Rigorous Expressions for the first, 3, moments For the Combinatorial Statistic, "Number of Occurences of the Pattern 1243 in the set of n-permutations" By Shalosh B. Ekhad Let X(n) be the combinatorial random variable "Number of Occurences of the Pattern 1243 in the set of n-permutations" The expected value is (n - 1) (n - 2) (n - 3) n ------------------------- 576 and in Maple input notation 1/576*(n-1)*(n-2)*(n-3)*n The variance is 3 2 (n - 1) (n - 2) (n - 3) n (718 n + 1137 n + 1829 n + 13065) ------------------------------------------------------------- 50803200 and in Maple input notation 1/50803200*(n-1)*(n-2)*(n-3)*n*(718*n^3+1137*n^2+1829*n+13065) The , 3, -th moment about the mean is 6 5 4 3 (n - 1) (n - 2) (n - 3) n (128144 n + 263509 n - 952375 n + 2462965 n 2 + 3911831 n + 5167246 n + 45006420)/548674560000 and in Maple input notation 1/548674560000*(n-1)*(n-2)*(n-3)*n*(128144*n^6+263509*n^5-952375*n^4+2462965*n^ 3+3911831*n^2+5167246*n+45006420) The square of the skewness to order , 2, is 201155838016 387216897617144 ------------- + ---------------- 10410362775 n 2 3737320236225 n and in Maple input notation 201155838016/10410362775/n+387216897617144/3737320236225/n^2 it indeed tends to 0, as it should This ends this exciting article, that took, 0.035, seconds to generate