125700, 346644, 956180, 2637228, 7274052, 20063076, 55337964, 152632628, 420990700, 1161174300, 3202745732, 8833798444, 24365342844, 67204377788, 185362809324, 511266858076, 1410173921804, 3889535296244, 10728098554604, 29590192617660]