The asymptotics of the average and variance for the number of Participating \ Edges in Tilings, by rectangles, of an m by n rectangle, in n, for m fro\ m 1 to 6 By Shalosh B. Ekhad The average number of edges in all tilings of an , 1, by n rectangle by rectangular tiles is, 2.500000000 n + 1.500000000 The asymptotic variance is , 0.250000000 n - 0.250000000 ----------------------------------------- The average number of edges in all tilings of an , 2, by n rectangle by rectangular tiles is, 4.013742011 n + 2.589811379 The asymptotic variance is , 0.57931604 n + 0.048285884 ----------------------------------------- The average number of edges in all tilings of an , 3, by n rectangle by rectangular tiles is, 5.527892467 n + 3.665725813 The asymptotic variance is , 0.87887199 n + 0.07485217 ----------------------------------------- The average number of edges in all tilings of an , 4, by n rectangle by rectangular tiles is, 7.037413436 n + 4.723344169 The asymptotic variance is , 1.16060104 n + 0.06415333 ----------------------------------------- The average number of edges in all tilings of an , 5, by n rectangle by rectangular tiles is, 8.543177849 n + 5.770581527 The asymptotic variance is , 1.43638924 n + 0.06292746 ----------------------------------------- The average number of edges in all tilings of an , 6, by n rectangle by rectangular tiles is, 10.04676512 n + 6.812876581 The asymptotic variance is , 1.7116355 n + 0.07656635 ----------------------------------------- ---------------------------- The asymptotics of the average and variance for the number of Participating \ Tiles in Tilings, by rectangles, of an m by n rectangle, in n, for m fro\ m 1 to 6 By Shalosh B. Ekhad The average number of tiles in all tilings of an , 1, by n rectangle by rectangular tiles is, 0.5000000000 n + 0.5000000000 The asymptotic variance is , 0.2500000000 n - 0.2500000000 -------------------------------- The average number of tiles in all tilings of an , 2, by n rectangle by rectangular tiles is, 1.046918161 n + 0.6601886205 The asymptotic variance is , 0.457751531 n - 0.4059748357 -------------------------------- The average number of tiles in all tilings of an , 3, by n rectangle by rectangular tiles is, 1.590102765 n + 0.7391500015 The asymptotic variance is , 0.634048845 n - 0.4674921018 -------------------------------- The average number of tiles in all tilings of an , 4, by n rectangle by rectangular tiles is, 2.125612546 n + 0.7886371868 The asymptotic variance is , 0.802185237 n - 0.4680960975 -------------------------------- The average number of tiles in all tilings of an , 5, by n rectangle by rectangular tiles is, 2.656076852 n + 0.8270397141 The asymptotic variance is , 0.971862088 n - 0.4352732041 -------------------------------- The average number of tiles in all tilings of an , 6, by n rectangle by rectangular tiles is, 3.183895210 n + 0.8613050963 The asymptotic variance is , 1.144945700 n - 0.3857805646 -------------------------------- ---------------------------- The Limit of the Correlation between the number of Horizontal Edges and th\ e Number of Vertical Edges, in tilings by rectangular tiles, of an m by\ n rectangle, an s goes to infinity, for m from 2 to 5 By Shalosh B. Ekhad For rectangular tilings of an, 2, by n rectangke: The limit correlation between the number of horiz. edges and the number of v\ ertical edges, as n goes to infinity is, -0.4201213667 -------------------------------- For rectangular tilings of an, 3, by n rectangke: The limit correlation between the number of horiz. edges and the number of v\ ertical edges, as n goes to infinity is, -0.5226003360 -------------------------------- For rectangular tilings of an, 4, by n rectangke: The limit correlation between the number of horiz. edges and the number of v\ ertical edges, as n goes to infinity is, -0.5744760548 -------------------------------- For rectangular tilings of an, 5, by n rectangke: The limit correlation between the number of horiz. edges and the number of v\ ertical edges, as n goes to infinity is, -0.6045010351 --------------------------------