Rigorous Expressions for the first, 6, moments For the Combinatorial Statistic, "Number of Occurences of the Pattern 132 in the set of n-permutations" By Shalosh B. Ekhad Let X(n) be the combinatorial random variable "Number of Occurences of the Pattern 132 in the set of n-permutations" The expected value is (n - 1) (n - 2) n ----------------- 36 and in Maple input notation 1/36*(n-1)*(n-2)*n The variance is 2 (n - 1) (n - 2) n (21 n + 78 n + 77) ------------------------------------- 21600 and in Maple input notation 1/21600*(n-1)*(n-2)*n*(21*n^2+78*n+77) The , 3, -th moment about the mean is 4 3 2 (n - 1) (n - 2) n (129 n + 3705 n + 5355 n + 8655 n + 11356) --------------------------------------------------------------- 12700800 and in Maple input notation 1/12700800*(n-1)*(n-2)*n*(129*n^4+3705*n^3+5355*n^2+8655*n+11356) The , 4, -th moment about the mean is 7 6 5 4 3 (n - 1) (n - 2) n (21609 n + 64222 n - 115572 n + 148390 n + 1137271 n 2 - 951092 n - 226348 n + 3420240)/7620480000 and in Maple input notation 1/7620480000*(n-1)*(n-2)*n*(21609*n^7+64222*n^6-115572*n^5+148390*n^4+1137271*n ^3-951092*n^2-226348*n+3420240) The , 5, -th moment about the mean is 9 8 7 6 (n - 1) (n - 2) n (327789 n + 9604185 n - 6812100 n - 83691420 n 5 4 3 2 + 82838997 n + 839145885 n + 9200750 n - 3948036330 n - 3402024236 n + 830425680)/3319481088000 and in Maple input notation 1/3319481088000*(n-1)*(n-2)*n*(327789*n^9+9604185*n^8-6812100*n^7-83691420*n^6+ 82838997*n^5+839145885*n^4+9200750*n^3-3948036330*n^2-3402024236*n+830425680) The , 6, -th moment about the mean is 12 11 10 (n - 1) (n - 2) n (454697031789 n + 385578225747 n - 4843089334017 n 9 8 7 + 76283809323855 n - 174203482681143 n - 1003656163437939 n 6 5 4 + 985378366003829 n + 10653529158652365 n + 16964728267380254 n 3 2 - 36340200072796508 n - 139572352062269512 n - 109134013046597920 n + 2889979231910400)/32986347467673600000 and in Maple input notation 1/32986347467673600000*(n-1)*(n-2)*n*(454697031789*n^12+385578225747*n^11-\ 4843089334017*n^10+76283809323855*n^9-174203482681143*n^8-1003656163437939*n^7+ 985378366003829*n^6+10653529158652365*n^5+16964728267380254*n^4-\ 36340200072796508*n^3-139572352062269512*n^2-109134013046597920*n+ 2889979231910400) The square of the skewness to order , 2, is 92450 31903850 -------- + ---------- 823543 n 2 5764801 n and in Maple input notation 92450/823543/n+31903850/5764801/n^2 it indeed tends to 0, as it should The kurtosis to order , 2, is 31475 39125 3 - ------ + -------- 7203 n 2 50421 n and in Maple input notation 3-31475/7203/n+39125/50421/n^2 It indeed tends to, 3, as it should be! The square of the , 5, -th alpha coeff. to order , 2, is 9245000 1151752205000 -------- + ------------- 823543 n 2 2092622763 n and in Maple input notation 9245000/823543/n+1151752205000/2092622763/n^2 It indeed tends to, 0, as it should be! The , 6, -th alpha coeff. to order , 2, is 53055125 2074664135875 15 - -------- + -------------- 823543 n 2 10716765059 n and in Maple input notation 15-53055125/823543/n+2074664135875/10716765059/n^2 It indeed tends to, 15, as it should be! This ends this exciting article, that took, 0.153, seconds to generate